8.1 INTRODUCTION

In this chapter¹ we present some models for market risk factors: all market variables, such as interest rates, credit spreads, FX rates, stock prices and commodity prices, are affecting the payoff of most of the contracts a bank has on its balance sheet. Effective and parsimonious models are required to allow simulation of the cash flows of contracts linked to the different market variables, which become risk factors in themselves.

We first introduce models that can be used to model the evolution of FX rates and equities, then we dwell on interest rate models and default models, which are the basis for the modelling of credit spreads. We do not focus on all market variables (e.g., we do not analyse inflation modelling), but hopefully we will cover the vast majority of the market risk factors that affect the cash flows of contracts.

The second part of the chapter, from Section 8.5 on, is devoted to application of the models to liquidity risk management.

8.2 STOCK PRICES AND FX RATES

A standard way to model the evolution of stock prices and FX rates is to assume that they are commanded by a geometric Brownian motion. A process x_t follows a geometric Brownian motion if it described by the dynamics

where μ and σ are, respectively, the constant drift and the volatility of the process, and W_t is a Brownian process. The differential equation (8.1) can be solved exactly. Given the value x_t at time t, the value of the process at any subsequent time T is

Equation (8.1) is a real-world measure process, where the expected trend (μ) and the volatility of the process (σ) are supposed to be parameters reflecting historical realization of the observed equity prices of FX rates: as such they can be estimated by statistical techniques from actual time series. The process can also be written under the risk-neutral measure,

where r_t is the deterministic time-dependent instantaneous interest rate. The risk-neutral measure is used for pricing so that the volatility σ_t is extracted from market quotes of options backing out the implied volatility from the formula we will present below. The expected value of x_t at time T, assuming a constant interest rate and volatility parameter, is

where, both here and in the following, we denote by E_t the expectation value with respect to filtration F_t under the risk-neutral measure (Q). In reality, we will see later (Chapter 12) that in many cases the proper drift under the risk-neutral measure (from a replication argument) is the repo rate in the case of equity, or the FX swap rate in the case of FX rates.

In the specific cases of an equity price S, equation (8.1) has to be extended to include the possibility of discrete dividends D_i paid on dates over a predefined observed period as well. The price of the equity can be assumed to be composed of two parts: a stochastic part similar to (8.1) plus a deterministic component equal to the present value of the future dividends paid by the stock:

where .

The evolution of the stock, as modelled by equation (8.5), evolves with jumps downwards on each dividend payment date with a a jump size equal to the lump sum paid. The evolution can also be applied to stock indices.

For FX rates there is no problem related to the payment of dividends, so that the dynamics in equations (8.1) and (8.3) do not need any adjustment. The risk-neutral dynamics have to be adjusted as follows:

where X is the FX rate and is the implied FX swap rate implied from levels quoted in the market. The expected value is modified accordingly.

It is useful to show how to price a call option with strike K on a stock or an FX rate following the process x_t and maturity at time T:

has a value today equal to

Here, we set and we define the function

where

while N(x) is the cumulative probability distribution for a standardized normal distribution,

Put options can be priced via put–call parity:

8.3 INTEREST RATE MODELS

We show here two one-factor short-rate models—the Vasicek model and the CIR model—as well as the Libor Market Model.

8.3.1 One-factor models for the zero rate

The (annual) risk-free zero rate r_t is the interest rate at which, at time t, money can be lent or borrowed for the infinitesimal amount of time dt: this is true if in the economy the default of economic agents is excluded. If r_t is modelled through a stochastic process, the price at time t of a zero-coupon bond maturing at time T is

where the expectation is taken under the risk-neutral measure Q. The price in equation (8.12) coincides with the discount factor in the period [t, T], with an instantaneous forward rate given by

The zero-rate is recovered as

The expected return on a zero-coupon bond is under the risk-neural measure:

The same expectation in the real measure is:

where π represents the market risk parameters and is the risk-premium required, which is proportional to the elasticity of the bond's value with respect to the risk factor r_t.

In one-factor models, the zero rate r_t is described by a single stochastic factor, depending on a Brownian process W_t. In the following, we briefly review two of the most popular one-factor models.

8.3.2 Vasicek model

In the Vasicek model [111], the zero rate follows the stochastic differential equation

The parameter σ defines the volatility of the process, θ is a parameter defining the long-term mean to which trajectories evolve, and κ is the mean reversion speed, describing the velocity at which trajectories approach the long-term mean. Moreover, the quantity σ²/2κ defines the long-term variance possessed by trajectories after a time .

The risk-adjusted dynamics including the market risk parameter as well are:

We assume that the market risk parameter π is zero, so that the zero-rate r_T can be obtained at future times T when the process is known at time t, by solving equation (8.15), obtaining

Although the dynamics of the Vasicek model enjoy some nice properties, such as mean reversion towards the long-term means θ, unfortunately it does not prevent short-rate means from going below zero and thus assume negative values. While this might be seen as completely unrealistic,² it many cases it is preferred to have a boundary at the zero level. The CIR model allows such a feature to be introduced.

8.3.3 The CIR model

We will describe the CIR model extensively because it will be the main building block of most of the models we present in what follows. In a generic “mean-reverting” process, the interest rate r_t tends to be pulled towards a long-term average θ whenever the process deviates from it.

In the CIR model the process for r_t is:

where θ is the long-term average, κ is a constant describing the speed of mean reversion, σ is the volatility of the process that causes deviations from a pure deterministic model and r₀ is the initial value (at time t = 0) of the interest rate.

In this case, we can also write the risk-adjusted dynamics we have to use when pricing contracts depending on the interest rates:

Furthermore in this case, we will assume in what follows that π = 0, so that drift under the risk-neutral and real measure is the same.

The short-rate dynamics in equation (8.17) were first suggested by Cox, Ingersoll and Ross in [55]. In the following, we indicate the CIR process in equation (8.17) as

When the parameters of the process satisfy the inequality

it is ensured that the process r_t thus generated is always positive. Since the CIR model is of the mean-reverting type, both the expected value and the variance of r_t tend to a constant value when time tends to infinity. In fact, at time t > 0, the average value of the CIR process CIR(κ, σ, θ, r₀, t) is

which shows explicitly that the long-term average of the CIR process tends to the value θ. To prove the formula in equation (8.20), one might take the average on both sides of equation (8.17), and use the fact that W_t has zero average. One is left with an ODE with initial condition r = r_s, leading to equation (8.20). With a similar technique, it can be shown that the variance of a CIR process is

Zero-coupon bonds

A closed-form formula is available for zero-coupon bonds, alternatively named discount bonds or discount factors, in the CIR model. For a process r_t the moment-generating function over the time interval [t; T] is defined as

In the case of the CIR process in equation (8.17), m(q; t, T) is known in a closed-form formula,

where the coefficients A(q; t, T) and B(q; t, T) are defined as

and with

The discount factor in the CIR model is then obtained from equation (8.23) when q = −1,

The explicit expression for the discount factor is given in CIR (see [55]):

with

and with .

Future and forward prices

Furthermore, futures and forward prices of zero-coupon bonds are available in closed-form formula in the CIR model.

At time t, the future price³ of a future contract with maturity date s > t on a discount bond paying one monetary unit at time T > s is (see [53] for details):

where

The forward price of a forward contract to deliver at the maturity date s > t a discount bond paying one monetary unit at time T > s is (see [53])

The future price of the bond can be seen as its expected value at time s > t, under the risk-neutral measure:

On the other hand, the forward price of the bond can be considered as the expected value at time s > t under the forward risk-adjusted measure, also called the s-adjusted measure (see [34]), in which equation (8.17) becomes:

where the notation is the same as that introduced above. The forward price of the bond is then:

Probability distribution for a CIR process

In the CIR model, the process r_t is characterized by a non-central chi-squared distribution. In particular, the probability distribution of the process in equation (8.18) is (see [55]):

where

and χ²(x; d, c) is the non-central chi-squared distribution, with d degrees of freedom and non-centrality parameter c.

It is also useful to give the CIR dynamics in the forward risk-adjusted measure, also called the t-adjusted measure: technically speaking, this is the measure under which the terminal payoff of a contract at time t is rescaled by dividing it by the value of a zero-coupon bond expiring at the time the probability distribution of the CIR process is:

where

As will be made clear in the models we will present, we will need both dynamics and distributions.

Options on bonds and interest rates

Following [55], the price of a European call option, with maturity s and strike K, on a zero-coupon bond with maturity T > s and with a short interest rate r_t = CIR(κ, σ, θ, r₀, t) is

where

and with the quantity r* expressed in terms of the strike K by

Note that K = A(s, T)e^−B(s,T)r* < A(s, T).

The value of the corresponding European put on the zero-coupon bond is found by put–call parity:

Although zero-coupon options are not actively traded in the market, equations (8.37) and (8.40) can be used to price much more common contracts such as cap and floor options on a discrete interest rate L(t, s, T) = L_T(t), simply compounded and applied for a period starting at s and ending in T with strike rate X (e.g., a cap on a 3-month Eonia).⁴

The formula to evaluate a caplet with unit notional amount is:

A floorlet's value is given by the formula:

where the strike is

Let us now consider a cap or floor option, with maturity T and fixing schedule t_i, i = {0, …, n − 1}, where n is the number of caplets or floorlets, and t_n = T. The cap option is a sum of the caplets between fixing dates t_i and payment dates t_i+1, hence:

where we define the time intervals T_i = t_i+1 − t_i, and

Similarly, the floor is

Even if quotes on caps and floors on Eonia or OIS rates are not very liquid in the current financial markets, equations (8.41) and (8.42) can be useful to drive the distributions of discrete period rates that form the risk-free (Eonia or OIS) component of a Euribor or Libor fixing, if we model these as F_t(t) = L_i(t) + S_i(t), (i.e., as the sum of the risk-free rate plus a credit spread S). We will dwell more on this later on in this chapter.

Summing two CIR processes

It is useful to study some properties of the CIR process since we will exploit them heavily in the models we present in this book. We start by analysing the sum of two CIR processes.

Let us consider two independent CIR processes,

We wish to know under which restrictions of the CIR parameters for and the process is still a CIR process.⁵ To answer this, we consider the two CIR processes in the form given in equation (8.17),

so that the expression for dr_t is, considering the two processes and as independent,

with the initial condition . We can write the expression for r_t in equation (8.49) in the form

if we constrain the parameters as

with the new Brownian motion W_t defined through

Summing up, if the CIR parameters for r_t are defined as in equation (8.51), then

Multiplying a CIR process by a constant

We now consider how the process r_t = CIR(κ, σ, θ, r₀, t) is related to αr_t, where α is a positive constant. For this, we multiply equation (8.17) by α,

or

Estimation of the CIR model using the Kálmán filter

Since we will also use the CIR model for risk management purposes and not only for pricing purposes (in which case we would just calibrate the model to market prices to infer risk-neutral parameters), we need a robust procedure to estimate the parameters according to historical prices. In this section, we outline a technique for estimating the parameters κ, θ and σ of the CIR model. For this, we use a maximum likelihood estimation based on the Kálmán filtering technique.

The basis of the procedure we present is to extract zero rates from market prices of zero-coupon prices: these prices are often embedded in coupon bonds that can be seen as portfolios of discount bonds. In the CIR model, the prices of a zero-coupon bond are related to the zero rate r_t (here referred to as the latent variable) through the discount factor in equation (8.27). Indicating the bond maturity date by T, and setting

where P(t, T) is the market price of a discount bond. The relation between the latent variable and z_t is

and is referred to as the observation equation. For the CIR model we have:

where A(t, T) and B(t, T) are the CIR factors defined in equation (8.28).

The Kálmán filter technique has become the standard tool for estimating the parameters of a short-rate model (not only the CIR model), given the term structure of a portfolio of bonds. Kálmán filters are based on time discretization of the equations describing the evolution of the interest rate r_t (the latent variable) and of the bond price. If the time series of bond prices is considered from a date t₀ to an end date t, we discretize the time interval [t₀, t] by introducing the time step δ = (t − t₀)/N, where N is the number of dates in the interval, and the dates are

Given the market price of the bond at time t_i as p^MKT(t_i, t), we define

The discretized version of equation (8.57) is

where z_i = z_ti, r_i = r_ti, α_i = α_ti and β_i = β_ti. A second equation is obtained by considering the time evolution of the rate, according to equation (8.17),

where, for a CIR process, C = θ (1 − e^−κδ) and F = e^−κδ, while ω_i is a Gaussian noise with mean equal to zero and variance equal to

With the Kálmán filter technique, equations (8.61) and (8.62) are simultaneously solved for r_i and z_i at step i, given the values of the variables at the i − 1-th step and the knowledge of z_i from the market data at time t_i, . As additional outcomes, the procedure produces values of the error , and the variance associated with the error Var z_i. At each step, we construct the logarithm of the likelihood function

The log-likelihood function of the process is the sum

The parameters κ, θ and σ are obtained by requiring that the function be maximized with respect to these parameters. In practical numerical codes, since minimization techniques are easier to implement, one looks for the parameters that minimize the function −.

8.3.4 The CIR++ model

The CIR model is a parsimonious way to model the entire term structure of interest rates and also prevents negative values, in contrast to the Vasicek model. Nonetheless, it is often not rich enough to allow a perfect fit to observed market prices. On the other hand, when parameters are estimated from the historical time series, the model is unable in most cases to exactly reproduce observed market prices, even if they could be perfectly matched to them by means of single-time calibration.

In these cases it is convenient to extend the CIR model so as to allow perfect fitting to the initial term structure of (risk-free) interest rates, and a time-dependent deterministic extension is the easiest approach to adopt.

The CIR++ process

In the CIR++ model,⁶ the short-rate dynamics for r_t are described as

where x_t is a CIR process,

and ψ_t is a deterministic function that can be chosen so as to exactly match the initial term structure of interest rates. We denote the CIR++ model in equation (8.66) as

The price at time t of a zero-coupon bond expiring at T under a CIR++ process is (see [35]):

where is the discount factor defined in equation (8.27).

The average value of the CIR++ process in equation (8.27) over the period [t, T] is

while, since ψ_t is a deterministic process, the variance of r_t is given by the variance of the CIR process x_t alone,

In a CIR++ model, we define the future in t, with expiry S and bond maturity T, as

where H_CIR(t, S, T) is the future of the CIR process defined in equation (8.29).

Typically, the deterministic function ψ_t will be stepwise constant: we will see below how to fit it to actual market prices.

Options on bonds and interest rates

In the CIR++ model, the price of a European call option with maturity s and strike K on a zero-coupon bond with maturity T > s and interest rate r_t = CIR++(ψ_t, κ, σ, θ, x₀, t) is

where ϕ and ζ are as defined in equation (8.38), while

The value of the corresponding put on the European bond is found from the formula

Caps and floors in the CIR++ model are related to the call and put bond options as in equations (8.41) and (8.42).

8.3.5 The Basic Affine Jump Diffusion Model

An extension of the CIR model, which can be applied also to the CIR++ model, is the inclusion of jumps in the process of the short rate r_t.⁷

A stochastic process Z_t is a basic affine jump diffusion (bAJD) model if it follows the process

where W_t is a standard Brownian motion, and J_t is an independent compound Poisson process with constant jump intensity l > 0, so that jumps occur more frequently with increasing l. We assume that each jump is exponentially distributed, with mean μ > 0: the larger μ is the larger the size for a given jump. We also require κθ ≥ 0 in order for the process to be well defined. We indicate the process Z_t in much the same way as the process in equation (8.76) as

Following [59] and [64], but with the notation as in [55], the moments of the bAJD in equation (8.77) are defined as

where

and where A(q; t, T) and B(q; t, T) are the CIR factors given in equation (8.24). In particular, all moments for the bAJD process reduce to the moments for the CIR process when either l = 0 or μ = 0, corresponding to the case in which either the intensity or the frequency of jumps is zero. In formulas, we have

as can be seen explicitly from the fact that when we set either μ = 0 or l = 0 in equation (8.79), the expression for m(1, q; t, T) reduces to m(q; t, T) in equation (8.23).

The discount factor in the bAJD model can be written in exponential form as

where

and where A(t, T) and B(t, T) are the CIR coefficients in equation (8.28).

8.3.6 Numerical implementations

We now focus on some numerical issues related to the CIR model and its extensions. More specifically, the CIR process has to be handled with care in its discrete version, used to simulate paths for the short-rate r_t in Monte Carlo simulations.

8.3.7 Discrete version of the CIR model

Consider a short-rate factor r_t described by a CIR process, as in equation (8.17), and a security that depends on r_t. In order to evaluate the price of a contract at a future time s > t, or to simulate its cash flows, the value of r_s has to be known. This can be done by numerically evolving r_t using equation (8.17) in steps, from time t to time s. In greater detail let us introduce N discretization points t_i, equally spaced by

so that

so that t₀ = t and t_N = s.

For the Brownian motion term dW_t, we first note that

where the notation A ~ B means that the two distributions A and B are equivalent, and N_0,1 is a Gaussian random variable of zero mean and variance equal to one. We generate a sample of N values ~N_0,1 and call them Z_i, so that

Setting r(t_i) = r_ti, equation (8.17) is discretized as

Given the initial value r(t₀) = r₀, the entire solution is reconstructed in steps. The procedure in equation (8.87) goes under the name of the Euler scheme. A major failure of the Euler scheme is that it cannot guarantee that the process r_t is always positive.

Of the various alternative schemes that have been proposed to mend this problem, [4] showed that the most suitable discretization for generating the process CIR(κ, σ, θ, r₀, t) is

which provides a positive-definite process whenever the CIR condition in equation (8.19), 2κθ > σ², is satisfied.

The same approach can be used to derive robust discretization for the forward risk-adjusted CIR process. A numerical implementation of the CIR process in this measure is (see [4]):

where .

We know that in the CIR++ model the short-rate r_t is modeled as a sum of a stochastic term x_t and a deterministic function ψ_t,

For the stochastic term, the same numerical procedures used for the CIR model can be employed. Instead, the deterministic function ψ_t can be modelled by a step function, with constant steps over some given time intervals. If the function is defined over the interval [t, T], we introduce a set of times T_i, i = {0, …, n}, with T₀ = t and T_n = T, and we discretize the function ψ_Ti = ψ_i with i = {1, …, n} so that ψ_i is constant over the corresponding step size T_i = T_i − T_i−1. An integral over [t, T] is then approximated by

A little more care is needed if we wish to discretize an integral on the subinterval [t₁, t₂], with t < t₁ < t₂ < T. In this case, we first need to find the integer n₁ for which T_n1 < t₁ ≤ T_n1+1 and similarly the integer n₂ for which T_n2 < t₂ ≤ T_n2+1. The integral is then discretized as

where

8.3.8 Monte Carlo methods

Consider a contract with maturity T and payoff H(r(T)). The value at time t of this contract is

where E_t is the expectation value in the risk-neutral measure, while is the expectation value taken in the T-forward risk-adjusted measure (see Section 8.3.3). The discount factor P^D(t, T) is obtained analytically, while the expected value of the payoff depends on the unknown value r(T) of the short rate at maturity.

To compute the value of , we use the following Monte Carlo method. We generate p different paths of the discretized short rate, each path obtained by using the method in equation (8.89) for the forward risk-adjusted measure. In particular, for each path we obtain the value of the short rate at maturity and call it r_j(T), with j ∈ {1, …, p}. The expectation value is approximated as

so that the price of the contract is

Pricing using a Monte Carlo method can be generalized to the case of path-dependent options. We consider the discretization of the short rate on the grid t_i, {i = 0, …, N}, with t₀ = t and t_N = T, as discussed in Section 8.3.7. Defining

the price of the path-dependent contract is

The expectation value is computed by considering p realizations of the short rate in the T-forward measure. For a given realization j ∈ {1, …, p}, the value of the short rate at the discretized time t_i is r_j(t_i). Setting

the price of a path-dependent contract is

8.3.9 Libor market model

The Libor market model (LMM) [31] approaches the problem by using forward rates that are directly observable in the market, such as Libor or Euribor fixings. While these rates could basically be considered as risk free up until 2008, in current markets they are actually traded as rates referred to default-risky countrparties, since even major banks are subject to this risk.

These considerations imply that the LMM can no longer be used to model Libor (or Euribor) fixing rates, but it is more appropriate to model Eonia forward rates, applying then a spread on these to derive the value of fixing rates. Anyway, for the moment we disregard this complication and we present the LMM in its standard version.

In the LMM, the quantities that are modelled are a set of forward rates rather than the instantaneous short rate or instantaneous forward rates. Consider a set of maturities {T_i|i = 1, …, N} describing the reset dates for traded caps or floors on the market. By including the additional date T₀ = t, we define the time intervals T_i = T_i − T_i−1, i = {1, …, N} and the forward rates L_i(t) observed at time t for the period (T_i−1, T_i]. In this model, the market has maturity T_N = T.

Here, for each date i, we model the Libor rate L_i(t) by a stochastic process under its forward measure, over the period [T_i−1, T_i]. Assuming that the Libor rate L_i(t) is a martingale under the T_i-forward measure, we have

where σ_i(t) is the volatility of L_i(t) at time t, and is an N-dimensional Brownian motion under the measure Q_Ti, with instantaneous covariance matrix ρ_t.

It is possible to link different T_i-forward measures with different values of i through a change of measure. As an example, we first consider the process for L_i(t) in the T₁-forward measure. Denoting by v_i(t) the volatility of the zero-coupon bond price P^D(t, T_i) at time t, we find

Forward rate agreement and caps&floors

In a forward rate agreement (FRA), at the inception time t, counterparties fix the expiry date S > t and the maturity of the contract T > S, to lock the interest rate over the period [S, T]. More specifically, at the maturity T, a payment with fixed interest rate K previously defined is exchanged in return for a floating payment based on the value of the Libor rate at time S, L(S, T). At time S, the value of the contract on a unit notional value is then

The value at time t of a FRA contract is

where the expectation value is taken in the T-forward measure, denoted by . It should be noted that the discount bond P^D(t, T) is taken from a curve different from the Libor (or Euribor) one. This numeraire bond is calculated out of a risk-free interest rate curve: in the current financial environment the best approximation to a risk-free rate is considered the OIS (or Eonia) rate, so that P^D(t, T) is in practice derived from the OIS (or Eonia) swap prices quoted in the market.

A forward rate agreed at time t is defined as the fixed rate to be exchanged at time T for the Libor rate L(S, T) so that the contract has zero value at time t. We denote the value of the forward rate as

When the Libor curve L(S, T) corresponds to the risk-free discount curve, we have an equality between the FRA rate and the forward rate computed out of the curve:

where P(t, ·) are zero-coupon bonds extracted by Libor fixing curves. However, as mentioned above, this has no longer been the case since 2008, and Libor curves are treated independently of the discount (OIS or Eonia) curves, see the discussion above.

Caps and floors are options of forward rates. We consider the set of payment dates {T_i|i = 1, …, N} and corresponding periods T_i = T_i − T_i−1, with i = 1, …, N, as discussed before, and we set T₀ = S.

An interest rate cap is a contract in which the buyer receives a flow of payments at the end of each period in which the interest rate has exceeded an agreed strike price K. The price at time t of a cap option can be viewed as the sum of individual caplets,

At each time T_i, the option pays off the difference between the Libor rate and the strike X. Each caplet has a payoff

When the expectation value is taken in the T_i-forward measure, denoted by , we find

The pricing of floor options relies on similar techniques to those used for pricing caps. A floor is a contract in which the buyer receives payments whenever the interest rate falls below a strike price X. Decomposing the floor option as a sum of floorlets,

where each floorlet has the value in the T_i-forward measure of

The price of cap and floor options can be given in a closed-form formula using Black's formula [23], assuming that forward rates are lognormally distributed under the risk-neutral measure. The price of the cap option is

where

and the function BlSc(F, K, Σ) is as defined in Section 8.2. The value of is retrieved from the market quote σ_MKT,i for each i, as

and determines the price of the caplet through equation (8.112).

Swaps and swaptions

An interest rate swap (IRS) is a contract in which two parties agree to exchange one stream of cash flows against another stream. In a fixed-for-floating IRS, one party agrees to exchange a payment stream at a fixed interest rate K (the fixed leg of the contract) with a counterparty who in turn agrees to pay a flow of floating amounts (the floating leg).

In the following, we consider the term structure of payments for the floating leg given by the set {T_i|i = 1, …, n} defined above, together with the set , j = 1, …, n^S, for the payment stream of the fixed leg of the contract. We define the time interval

and

for the floating leg. Usually, the floating rate is fixed at the beginning of each period and both payments are due at the end of the period.

At time T_i, the floating leg pays the Libor rate L_i(T_i−1), where we set

while the fixed leg pays a fixed rate. The value of the swap contract at time t is

The forward rate swap S_1,n(t) is the unique fixed rate for which the two payment flows from the floating and the fixed legs in equation (8.118) coincide,

Defining the weights

the price of the swap in equation (8.119) is written as a linear combination of Libor rates,

A European payer swaption is an option on interest rates swaps giving the right to enter an interest rate swap contract at a future time T_e, when the fixed rate is K. In contrast, a receiver option gives the right to enter a receiver swap at time S. The payer swaption payoff at time S is

where we have defined the annuity

Using equation (8.119), we rewrite the swaption payoff as

Contrary to the cap option case, a swaption payoff cannot be decomposed into a sum of options on each period T_i. Moreover, in [35] it is shown that the price of a swaption depends not only on the evolution of each Libor rate, but also on their joint behaviour.

To price the payoff in equation (8.124), we consider the measure Q_C in which the annuity is the numeraire. In this measure, the formula for the price of a swaption of expiry S and maturity T is (see [89])

A closed-form formula is in theory unavailable. In practice, one may resort to the assumption of a lognormal distribution for the forward swap rate and thus obtain a pricing formula for payer swaptions of the type:

For a corresponding receiver swaption Swpt(t, T_e, T_n, S_e,n(t), K, σ_e,n, −1) we can resort to put–call parity.

Modelling the Libor rate with a spread over the forward rate

As already anticipated above, after the 2008 crisis, a spread S_i(t) at time t between the risk-free rate (approximated by the OIS or Eonia rate) and the Libor or Euribor fixing rate is experienced in the market,

where F_i(t) = F(t, T_i−1, T_i) is the Libor (or Euribor) forward (FRA) rate and F_i(t) = F(t, T_i−1, T_i) is the risk-free OIS (or Eonia) forward rate. In general, we can decide to model using two stochastic processes two variables from S_i(t), F_i(t) and L_i(t), leaving the third determined by equation (8.127).

For swap rates, we use the fact that the swap s(t) is written in equation (8.121) as a linear combination of Libor foward rates. Writing the Libor rate as a sum of the forward rate plus a spread as in equation (8.127), we obtain

where we defined the processes

In particular, is a martingale under the measure Q_C. Following [89] and [68], the process can be treated, under generic assumptions, as a driftless process. In contrast, the pricing of the spread part needs an additional assumption, since it depends on both the collection of spread rates S_i(t) and on discount factors as a result of weights ω_i(t). For this, we approximate the process as

where we replaced the time dependence on the weight with the value at t₀. With this approximation, is a martingale under Q_C and can be described, analogously to , by a driftless process.

8.4 DEFAULT PROBABILITIES AND CREDIT SPREADS

In the credit risk literature, two different approaches to evaluating the probability of default have been developed; namely, the structural model and the reduced model. In a structural approach, the performance of the firm is governed by structural variables like the asset or the debt value, and a default is the result of poor operations of the firm. In reduced (or statistical) approach, the default intensity is modelled by a stochastic process, which might include jumps in the intensity of default λ_t.

The structural model and the reduced model are also referred to as endogenous and exogenous approaches, respectively, because in structural models the time of default is determined through the value of the firm, while in reduced models it is the jump of a stochastic process that determines bankruptcy.

We will mainly use reduced models, but we also quickly review structural models.

8.4.1 Structural models

In structural approaches, we model the structural variables of a firm (i.e., the assets and the debt value) using a stochastic or a deterministic process, in order to determine the time of default. Earlier literature on the subject includes [25], [91] and [24]. For example, in Merton's model, a firm defaults if its assets are below its debt when servicing the debt. In the Black and Cox model, a default occurs when the value V of the firm reaches a default boundary K.

Merton formula

Merton [91] proposed a model in which the firm value is treated as an option on the asset V. Suppose for simplicity that, at time t, the firm issues a bond with maturity T. We define the variables and parameters in Merton's model as in Table 8.1.

Table 8.1. Variables and parameters in Merton model

In the Merton model, the firm is financed through a single bond paying no coupons and a single equity issue. At any intermediate time t < s < T, the asset of the firm is

We assume that the asset follows the geometric Brownian motion described in Section 8.2,

where μ is the instantaneous expected rate of return, and W_t is a Brownian motion.

At time T, the firm repays its debt D. When V_T < D, the firm defaults, and the value of the equity is zero. Conversely, if V_T > D, the firm repays the debt and the equity has value E_T = V_T − D. At time T, the value of the equity is then

which resembles the payoff of a call option of the firm's asset V_t with strike price D. Using the Black–Scholes formula, we find the value of the equity at time t as

where r is the risk-free rate, N(x) is the cumulative probability distribution function for a normal distribution, and

Setting and using the notation in Section 8.2, we have

Although the structural approach is theoretically fascinating, it is hard to be satisfactorily calibrated to all available market data. The reduced-form approach we will sketch below is much more flexible, even if less linked to the microeconomic factors triggering the default event.

8.4.2 Reduced models

In a reduced-form approach, the event of default is modelled via an intensity of default λ. In this perspective, default intensities are modelled similarly to default-free interest rates, allowing us to use the same results and formulae previously described. In fact, it can be proved (see [62]) that, remarkably, defaultable bonds can be priced by adjusting the discount rate: this effective rate is the sum of the risk-free rate r_t and the intensity of default λ_t, and both components can be treated with the mathematical tools we have shown before.

Given that the firm has not defaulted up to time t, the time of default T is defined as the probability of a default occurring in the next instant dt,

The intensity of default can be modelled differently as a

• constant (T is the first jump of a time-homogeneous Poisson process);
• deterministic function (T is the first jump of a time-inhomogeneous Poisson process);
• stochastic function (T is the first jump of a Cox process).

In particular, the second and third approaches can be used to model the term structure of credit spreads, and the third approach can be used to model credit spread volatilities.

The probability that the firm survives to time T, given we are at time t, equates

This definition is consistent with the expression in equation (8.137), by setting

In fact, equation (8.137) can be proved using the definitions in equations (8.138) and (8.139).

Modelling default intensities with a CIR process

We model the intensity of default by a CIR process

where we impose the additional constraint on the parameters 2κθ > σ² in order to ensure positiveness of the process.⁸ At time t, the probability that the firm has not defaulted up to time T is given by equation (8.138) which, in the CIR model, can be written in the form of equation (8.27)

with the factors A(t, T) and B(t, T) as defined in equation (8.28). The corresponding probability of default is PD(t, T) = 1 − SP(t, T).

We can also suppose that the intensity of default is a CIR++ process, with the dynamics and the formula for the SP modified accordingly as described in Section 8.3.4.

Multiple defaults of correlated firms

For multiple issuers, we can assume that each default intensity process is the sum of an idiosyncratic component plus a common intensity of default. In this model, the default of the i-th issuer can be triggered by either the i-th idiosyncratic component or by the common component. Both the idiosyncratic and the common parts are described by independent CIR processes

where m is the total number of firms, and

The CIR process for the i-th firm is

where p_i ∈ [0, 1] operates as a correlation of the i-th issuer with the probability of a common default: this approach to modelling is termed “affine correlation”.

The intensity is decomposed into an idiosyncratic term and a common intensity of default . The idiosyncratic term is related to the probability that the default of the i-th issuer occurs independently of all other firms, while the process accounts for the probability that all firms default simultaneously. A common default affects the i-th issuer with a probability p_i.

The sum of the two CIR processes in equation (8.144) does not automatically imply that λ_t,i is a CIR process. To find the constraints on the parameters, we first use the results in Section 8.3.3, to write the common process of default as

and (again from Section 8.3.3) the sum of the two CIR processes and is a CIR process λ_t,i if we impose the constraints in equation (8.51),

For the correlated default intensities considered, the constraint in equation (8.19) reads

Likewise, the intensity process can follow the CIR++ process in equation (8.67). The deterministic part of λ_i,t is a function ϕ_i,t, and we have

8.4.3 Credit spreads

In Chapter 7 we quickly sketched how to set a fair credit spread for a loan contract, given the counterparty's probability of default and the loss given default. In this section we will try to show in more depth how to model credit spreads.

The modelling of credit spreads critically depends on assumptions made about recovery from the borrower's default: recovery is just complementary to loss given default. Let us assume that the loan has a face value of 1 which the borrower has to repay at expiry T, with (simply compounded) interest (r + s) × T, where r is the risk-free rate and s is the credit spread. The default can occur at any time between the evaluation time t = 0 and the expiry t = T, but is only observed by the lender at expiry when the repayment should be made.

There are different possible choices for recovery, two of which are most relevant for practical purposes:

• Recovery of market value (RMV): upon the borrower's default, the lender recovers a fraction R = 1 − L_GD of the market value of the loan. This means that the expected value V_T at expiry (i.e., (1 + (r + s)T)) is:

  

  At time 0, the expected value is:

  

  Since the face value of the loan is 1 which is also the fair value it should have at inception when the amount is lent to the borrower (V₀ = 1), from equation (8.150) we have:

  

  Setting L_GD¹ = L_GD(1 + rT), recalling that , and using the approximation e^c = 1 + c for small c, after a few manipulations we obtain:

  

  where P(L_GD¹λ_t; 0, T) is the price of a zero-coupon bond expiring in T and with a discounting rate equal to L_GD¹λ_t. If λ_t follows a CIR model, it is possible to use formula (8.27) for a modified CIR process as shown in equation (8.54).

• Recovery of face value (RFV): upon the borrower's default, the lender recovers a fraction R = 1 − L_GD of the face value (in our case, 1) of the loan. The expected value V_T is:

  

  At time 0, the expected value is:

  

  Furthermore in this case, since V₀ = 1, after a few manipulations we get:

  

  Setting L_GD² = L_GD + rT, we can write the spread as

  

For loans with a short maturity and for a counterparty with a reasonably low PD (say, 3%), the denominator of both equations (8.151) and (8.154) can be set approximately equal to 1. Moreover, it is also easy to check that for small rT, L_GD¹ ≈ L_GD², so that for short-term loans, such as deposits in the interbank market, the two assumptions produce very similar spreads.

In conclusion, the credit spread between time t and T can be written as:

Equation (8.155) is a good approximation for both assumptions regarding recovery.

8.5 EXPECTED AND MINIMUM LIQUIDITY GENERATION CAPACITY OF AVAILABLE BONDS

Liquid assets are used to generate BSL and they can be considered as a component of the liquidity buffer, as seen in Chapter 7. We introduced the TSAA in Chapter 6 and showed how it is built and its connection with the TSLGC. We have left the problem of how to determine the future expected value of the assets, the unexpected minimum (stressed) values and the haircuts that can be applied unsolved. All this information is useful to the LGC because it is affected by the actual price of the assets on the balance sheet that can be used to extract liquidity and to match negative TSECCF.

In what follows we show how to monitor the LGC of one bond's holding and of a bond portfolio. Other assets, such as stocks, are less important as far as the LGC is concerned and they also require less sophisticated modelling. The main tool to monitor the LGC related to bonds is to build a term structure of minimum liquidity that can be generated at future dates within a chosen period at a given confidence level (e.g., 99%).

To build a term structure, we take the following steps:

1. Divide the chosen period into a number of M subperiods;
2. At the end of each subperiod t_m compute the minimum value of the bond's holding or of the portfolio of bonds;
3. Include the haircut either according to the approach outlined above or according to some predefined rules (e.g., ECB haircuts).

The building blocks we need to price the bonds and to compute the expected value and the stressed levels at future dates are:

• An interest rate model—we opt for the CIR++.
• A default model for multiple issuers—we opt for a reduced-form approach, with the default intensities depending on idiosyncratic and common factors (to account for the correlation between defaults and credit spreads). All intensities have CIR dynamics and a deterministic time function is added.
• A model for haircuts.
• A parameter to account for the specialness of single bonds.

8.5.1 Value of the position in a defaultable coupon bond

A defaultable coupon bond is a bond issued by an agent m that can default at some date in the future. Consider a bond held by the bank with the following characteristics:

• notional amount N;
• annual coupon rate c;
• coupon calendar {t_i}, with coupon periods T_i = t_i − t_i−1;
• maturity T;
• loss-given-default rate L_GD.

The value of the position in the bond with the specifics above is acquired at time t. We define the accrued time as the time between the last coupon payment and the acquisition,

The price of the coupon bond is

In the first line we have the sum of all coupon flows, discounted by the risk-free discount factors over the period [t, t_i]. The third term accounts for the recovery paid in case of default, while the last term subtracts the accrual amount, which is the amount paid from the last coupon date to the time of acquisition. Obviously, setting N = 100 we get the market price of the bond.

We assume that the short rate follows CIR++ dynamics as in equation (8.66), so the function is given by the formula:

where is a bond-specific parameter to capture liquidity specialness and is defined in equation (8.27).

8.5.2 Expected value of the position in a coupon bond

The expected value of the bank's position in a coupon bond can be easily computed, since it is the future price calculated according to the model we use for interest rates.⁹ We are interested in also computing the expected value net of the haircut, so we write the expected value of the position in a coupon bond as

where H_t(s) is the expected haircut at time s, see equation (8.166), while the future without including the haircut is

and

In this case is also given by the formula:

where is the bond-specific parameter to capture liquidity specialness and H_CIR++(r; t, s, T) is defined in equation (8.72).

Equations (8.158) and (8.159) do not consider the possibility of default of the bond's issue between t and s. If the issuer goes bankrupt, the bank will not have a bond worth CH(t, s, T) but CH(t, s, T) − L_GD, where L_GD is the amount lost given the default. If we introduce the assumption the L_GD = x%N, or a constant fraction of the par value of the bond, times the notional amount, then it is easy to consider the loss potentially suffered as well since in this case the expected value at time s would be:

or, put into words, the expected value including the default event CH^DI(t, s, T) is equal to the corresponding expected value without considering the default CH(t, s, T) minus the expected loss on default L_GDPD(t, s).

The SP and the PD are modelled by a reduced-form approach where the default intensity of the n-th issuer is λ_t follows a CIR++ process too, so that formula (8.66) can also be used in this case:

When a portfolio of bonds is considered, then the intensities of default are defined as in equation (8.148), where an idiosyncratic and common intensity and a deterministic time function ϕ_t appear.

We would like to stress the fact that for some issuers (typically sovereign) the PD can be zero and the bonds issued by them may even incorporate a liquidity discount (i.e., ), so that their yield could even end up being lower than a corresponding risk-free bond.

8.5.3 Haircut modelling

At the present time t, the value H(t) of the haircut to be applied to the bond's market value is typically a function of the probability of default PD of the issuer. This can be inferred from the bond's price or from the rating the bond's issuer has.

An approach based on the maturity T and credit rating has also been adopted by the ECB.¹⁰ For a given maturity, the bond can be Step 1 and 2 if the harmonized rating is between AAA to A−, so that it has a haircut depending on the maturity and the category H₁; otherwise, the bond can be Step 3 if the rating is between BBB+ and BBB−, with a haircut H₂; finally, the bond can be out of the eligible set that is accepted by the ECB as collateral and implicitly has a haircut H₃ = 100%, since no liquidity can be extracted from it by a repo transaction with the central bank.

A similar approach can be followed when simulating the haircut of a bond, since in the end a bond can always be pledged in a collateralized loan with the ECB. For the haircut to be applied at a future time s, we model H(s) with a three-step function. Given two levels of the survival probability in the next period (say, 1 year) K₁ and K₂ < K₁, the bond at time s is Step 1 and 2 (first quality according to the ECB's classification) if its default probability is below 1 − K₁; it is Step 3 if its default probability is between 1 − K₁ and 1 − K₂; finally, it is considered ineligible if its default probability is higher than 1 − K₂.

We adopt a reduced-form approach to model the default probability of the issuer, and we assume that the default intensity follows a CIR process as in equation (8.140), using equations (8.139) and (8.141):

where ϕ_t is a deterministic function and the CIR stochastic function is defined as

We can derive the probability that the bond falls in the first, second or third of the groups described above. In particular, the probability at time t that the bond at time s falls in the first (Step 1 and 2) group is

where, . We define the parameters

and, by making the trigger K₁ equal to the discount factor on [s, s + T_Hor] computed from the CIR++ intensity λ_t, we find

where T_Hor sets the default time horizon from the time at which we compute the haircut. We can set T_Hor = 1 year, for example, or we can link the function to a longer term PD by setting T_Hor = 5 years.

Similarly, the probability that the PD at time s is between 1 − K₁ and 1 − K₂ is

with

Finally, the probability that the PD lies above 1 − K₂ is p₃(t) = 1 − p₁(t)− p₂(t).

The expected haircut at time s is then

Since we presented a model for stochastic default intensities in Section 8.4, we can use it to model stochastic haircuts evolving in the future according to changes in the issuer's PD.

8.5.4 Future value of a bond portfolio

We are able to calculate the expected value of a bond's position at future dates; it is not a major problem to compute the expected value of a portfolio of bonds as well, since it is the sum of the single expected values. We are interested in finding a way to determine which is a stressed minimum level that the bond, or the portfolio of bonds, can reach at different dates in the future at a defined confidence level, say, 99%. In this case the correlation between issuers plays a major role in reducing the distance between the expected and the stress level, or the unexpected variation of the bond or portfolio of bonds.

8.5.5 Calculating the quantile: a Δ − Γ approximation of the portfolio

We consider a portfolio comprising N_B bonds and issued by N_f issuers. For each issuer i, there are N_i bonds in the bank's portfolio; each bond's position has value Bⁱ(t, T_j) and expected CHⁱ(t, s, T_j), 1 ≤ j ≤ N_i. For numerical reasons, if a liquidity discount for the i-th bond exists (i.e., ), then the corresponding value of the process λ_i,t for the i-th issuer is theoretically zero, although we set it to 1e-10, and so is the value of the function ϕ_t.

Defining the vector of intensities¹¹

the expected value of the position for all bonds at time s is

We perform a Δ − Γ approximation of the value of the portfolio at time s, around the value Y_s, as

The vector Δ and the symmetric matrix Γ are computed by taking the numeric first and second derivatives of the portfolio V(Y_t) with respect to r_t and λ_i,t. We recall that, given a function f(x, y), the first derivative along x is approximated as

the second derivative along x is

and the mixed derivative is

The dimension of Δ and the rank of Γ both equate to N_f + 1. Setting

and

we rewrite equation (8.169) as

This assumes that Y_s has a Gaussian distribution, with mean M = (M^r, M^λ)^T and a covariance matrix

For the CIR++ processes in equation (8.67), and the intensity decomposed as in equations (8.142), and (8.143), at time s we have the mean

while the covariance matrix is

We recall that and . For N_f firms, the dimension of the vector M and the rank of the covariance matrix Ω are both N_f + 1.

Now we are able to compute the value of the quantile v for which there is a probability p = P(V(Y_s) < v) that the value of the portfolio at time s > t falls below v. To obtain P(V(Y_s) < v), we first Fourier-transform V(Y_s), defining the function

The probability relates to ϕ(u) according to the formula of Gil-Pelaez [72],

To perform the integration numerically, we first rewrite it as

then we introduce a grid with equally spaced abscissae u → u_j = (j + 1/2)δx, with step size δx and j running from one to an integer K. The integral is then truncated from zero to K δx, and equation (8.184) is approximated by

Fixing the time horizon s and the probability p that the portfolio falls below the level v, we invert equation (8.183) to obtain the quantile v; namely, we invert

To find v numerically, we use a bisection method.

Since v only accounts for the market value of the portfolio of bonds given that no default of the issuers occurs during the period [t, s], if we are interested in considering the quantile that also includes the losses suffered if one or more issuers go bankrupt we can compute the expected loss for the portfolio in the event of defaults,

so that the quantile with default included is:

The minimum level of a portfolio of bonds at the 99% c.l. corresponds to the first quantile of the distribution and is derived as described above.

8.5.6 Estimation of the CIR++ model for interest rates

To estimate the parameters of the process x_t in equation (8.66) (the process for the instantaneous interest rate) from market data, we can consider a series of discount factors bootstrapped from the Eonia swap quotes. We mentioned above that the Eonia rate can be considered virtually risk free, since it embeds the credit risk related to lending money to a bank for one day; moreover, swap rates with the Eonia rate as the underlying are assumed to be fully and continuously margined, so that counterparty credit risk is almost totally eliminated. In the end the Eonia rate and swaps on Eonia can be supposed to be the contracts from which it is possible to extract the risk-free term structure of interest rates.

The estimation involves two steps:

1. Estimate the parameters of the process x_t with the Kálmán filter, as explained in Section 8.3.3.
2. Calibrate a time-dependent function ψ_t with the market data of each date of the time series, so as to have a best fit of the model.

Let us now provide a practical example of estimating the parameters using real market data.

Example 8.5.1. We consider a panel of bootstrapped discount factors, obtained from the Eonia swap rate quotes with maturities given in Table 8.2. The time series of the term structures of the swap rates is considered in the period running from 31/12/2010 to 4/6/2012. We plot the 1W, 1Y, 5Y, 10Y and 30Y swap rates in Figure 8.1.

Table 8.2. Maturities of Eonia swap rates used in the estimation procedure

Discount factors on the same set of maturities as in Table 8.2 are derived for each date and used for the estimation with the help of the Kálmán filter technique, obtaining the results in Table 8.3.

Figure 8.1. Eonia swap rates from 31/12/2010 to 4/6/2012

Table 8.3. Values of the parameters of the CIR model for the zero rate, obtained using the Kálmán filtering technique described

The deterministic function in equation (8.66) is found by requiring, for any date t_m in the panel data, a perfect match between the swap rate data of the time series with maturity T_i, P^data(t_m, T_i), and the value of the discount factor P(x; t_m, T_i) from equation (8.69), imposing

P_CIR(x; t_m, T_i) is known since we have already calibrated the CIR process for x_t (we approximate the value of x₀ using the Eonia overnight). We use the other N_ϕ = N_Eonia − 1 series of Eonia maturities to construct the function ψ_t. We define a series of N_ϕ time steps ΔT_i as

We ignore the first value T₁ that refers to the maturity at one week. Assuming that ψ_t is a step function, of step size ΔT_i, the integral of ψ_t is

Figure 8.2. The function ψ_t, obtained with the method described in the main text, as a function of maturity

the values for ψ_t are then built iteratively. The first value of the step function is

while the i-th value is

Note that, if there are M days of observation, this technique results in M step functions ψ_t.

Example 8.5.2. We use the panel data described in Example 8.5.1 with the calibrated parameters for the zero rate in Table 8.3 to construct the function ψ_t. We show a plot of ψ_t as a function of maturity in years in Figure 8.2.

8.5.7 Estimation of the CIR++ model for default intensities

In order to calibrate the parameters describing the intensities of default, we need to modify somewhat the Kálmán filter procedure described in Section 8.3.3 for the zero rates. Intensities of default are calibrated using a time series of coupon bonds, in which we consider a total of N_B coupon bonds and N_f issuers.

The number of CIR processes to be calibrated is N_f + 1, corresponding to the number of idiosyncratic processes N_f (one for each firm), plus the common process. For each day t, we store this information into the vector y_k,t, where the first entry y_0,t is for the common intensity of default, and the remaining N_f entries are for each idiosyncratic process. The total number of parameters to be calibrated is

Table 8.4. Specifics for the bonds issued by Italy, France and Spain

corresponding to κ, θ^C, σ^C plus p_i and for each firm i.

Example 8.5.3. We consider a portfolio of 15 bonds, issued by Italy (ITA), France (FRA) and Spain (SPA), respectively, over the period from 31/12/2010 to 25/05/2012 and with the specifics in Table 8.4. For each bond, we take into account the time series of prices from the beginning of the period, or the issue date if later, to the end of the period. We estimate the values of the parameters from the time series of bond prices using the Kálmán filter technique, obtaining the results in Table 8.5. In Figure 8.3, we plot the intensity of default λ_t for each issuer ITA, FRA, SPA, over the period 31/12/2010 to 25/5/2012, obtained by the Kálmán filter.

Having calibrated the parameters of the CIR processes commanding the intensity of default of each issuer, we need to calibrate at the reference time t = t₀. We choose to use the model of the value of the intensity λ_0,i for the i-th issuer, which can be decomposed into an idiosyncratic term and a common term as

Defining the total number of bonds in the portfolio N_B as in equation (8.168), we construct the function

Table 8.5. Values of the parameters of the CIR model for the intensity of default process for the three issuers Italy, France and Spain, obtained with the Kálmán filtering technique

Figure 8.3. The value of the default intensity as a function of time, for the three issuers Italy, France and Spain, obtained from the Kálmán filter technique

where B_j(MKT) is the market price for the j-th bond and B_j is the model price for the same bond according to equation (8.157). At this stage, we have set to zero the values of the functions ϕ_i,t for all firms, while we consider the calibrated function ψ_t for the zero-rate process. The values of and are found by requiring that the above function be minimized.

After estimating the initial values λ_0,i, for each issuer i, we model the time-dependent ϕ_i,t with a step function of step size ΔT_ϕ, for a total of N_ϕ steps. For the i-th issuer, N_ϕ is fixed by the bond with the longest maturity, say T_J, so that

where t₀ is the last day considered and [x] is the integer part of x.

Figure 8.4. The value of the function ϕ_t as a function of time, for the three issuers Italy, France and Spain

The function ϕ_i,t is calibrated to the market data by a least squares procedure. We consider the function

In equation (8.197), the sum runs over the N_i bonds issued by the i-th firm, B_j is the price of the j-th defaultable coupon bond according to equation (8.157) and B_j(MKT) is the corresponding market price for the same bond.

We demand the function f_i(ϕ_i,t) to be minimized with respect to ϕ_t, under the additional constraint that the second derivative of ϕ_i,t be minimized at each step as well.

Example 8.5.4. We consider the market bond prices of the portfolio described in Example 8.5.3, on the reference date 25/5/2012. Using CIR parameters for the zero rate and the default intensities in Tables 8.3 and 8.5, and using the function ψ_t obtained in Example 8.5.2, we obtain the model price in equation (8.157) for each issuer as a function of ϕ_i,t for that issuer. Using the methods described in this section, we calibrate the functions ϕ_i,t with the market data, obtaining the results shown in Figure 8.4.

The model bond prices resulting from the estimated parameters and the deterministic function are shown in Table 8.6: for each bond we also show the theoretical price obtained only by the CIR component of the default intensity model and the price obtained by adding the time-dependent function.

Example 8.5.5. We consider the market bond prices of the portfolio described in Example 8.6.1, on the reference date 4/6/2012. Using CIR parameters for the zero rate and the default intensities in Table 8.3 and 8.21, and using the function ψ_t obtained in Example 8.5.2, we obtain the model price in equation (8.157) for each issuer as a function of ϕ_i,t for that issuer. Using the methods described in this section, we calibrate the functions ϕ_i,t with the market data, obtaining the results shown in Figure 8.5.

Table 8.6. Model bond prices in Example 8.5.3

Figure 8.5. The value of the function ϕ_t as a function of time, for the three issuers Italy, Unicredit and Intesa Sanpaolo Bank

Finally, we need to compute the bond-specific parameter to perfectly match market bonds priced with model prices obtained with the estimated parameters. The value of can easily be found with a numerical procedure like the Newton–Rahpson method or bisection.

Example 8.5.6. We refer to the portfolio described in Example 8.5.3. Using the method described in this section, we find the liquidity parameter for each bond as in Table 8.7.

Table 8.7. Liquidity parameters for each bond considered in Example 8.5.3

It is worthy of note that not all issuers have to be given a default probability. In some cases the bond prices show that the theoretical price can match market quotes only if the default intensity is set equal to zero and the process does not depart from this value. It is also likely that in these situations a liquidity premium (i.e., ) is implied in market quotes, as we will see in the next example.

Example 8.5.7. To the portfolio in Example 8.5.3 we add five additional bonds issued by Germany (GER). the specifics for the GER bonds are given in Table 8.8. Since the market price of these bonds is above the price obtained using the bond formula in equation (8.157) with the intensity parameter λ_t = 0, there is only a liquidity premium to be attached to these bonds. We obtain the liquidity parameter for each bond in Table 8.9.

Table 8.8. Specifics for the GER bonds considered

Table 8.9. Liquidity parameter for the GER bonds in Table 8.8

Example 8.5.8. We consider the portfolio of bonds with the specifics in Table 8.20. Using the method described in this section, we find the liquidity parameter for each bond as in Table 8.10.

Table 8.10. Liquidity parameters for each bond considered in Example 8.6.1

8.5.8 Future liquidity from a single bond

We now have all the information to forecast the expected and minimum (at a given confidence level) liquidity that can be extracted form a bond: in fact, we can obtain liquidity as analysed in Chapter 6 when describing the TSAA, by selling or pledging the bond in a collateralized loan (which is in practice the same as repoing it).

If we know the expected and minimum levels of the price, with or without the haircut, we can build a term structure of liquidity that can be generated by the bond by adding this information to the TSAA. In the next example we show how the model we have calibrated in this section can help produce these data.

Example 8.5.9. We consider a portfolio of EUR30 million notional containing only the bond issued by Italy maturing in 2017, see Example 8.5.3. We compute a term structure of expected price levels and of minimum levels at the 99% quantile over a period of one year, in monthly steps. We also compute this stressed minimum level by considering and excluding the default event: in the latter case the intensity of default affects the price but does not trigger any jump event. When a haircut is not applied, we obtain the values in Table 8.11 and Figure 8.6 indicating the different levels of future liquidity.

Table 8.11. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in the bond issued by Italy and maturing in 2017. The term structure is considered over one year, in monthly steps. No haircut has been applied

The information contained in Table 8.11 can be used when building the TSAA and the TSLGC to estimate how much liquidity can be obtained by selling the bond in the next year.

The same type of analysis can be conducted by including the haircut so as to forecast the potential liquidity obtainable by repo transactions. We assume that the haircuts are determined by an approach such as the one explained above, and that the bond in the Step 1 and 2 group of the PD over one year is below 5%, whereas it falls in the Step 3 group when the PD is below 15% (above 15% it is no longer eligible for repo transactions). We obtain the term structure of expected and stressed levels at the 99% c.l. in Table 8.12 and Figure 8.7.

We can also set the trigger for the passage from Step 1 and 2 to Step 3 at lower PD levels. For example, we can leave the first trigger at 5% PD over one year and lower the second trigger to 7.5%. In this case we obtain the term structure in Table 8.13 and Figure 8.8.

8.5.9 Future liquidity from more bonds

The framework we have designed can also cope with a portfolio of many bonds with different maturities: in this case the bank can take advantage of the diversification effects that can be attained by buying bonds with different exposures to interest rates and to the evolution of the PD. It is clear that in this case the TSAA needs to be built while considering the aggregated position the bank has on a single issuer as well, since information on just one bond can be useful but is only a small part of the entire picture of the BSL.

Table 8.12. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in the bond issued by Italy and maturing in 2017. The term structure is considered over one year, in monthly steps. We applied a haircut with triggers at 5% and 15% for the PD over one year

Table 8.13. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in the bond issued by Italy and maturing in 2017. The term structure is considered over one year, in monthly steps. We applied a haircut with triggers at 5% and 15% for the PD over one year

Example 8.5.10. Consider a portfolio of EUR30 million, equally distributed among the five bonds issued by Italy and with the specifics given in Example 8.5.3. The term structure of expected price levels and of minimum levels at the 99% quantile over a period of one year, in monthly steps, can be computed with the approach described in the main text. In this example we also compute this stressed minimum level by considering and excluding the default event, as in Example 8.5.9. When a haircut is not applied, we obtain the liquidity that can be generated by selling the portfolio at future dates, both at an expected and at a minimum level (99% c.l.), as shown in Table 8.14 and Figure 8.9.

Figure 8.6. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in the bond issued by Italy and maturing in 2017, as in Table 8.11

Figure 8.7. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in the bond issued by Italy and maturing in 2017, as in Table 8.12

Moreover, bond portfolios can be repoed out, so we can compute the expected and minimum liquidity by including the haircuts too. As in Example 8.5.9, we assume two levels of PD triggering the passage from the ECB's Step 1 and 2 to Step 3 at 5%, whereas above 15% the bonds are no longer eligible for repo transactions. Table 8.15 and Figure 8.10 show the results.

Figure 8.8. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by 30 million euros invested in a portfolio of five bonds issued by Italy, as in Table 8.13

Table 8.14. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in a portfolio of five bonds issued by Italy and maturing. The term structure is considered over one year, in monthly steps. No haircut has been applied

Haircuts play a major role on the liquidity that can be obtained by repo transactions when a portfolio of bonds is also considered. In fact, when a higher haircut is triggered with higher probabilities, the minimum liquidity is strongly affected as shown in Table 8.16 and Figure 8.11: in this case triggers are set at 5 and 7.5% of the PD at one year.

Table 8.15. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million euros invested in a portfolio of five bonds issued by Italy and maturing. The term structure is considered over one year, in monthly steps. We applied a haircut with triggers at 5% and 15% for the PD over one year

Table 8.16. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in a portfolio of five bonds issued by Italy and maturing. The term structure is considered over one year, in monthly step. We applied a haircut with triggers at 5 and 7.5% for the PD over one year

The BSL needs to be considered at an aggregated level when bonds are issued by several debtors as well. In this case we should also take into account the correlations between the probabilities of default that affect not only the losses suffered by the holder (the bank in our case) when the issuers go bankrupt, but also the price of the bonds in the portfolio.

Figure 8.9. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in a portfolio of five bonds issued by Italy, as in Table 8.14

Figure 8.10. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in a portfolio of five bonds issued by Italy, as in Table 8.15

Figure 8.11. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in a portfolio of bonds issued by Italy, as in Table 8.16

The model we have introduced is capable of accounting for the correlation by using the composite CIR++ process: this correlation affects the liquidity that can be generated by the portfolio in two ways:

1. Prices are more correlated if the default intensities of the issuers are more correlated.
2. The haircuts of the bonds increase jointly with higher probability if the intensities are more correlated, since they are linked to the credit rating and we assumed that this is dependent on the probability of default, as explained above.

In both cases the liquidity that can be generated at future dates by the portfolio of bonds is smaller.

Example 8.5.11. We consider a portfolio of EUR30 million, equally distributed among 20 bonds, 15 of which are those considered in Example 8.5.3, while the other 5 are the GER bonds in Example 8.5.7. We compute the term structure of the expected and minimum liquidity (99% c.l.) that can generated by the portfolio. We start by considering the case the when the portfolio is sold, so we exclude the haircuts. The results are in Table 8.17 and Figure 8.12.

If the bank wants to compute the expected and minimum liquidity that can be extracted from the bond portfolio by repo transactions, the haircut has to be included. We assume that the levels at which the PD triggers passages between haircuts, modelled on the ECB's approach as explained in the main text, are set at 5 and 15% for all issuers. We obtain the term structure of liquidity in Table 8.18 and Figure 8.13.

When higher haircuts are more likely, such as when they are triggered by PD levels set at 5 and 7.5%, we obtain the term structure of liquidity in Table 8.19 and Figure 8.14.

Table 8.17. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in a portfolio equally distributed among 20 bonds issued by ITA, GER, FRA, SPA. The term structure is considered over one year, in monthly steps. No haircut has been applied

Table 8.18. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in a portfolio equally distributed among 20 bonds issued by ITA, GER, FRA, SPA. The term structure is considered over one year, in monthly steps. We applied a haircut with triggers at 5 and 15% for the PD over one year

8.6 FAIR HAIRCUT FOR REPO TRANSACTIONS AND COLLATERALIZED LOANS

In the previous section we introduced a tool to monitor the LGC of a portfolio of bonds held by the bank. In the case of liquidity generated by repo transactions, we have modelled the haircut to apply to the market price of the bonds following the approach adopted by the ECB since, first, the central bank is the main counterparty for European banks¹² and, second, because the ECB's haircuts are often used as a guide to set haircuts for transactions not involving the central bank itself.

Figure 8.12. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in a portfolio of bonds issued by ITA, GER, FRA, SPA, as in Table 8.17

Table 8.19. Term structure of expected and minimum (99% c.l.) levels of liquidity (million), with and without credit event, generated by EUR30 million invested in a portfolio equally distributed among 20 bonds issued by ITA, GER, FRA, SPA. The term structure is considered over one year, in monthly steps. We applied a haircut with triggers at 5 and 7.5% for the PD over one year

Figure 8.13. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in a portfolio of bonds issued by ITA, GER, FRA, SPA, as in Table 8.18

Nonetheless, we would like to design an approach to set fair haircut levels for repo transactions and collateralized repos which is somehow different and possibly more robust than that devised by the ECB. In fact, one of the main flaws of the ECB's approach to set haircuts is that it does not consider wrong-way risk in any way: this increases the probability of there being a joint default of the counterparty borrowing money in the deal and the issuer of the bond pledged as collateral. This situation clearly makes the collateral lose its risk mitigation feature and the credit risk to the lender approaches that of a non-collateralized loan.

We would like to stress the fact that wrong-way risk is very common for the following reason: collateral bonds are often Treasuries issued by the government of the country of the borrower bank. Typically, there is a strong correlation between the credit spreads (which reflects the PDs) of the Treasuries and the banks, for the circular fact that banks have huge quantities of Treasury bonds on their balance sheets to build the liquidity buffers. So, when crises occur with sovereign debts, such as the one experienced in 2011–2012 in Europe,¹³ the PD of the banks of the countries under pressure increase as a result of the higher probability of default of the sovereign debt they hold as assets. Banks of countries such as Greece, Spain, Portugal, Ireland and Italy pay high credit spreads over the risk-free rate (however we define it) because they are perceived as high risk for the Treasury bonds they hold (and for the more general country risk).

Figure 8.14. Term structure of expected and minimum (99% c.l.) levels of liquidity, with and without credit event, generated by EUR30 million invested in a portfolio of bonds issued by ITA, GER, FRA, SPA, as in Table 8.19

The framework we have introduced above is rich enough to effectively cope with this problem as well. Assume that counterparty d posts a portfolio of N_B bonds, issued by N_f issuers as collateral for a loan (or for a repo transaction). For each issuer there are N_i bonds in the portfolio: the notional of the loan at inception is the same as the value of the portfolio minus a fair haircut. This includes, at a given date T:

• market risk: the probability of having, on the counterparty's default, a value of the portfolio lower than the loan still outstanding because of market risk factor movements (interest rates and credit spreads);
• credit risk: the probability of having, on the counterparty's default, a value of the portfolio lower than the loan still outstanding because one or more issuers went bankrupt;
• wrong-way risk: as defined above, the higher probability that the counterparty's default and the issuers' default occur simultaneously.

The default of the counterparty i is modelled by a composite CIR++ process and a deterministic time-dependent function as seen in Section 8.5. We introduce an indicator function D_d(0, T) equal to 1 if the default of the counterparty occurred between times 0 and T; it is equal to 0 otherwise. We take the approximation

where is an indicator that the idiosyncratic default process for d has jumped by T, whereas is the indicator that the common default process has jumped by T; finally, ξ_i is the indicator of the event that d goes bust at the first common credit event.¹⁴ In the framework we introduced above P(ξ_d = 1) = p_d, where p_d is the factor linking the intensity of the common default event to the total intensity of the counterparty d, and

where and . It is worth noting that we included the effect of the time-dependent function ϕ_d,t within the idiosyncratic default event.

Let V^Coll(t) be the value of the portfolio of bonds used as collateral, formed as:

In a repo transaction or collateralized loan, the lender seeks to make the expected loss equal to zero over a given period from 0 to T. The expected loss (EL) is equal to the expected exposure at default (EAD) minus the value of the collateral, in the event of default of the counterparty i. The EAD is simply the amount lent (assuming no scheduled repayments between 0 and T), so that:

It is straightforward to check that:

If the amount lent is L then E[EAD] = L. Then we have:

where D_Coll(0, T) is the indicator function equal to 1 if at least one issuer of the bonds in the collateral portfolio has gone bankrupt by time T, and L_GD is the loss generated by the default events. The equation can be rewritten as:

We split this equation into two parts. In this case it is also clear that:

since V^Coll(T) is the value of the portfolio at time T given that no default event has occurred. CHⁱ(T, T_j) is the expected price at time T of a coupon bond issued by i expiring at time T_j > T, defined as in (8.158). We assume that an issuer does not have more than one bond expiring on a specific date. The second part is:

The first part of the right-hand side considers the expected value of the loss on bonds in the event of the counterparty's default, given that the common default event is not triggered (the amount is weighted by the probability that the common event does not occur: 1 − PD^C(T)(0, T)); the second part is the expected value of the loss when the counterparty goes bankrupt and a common default event occurs (the amount is weighted by the probability that the common event occurs: PD^C(T)(0, T)).

Making the simplifying assumption that with equal to the notional amount in the portfolio is a constant percentage of the par value of the bond, then we can rewrite the equation for the expected loss on the repo as:

By the same token we can also compute the maximum loss at a defined confidence level, say 99%. This is quite easy to obtain since we already showed how to compute the minimum level of a portfolio of bonds at a given quantile by a Δ − Γ approximation; let the minimum value at the 99% c.l. of the portfolio at time T be:

where v is the quantile calculated as shown in Section 8.5. Then the maximum loss is:

The fair haircut to apply to the value of the portfolio of bonds at the inception of the repo or collateralized loans is the level that makes the EL or the ML zero, depending on whether the bank wants to be more or less conservative. If the bank chooses to make the EL zero, considering that the amount lent at time 0 is equal to the value of the portfolio V(t) minus the haircut, L =(1 − H)V(0), the level of H is chosen so that:

If the bank wants to compute the haircut based on the minimum level of the portfolio of bonds, then:

where v^DI is the quantile of the portfolio including default events as defined in Section 8.5.5.

It is interesting to measure the degree to which wrong-way risk impacts on the haircut. To this end we have to apply the formulae above assuming that no correlation exists between the counterparty and the issuer of the bond. In our setup this means that the parameter p is equal to zero for each debtor, or that the probability of a common credit event PD^C = 0. Nonetheless, we still want the total PD of each debtor to be the same as that produced by the combined effect of both the idiosyncratic and common factors. This means that in the absence of wrong-way risk it is

where SP is the zero-coupon bond in equation (8.27). Equation (8.199) modifies as follows:

The haircut is then computed as indicated in equations (8.201) and (8.202).

Example 8.6.1. We consider a portfolio of Italian bonds, used as collateral for two firms (UCG, ISP), on June 6, 2012. The specifics for the ITA bonds (collateral) and the UCG, ISP bonds (portfolios) are given in Table 8.20.

Assume that the bank has to lend money to another bank and that it accepts as collateral Treasury bonds. Let the counteparty borrowing money be either Intesa Sanpaolo (ISP), an Italian bank, or Unicredit Group (UCG), a European international bank. Moreover, we assume that the collateral comprises bonds issued by the Italian Treasury (ITA). The collateral is posted at the start of the contact and is not updated until maturity.

The interest rate process follows a CIR++ model and the parameters are those estimated in Example 8.5.1. To estimate the parameters of the default intensities of the two possible counterparties and of the Italian Treasury, we consider the prices of the bonds issued, respectively, by ITA, UCG and ISP, with the specifics in Table 8.20, over the period from 12/31/2010 to 6/4/2012; we set the reference date as 4/6/2012. For each bond, we take into account the time series of prices from the issue date to the reference date. We calibrate the values of the parameters with the term structure of bond prices using the Kálmán filter technique, obtaining the results in Table 8.21.

In Figure 8.15, we plot the intensity of default λ_t for each issuer ITA, UCG and ISP, over the period 12/31/2010 to 6/4/2012.

In Tables 8.22–8.24, we show the fair haircut (based on the EL and the ML) that the bank should apply for a repo or a loan secured with UCG or ISP maturing, respectively, in s = 3M, 6M and 1Y, for the three cases in which the collateral is a portfolio of the five Italian bonds above, the Italian bond maturing on 01/05/2017, and the other Italian bond maturing on 01/03/2022. We show the results both when we include wrong-way risk (WWR) and when we exclude it.

Haircuts are typically revised periodically to get the credit risk of the repo or collateralized loan back to as close to zero as possible. If the life of the contract is the period [0, T], divided into n subperiods of length [T_i−1, T_i], for 0 ≥ i ≤ n and T₀ = 0, then the haircut has to be calculated every time according to the methodology we have sketched above, with the default risk referring to the subperiod [T_i−1, T_i], instead of the total period [0, T_i]. So, at the inception of the contract, the haircut is obtained, according to the expected or maximum loss, by solving the equations:

Table 8.20. Specifics for the bonds issued by the Italian Treasury, Unicredit and Intesa Sanpaolo bank

Table 8.21. Values of the parameters of the CIR model for the intensity of default process for the issuers Italy, Unicredit and Intesa Sanpaolo Bank, obtained with the Kálmán filtering technique

Figure 8.15. The value of the default intensity as a function of time, for the three issuers Italy, Unicredit and Intesa Sanpaolo bank, obtained from the Kálmán filter technique described

Table 8.22. Haircuts based on the EL and ML. The collateral is the portfolio of Italian bonds

and

Then in T₁ the haircut is revised using new-market data and PD data by computing EL(T₂) or ML(T₂), and so on until the end of the contract.

Table 8.23. Haircuts based on the EL and ML. The collateral is the bond maturing on 01/05/2017

Table 8.24. Haircuts based on the EL and ML. The collateral is the bond maturing on 01/03/2022

8.7 ADJUSTMENTS TO THE VALUE OF ILLIQUID BONDS

Banks do not only have liquid bonds on their balance sheet for liquidity management purposes. Often some assets are liquid and actively traded in the market in the first part of their life, but afterwards the trading activity becomes scant and they no longer can be considered liquid. In these cases the bank will likely hold the assets until their maturity, but it is always possible that they can be sold to extract liquidity, although the trading process takes much longer to complete (maybe weeks or even months).

The problem with illiquid assets is twofold:

1. A fair value has to be attached to them which cannot directly be retrieved from the market, for the very fact that they are not actively traded. The fair value does not consider any difficulty in trading the bond such that it only refers to the case when it is perfectly liquid. So, attributing a fair value to an illiquid asset can only be carried out by disregarding any liquidity effects and by pretending that the asset is liquid.
2. An adjustment should be added to the fair value to reflect the actual capacity of the asset to generate liquidity, in terms of the amount and time needed.

Limiting our analysis to bonds, the fair value can easily be computed if other liquid bonds issued by the same debtor are actively traded in the market. In fact, based on them, it is possible to estimate the parameters of a model of the credit risk run by the issuer, such as the composite intensity framework we have used above to predict the LGC of a bond portfolio. Applying the credit model to illiquid bonds jointly with an interest rate model that has been also properly calibrated, allows computing the fair value with a relatively good degree of accuracy and reliability.

The liquidity adjustment to the fair value requires more care. The adjustment, as will be clear from what follows, is not an objective correction to the value, meaningful for any holder. It depends on subjective factors, such as the funding spread of the holder, the amount of the bond held on the balance sheet and, finally, the probability of selling the bond before its expiry. On the other hand, objective factors also play a role in defining the adjustment to the fair value: the trading activity of the bond, identified by the number of times it trades in the market in a given period and average amount. The combination of these two types of factors determine the liquidity adjustment to the fair value of the bond that refers to the specific holder—not a quantity that can be applied “objectively” to the bond.

The model we present hinges on two considerations:

1. A long position in an illiquid bond that cannot be sold immediately, but only over a certain period that depends on its trading activity, can be made comparable, in terms of generation liquidity, with a position in a liquid bond that can be immediately sold, if the bank borrows the amount that it would receive if the position were in the liquid bond, and then repay the loan as it receives the money from the selling of the illiquid bond during the period that is required to complete the trade. This strategy has clearly a cost that depends on the funding spread that the bank has to pay for the period needed to fully dismantle the position. We name the correction to the fair value of the bond due to this cost as liquid equivalent adjustment (LEA).
2. During the time required to completely sell the amount of the bond, the price of the bond can move up and down. The bank is interested in forecasting adverse movements and then computing the maximum loss (at a certain confidence level) it could suffer for not being able to sell immediately the amount held. The loss is given by the difference between the value of the position in the bond at the time the bank decides to sell it, and the minimum value of the position during the period required to completely sell it. We term the correction to the fair value of the bond due to this loss as price volatility adjustment (PVA).

Both adjustments have to take into account the fact that the costs (for LEA) and the loss (for PVA) are borne by the bank only if it decides to sell the position in the future. So they have to be weighted by a selling probability that will then be included in the model. The sum of the two adjustments is the liquidity adjustment (LA) to the fair value. The adjusted price is the liquidity that can be expected to be extracted from the position in an illiquid bond.

Remark 8.7.1. The model we present produces the LA assuming that the bank wants to sell the entire amount held when it decides to do so. In reality, it is possible to consider cases when only partial liquidation of the position is planned as well. In this case the adjustment has to be computed with respect to this amount, smaller than the total amount on the balance sheet, so that it will be different. It is then possible to calculate a range of LAs depending on the different fractions of the total amount that have to be sold.

8.7.1 Liquid equivalent adjustment

Let us focus on the LEA component of the LA. Assume that at time t we wish to compute the liquidity adjustment for a bond maturing at T. Let B(t, T) be the fair value at time t; the expected price of the bond at a future date s is CH(t, s, T). We work within the framework described in Section 8.5 to evaluate the bond and we use the same notation. The amount held by the bank on its balance sheet is N.

The bond is not actively traded in the market: we first need to model the trading activity. Let the average traded amount of the notional of the bond be , and let the trades occur with a trading intensity at time t equal to λ^T(t). Trading intensity indicates how many trades occur in a short period dt; the inverse of the intensity 1/λ^T(t) gives the expected time elapsing before a new trade occurs.

For example, assume that and that λ^T(t) = 365. This means that the bank can expect to trade 1,000,000 notional every λ^T(t) = 1/365 of a year, or every day. If λ^T(t) = 182, then 1,000,000 notional is expected to trade approximately every 2 days.

Trading intensity is modelled as a deterministic function of the kind

where is the trade intensity at t = 0, λ^−T is the long-term average trading intensity. Usually, bonds start with good trading activity after their issuance, then the number of deals gradually fade away during the life and, finally, the market becomes scant as the maturity approaches. This pattern can be modelled by setting a high trading intensity just after issuance of the bond (e.g., λ^T(0) ≥ 365 for trades occurring at least once a day), and a low long-term average intensity (e.g., λ^−T ≤ 25 for trades occurring less than once a fortnight). The speed at which trading activity is pushed towards the long-term average is commanded by the parameter κ^T.

The probability that the bank decides to sell the bond can be modelled by an intensity process as well: in this case we want to make the probability dependent on the default probability of the bank, since when the latter increases there is a higher probability that the bank may wish to sell less liquid assets to avoid paying high credit spreads over the risk-free rate. Thus, the selling intensity is:

where f(t) is a deterministic function and is the bank's default intensity, which is in turn a composite intensity as in Section 8.5. This formulation will also allow wrong-way risk to be considered (i.e., a higher probability to sell the bond when its value is lower due to the higher default probabilities of the issuer). Using the property of the CIR process, we write the selling intensity as

so that μ(t) is a process. Here, for the function f(t), we take a deterministic function of the type

Figure 8.16. The probability of selling as a function of time (in years) for the parameters given in Table 8.25

so that the integral is

It is also easy to check that when the PD of the bank (i.e., the holder of the bond) increases, selling intensity increases as well by a factor α. The probability of selling the bond between two times s and T is

where PNS(t₀, t) is the probability that the bond is not sold within [t₀, t], which is available in a closed-form formula in our setting:

where is given in (8.27). In Figure 8.16 we show the probability of selling as a function of time for the selling parameters in Table 8.25.

Table 8.25. The selling parameters used for Figure 8.16

Given this setting, at time t the bank has to operate an expected number of transactions to sell the entire position it holds; with selling intensity equal to λ^T(t), the average time Tl(t) to close out the position is:

If the bank wants to have the money it would receive if the bond were liquid immediately, it has it to borrow an amount of money equal to N × B(t, T). The borrowed amount is gradually paid back as the selling process completes; meanwhile the bank pays interest on the outstanding debt it has. The total interest rate the bank pays is composed of the risk-free rate plus a funding (credit) spread; on the other hand, the bank still earns the yield on the part of the position in the bond that has not yet been sold (which equals the outstanding debt). For equilibrium the yield on the bond's position should match the risk-free rate so we assume that on an expectation basis the net cost paid by the bank is the funding spread.

Over the period [t, t + Tl(t)] the annualized funding spread required of the bank is:¹⁵

where L_GD is the loss given default percentage (e.g., L_GD = 60%) and PD_B is the default probability of the bank.

The LEA is the present value of the expected costs paid by the bank in case it decides to sell the bond, weighted by the probability of selling:

where s^B(u, u + Tl(t)) is the stochastic spread rate applied for the period [u, u + Tl(t)] and P^D(t, u) is the discount factor for the short-rate r_t over the period [t, u].

Over the period [t, t + Tl(t)], we approximate the spread and the price of the bond with its value in t. This is a good numeric approximation, since the liquidation time ranges from one day to a few months, and the integral over du can be performed numerically by using a few integration points. The integral in equation (8.213) is then approximated as

where s^B(t) = s^B(t, t + Tl(t)). The integration over du is

The liquidity adjustment in equation (8.214) is then

Numerical evaluation

To compute the value of the LEA, we consider a set of n dates T_i at which the selling of the bond might occur. We define the liquidation time at T_i with Tl_i = Tl(T_i). The liquidity adjustment in equation (8.216) reads

Taking the expression for the spread as in equation (8.212), the expression for LEA(t₀, T) reads

Defining the function

the expression for the LEA(t₀, T) is

The expression for SB(t₀, T_i, T) can be cast in a closed form (details are given in Appendix 8.A).

8.7.2 Price volatility adjustment

The second component of LA is the adjustment due to the losses the bank can incur during the liquidation period, since the bond price can divert from the level at the time the selling process starts. We define this as Price Volatility Adjustment (PVA). Consider a bond of maturity T. At time t₀, let the expected (future) in t be CH(t₀, t, T); the value of the portfolio can also drop, at time t, to V_p(t₀, t) for a certain confidence level p.

To compute PVA, we cannot apply the procedure in Section 8.7.1 to separate the expected value of the product from the probability of selling and the quantile. In the present case, we need to compute the expectation value in the forward risk-adjusted measure with respect to the short rate r and to the default intensity processes. Let us define PVA as

Similarly to the case of LEA(t₀, T), we assume that the selling of the bond can happen only at specific dates T_i. The expression in equation (8.220) reads

To evaluate PVA(t₀, T) numerically, we first consider the expectation value appearing in equation (8.221). We take this expectation value in the forward measure E^{T_i, T_i}, where the first index refers to the T_i-forward measure with respect to r_t, and the second index refers to the T_i-forward measure with respect to λ. We obtain

where CG(t, s, T) indicates the forward at time t for a bond with expiry s and maturity T, in the measure described above. Explicitly, it is

where is the default intensity of the bond's issuer, which follows the usual composite CIR++ dynamics, and the accrued time is

and is given by

where is the bond-specific parameter to capture liquidity specialness and G is the zero-coupon bond forward price given in equation (8.31). The expression in equation (8.221) reads

To compute the quantile, we refer to the procedure outlined in Section (8.5.5), with the following modification. First, all futures appearing in the formulae are replaced by the corresponding expressions for the forwards in equation (8.223). Second, the mean and the covariance matrix of the short rate and of the default intensities are taken in the forward risk-adjusted measure. Following the result in Appendix 8.A, the mean is

while the covariance matrix is

We defined the integrals

while and are the same as the CIR parameters for the common intensity of the default process λ^C(t) used. Explicit formulae for the integrals are given in Appendix 8.A.

Numerical evaluation of the integral in equation (8.224)

In computing the value of PVA(t₀, T), we cannot approximate the function CG(T_i, u, T) − V_p(T_i, u) appearing in equation (8.224) to its value in T_i, because at the selling time the quantile equals the value of the bond and the expected loss is zero. In order to include the expected loss in the numerical computation of the integral over du, we use the following procedure

• we divide the period [T_i, Tl_i] into m fractions;
• we introduce the time step

  

• we introduce the integration points

  

• At the integration point u = t_k, we substitute

  

The integral over du in equation (8.224) is then approximated by

With this approximation, the expression in equation (8.224) reads

Total liquidity adjustment for the bond is then:

Example 8.7.1. In the following example, we consider a bank with characteristics similar to Unicredit (UCG, see above) wants to value a bond issued by Intesa Sanpaolo (ISP) with the specifics in Table 8.26. In Table 8.27, we show the CIR parameters used to describe the default intensity of UCG and ISP. We assume that the bond can be sold only at specific dates T_i, with constant semiannual intervals T_i − T_i−1 = 6M. In Table 8.29, we show results for the term structure of the probability of selling at each T_i the LEA, the liquidation time in days, the expected loss over [T_i, T_i + Tl_i] and the values of PVA and total liquidity adjustment LA = LEA + PVA, when the parameters in Table 8.28 are used.

Table 8.26. Specifics of the ISP bond for which the liquidity adjustment is computed

Table 8.27. Parameter values of the model for the intensity of default process for UCG and ISP

Table 8.28. Parameters used for the results in Table 8.29

Table 8.29. The term structure of liquid equivalent adjustment (LEA), liquid volatility adjustment (PVA) and total liquidity adjustment LA = LEA + PVA, for the parameters used in Table 8.28

Example 8.7.2. Keeping Example 8.7.1 in mind, we consider the case in which the liquidation time is much shorter when the bond is sold in the first years. Table 8.30 gives the entire set of parameters used. λ^T = 730 indicates that the bond trades in the market on average three times a day, in an amount of . The results are in Table 8.31.

Example 8.7.3. Keeping Example 8.7.1 in mind once again, we consider the case in which the parameter of mean reversion governing the selling probability of the bond is higher. This means that before expiry of the bond, there will be more chances that the bond will be sold by UGC. Table 8.32 gives the parameters and the results are shown in Table 8.33.

Table 8.30. Parameters used for the results in Table 8.31

Table 8.31. The term structure of the liquid equivalent adjustment (LEA), the price volatility adjustment (PVA), and the total liquidity adjustment LA = LEA + PVA, for the parameters used in Table 8.33

Example 8.7.4. Keeping Example 8.7.1 in mind yet again, we show the impact of parameter α, which regulates the impact of the stochastic component of the probability of selling depending on the probability of default of the bank (UCG in this case). The results are shown in Figure 8.17 for the parameters in Table 8.34.

Conditioned selling probability

In this section we will show how the results presented in the previous examples modify when we assume the bank will definitely sell the bond within the maturity T. To consider this case, we introduce the probability of selling conditional to the selling of the bond before expiry. We impose that the probability of not having sold the bond within T is given the value of zero or, equivalently, the probability of selling before T the value of one. To achieve this, we divide the probability of selling presented in equation (8.209) by the probability of selling the bond after T, so that it becomes a conditional probability:

Table 8.32. Parameters used for the results in Table 8.33

Table 8.33. The term structure of liquid equivalent adjustment (LEA), price volatility adjustment (PVA) and total liquidity adjustment LA = LEA + PVA, for the parameters used in Table 8.32

The conditional probability of selling thus defined satisfies

Figure 8.17. LA, LEA and PVA as a function of parameter α (x-axis). The other parameters are given in Table 8.34

Table 8.34. The selling parameters used for Figure 8.17

Example 8.7.5. In this example we keep the results presented in Examples 8.7.1, 8.7.2 and 8.7.3 in mind, where UCG computes the liquidity adjustment on an ISP bond using the specifics in Table 8.26.

When the conditional probability of selling PS′(t₀, t) is used instead of PS(t₀, t), the results obtained in the examples above modify as follows. In Table 8.35, we show the results when the trading and selling parameters are the same as in Table 8.28. In Table 8.36, we show the results when the trading and selling parameters are the same as in Table 8.30, where the initial value of trading intensity is higher than in the first example. Finally, in Table 8.37, we show the results when the trading intensity and the selling parameters are as given in Table 8.32. In Figure 8.17, LA, ELA and PVA are plotted as a function of parameter α.

Table 8.35. The term structure of liquid equivalent adjustment (LEA), price volatility adjustment (PVA) and total liquidity adjustment LA = LEA + PVA, for the parameters used in Table 8.28

Table 8.36. The term structure of liquid equivalent adjustment (LEA), price volatility adjustment (PVA) and total liquidity adjustment LA = LEA + PVA, for the parameters used in Table 8.30

Table 8.37. The term structure of liquid equivalent adjustment (LEA), liquid volatility adjustment (PVA) and total liquidity adjustment LA = LEA + PVA, for the parameters used in Table 8.32

APPENDIX 8.A EXPECTATION VALUE OF THE BOND WITH SELLING PROBABILITY AND SPREAD

In this section, we compute the expectation value in equation (8.219). We write this expression as

In equation (8.233), we explained the expressions for P^D(t₀, T_i) in terms of the stochastic short-rate r(t), and the expressions for PD_B(T_i, T_i + Tl_i) and PNS(t₀, T_i) in terms of default intensities. Moreover, B(T_i, T) is the expression for a defaultable coupon bond, see equation (8.157), before the expectation value is taken,

The quantity in equation (8.234) is the sum of four terms, so that SB(t₀, T_i, T) in equation (8.233) can be split into four terms as well. For example, we show explicitly the computation involving the term

and the part in equation (8.233) containing this term reads

where we set the integral of the deterministic selling function F_i = F(t₀, T_i). The expression in equation (8.235) is the sum of four distinct terms, coming from cross multiplication of the default probability times the probability of selling the asset over the period considered. For example, the first term in the sum in equation (8.235) is

We split processes λ_B and λ_A into an idiosyncratic and a common component and we arrange terms, thus obtaining

We take the expectation value in the t_j-forward measure with respect to r_t and , so that the expression above is

Using the result

we obtain

and

Summing up, the expectation value is

Computing all four expectation values in equation (8.235), and repeating the computation for all four terms in the expression for the coupon bond in equation (8.234) and rearranging, we finally obtain the expression for the bond with the probability of selling and the spread in equation (8.219) as

where

and we define the coupon bond when the selling probability is included as

We now introduce for notational convenience the function

with, if t_j > T_i + Tl_i,

or, if t_j ≤ T_i + Tl_i,

Eventually, LEA is written in the more readable form

The forward risk-adjusted measure

In the T-forward measure, a CIR process reads

Brownian motion in the T-forward measure is

and the mean reversion speed is

with the CIR factor

and .

Mean in the FRA measure

In the following, we indicate the mean value of r_t in the T-forward measure with

Taking the expectation value of equation (8.248) in the T-forward measure, we find that E₁(t) satisfies the differential equation

To solve for E₁(t), we first compute a solution to the homogeneous part of the differential equation,

or

where r₀ is the value at time t = 0 of the interest rate. To find a complete solution to equation (8.250), we use the method of the variation of constants. For this, we write

where the function r₀(t) follows a differential equation which is obtained by plugging equation (8.253) into equation (8.250), giving

with the formal solution

Finally, a solution to the inhomogeneous equation (8.250) is given by the sum of the solution found by the methods of the variation of constants plus the homogeneous solution,

As a check, we impose a constant , obtaining

which is precisely the expression for the average value of r_t in the risk-neutral measure.

For the CIR model, this expression can be cast in a closed-form formula, since

It can be shown that, in the limit σ → 0, the expression above reduces to κ t. We have

and

The complete expression for the mean of the CIR process in the T-forward measure is

Volatility in the FRA measure

Let us try to find the value of the quantity

Setting

we have

Using Ito's lemma, the quantity r(t)² follows the process

where in the last equality we used the expression in equation (8.248) for the CIR process in the T-forward measure. Taking the expectation value on both sides, we obtain

The solution to the homogeneous equation is

The solution to the inhomogeneous equation is the sum

where the function v(t) is fixed by using the method of the variation of constants. We find

with the formal solution

Setting

and

we find that the volatility is

However, the second term is unnecessary, since we find that

and

so 2I₂(t) − I(1)² = 0. The volatility in the T-forward measure is then

When σ → 0, and γ → κ, this equation reduces to the usual expression for the volatility in the risk-neutral measure,