Stationary Distribution

In certain cases, the solution of an SDE admits a stationary distribution, but in general, this distribution is not available in explicit form. Wong (1964) showed that, for model ( 11.1) in the univariate case ( p = d = 1) with σ(·) ≥ 0, if there exists a function f that solves the equation

11.4

and belongs to the Pearson family of distributions, that is, of the form

11.5

where a <  − 1 and b < 0, then Eq. ( 11.1) admits a stationary solution with density f .

Euler Discretisation of a Diffusion

Diffusion processes do not admit an exact discretisation in general. An exception is the geometric Brownian motion, defined as a solution of the real SDE

11.12

where μ and σ are constants. It can be shown that if the initial value x ₀ is strictly positive, then X _t  ∈ (0, ∞) for any t > 0. By Itô's lemma, ² we obtain

11.13

and then, by integration of this equation between times kτ and (k + 1)τ , we get the discretised version of model (11.12),

11.14

For general diffusions, an explicit discretised model does not exist, but a natural approximation, called the Euler discretisation, is obtained by replacing the differential elements by increments. The Euler discretisation of the SDE ( 11.1) is then given, for the time unit τ , by

11.15

The Euler discretisation of a diffusion converges in distribution to this diffusion (Exercise 11.2).

Convergence of GARCH‐M Processes to Diffusions

It is natural to assume that the return of a financial asset increases with risk. Economic agents who are risk‐averse must receive compensation when they own risky assets. ARMA‐GARCH type time series models do not take this requirement into account because the conditional mean and variance are modelled separately. A simple way to model the dependence between the average return and risk is to specify the conditional mean of the returns in the form

where ξ and λ are parameters. By doing so we obtain, when is specified as an ARCH, a particular case of the ARCH in mean (ARCH‐M) model, introduced by Engle, Lilien, and Robins (1987). The parameter λ can be interpreted as the price of risk and can thus be assumed to be positive. Other specifications of the conditional mean are obviously possible. In this section, we focus on a GARCH(1, 1)‐M model of the form

11.16

where ω > 0, f is a continuous function from ℝ⁺ to ℝ, g is a continuous one‐to‐one map from ℝ⁺ to itself and a is a positive function. The previous interpretation implies that f is increasing, but at this point, it is not necessary to make this assumption. When g(x) = x ² and a(x) = αx ² + β with α ≥ 0, β ≥ 0, we get the classical GARCH(1, 1) model. Asymmetric effects can be introduced, for instance by taking g(x) = x and a(x) = α ₊ x ⁺ − α ₋ x ⁻ + β , with x ⁺ = max(x, 0), x ⁻ = min(x, 0), α ₊ ≥ 0, α ₋ ≥ 0, β ≥ 0.

Observe that the constraint for the existence of a strictly stationary and non‐anticipative solution (Y _t ), with Y _t  = X _t  − X _t − 1 , is written as

by the techniques studied in Chapter 2.

Now, in view of the Euler discretisation (11.15), we introduce the sequence of models indexed by the time unit τ , defined by

11.17

for k > 0, with initial values , and assuming The introduction of a delay k in the second equation is due to the fact that σ _{(k + 1)τ} belongs to the σ ‐field generated by η _kτ and its past values.

Noting that the pair Z _kτ  = (X _{(k − 1)τ} , g(σ _kτ )) defines a Markov chain, we obtain its limiting distribution by application of Theorem 11.1. We have, for z = (x, g(σ)),

11.18

This latter quantity converges if

11.20

where ω and δ are constants.

Similarly,

11.21

which converges if and only if

11.23

where ζ is a positive constant. Finally,

11.24

converges if and only if

11.25

where ρ is a constant such that ρ ² ≤ ζ .

Under these conditions, we thus have

Moreover, we have

We are now in a position to state our next result.

Stochastic Discount Factor

We start by considering a general setting. Suppose that we observe a vector process Z = (Z _t ) and let I _t denote the information available at time t , that is, the σ ‐field generated by {Z _s , s ≤ t}. We are interested in the pricing of a derivative whose payoff is g = g(Z _T ) at time T . Suppose that there exists, at time t < T , a price C _t (Z, g, T) for this asset. It can be shown that, under mild assumptions on the function g ↦ C _t (Z, g, T), ⁶ we have the representation

11.37

The variable M _{t, T} is called the SDF for the period [t, T]. The SDF introduced in representation (11.37) is not unique and can be parameterised. The formula applies, in particular, to the zero‐coupon bond of expiry date T , defined as the asset with payoff 1 at time T . Its price at t is the conditional expectation of the SDF,

It follows that Relation ( 11.37) can be written as

11.38

Forward Risk‐Neutral Probability

Observe that the ratio M _{t, T} /B(t, T) is positive and that its mean, conditional on I _t , is 1. Consequently, a probability change removing this factor in formula (11.38) can be done. ⁷ Denoting by π _{t, T} , the new probability and by the expectation under this probability, we obtain the pricing formula

11.39

The probability law π _{t, T} is called forward risk‐neutral probability. Note the analogy between this formula and Relation ( 11.34), the latter corresponding to a particular form of B(t, T). To make this formula operational, it remains to specify the SDF.

Risk‐Neutral Probability

As mentioned earlier, the SDF is not unique in incomplete markets. A natural restriction, referred to as a temporal coherence restriction, is given by

11.40

On the other hand, the one‐step SDFs are constrained by

11.41

where S _t  ∈ I _t is the price of an underlying asset (or a vector of assets). We have

11.42

Noting that

we can make a change of probability such that the SDF vanishes. Under a probability law , called risk‐neutral probability, we thus have

11.43

The risk‐neutral probability satisfies the temporal coherence property: is related to through the factor M _{T, T + 1}/B(T, T + 1). Without the restriction (11.40), the risk‐neutral forward probability does not satisfy this property.

Pricing Formulas

One approach to deriving pricing formulas is to specify, parametrically, the dynamics of Z _t and of M _{t, t + 1} , taking the constraint (11.41) into account.

Now consider a general GARCH‐type model of the form

11.44

where μ _t and σ _t belong to the σ ‐field generated by the past of Z _t , with σ _t  > 0. Suppose that the σ ‐fields generated by the past of ε _t , Z _t , and η _t are the same, and denote this σ ‐field by I _t − 1 . Suppose again that B(t, t + 1) = e ^−r . Consider for the SDF an affine exponential specification with random coefficients, given by

11.45

where a _t , b _t  ∈ I _t . The constraints ( 11.41) are written as

that is, after simplification,

11.46

As before, these equations provide a unique solution (a _t , b _t ). The risk‐neutral probability π _{t, t + 1} is defined through the characteristic function

The last two equalities are obtained by taking into account the constraints on a _t and b _t . Thus, under the probability π _{t, t + 1} , the law of the process (Z _t ) is given by the model

11.47

The independence of the follows from the independence between and I _t (because has a fixed distribution conditional on I _t ) and from the fact that

The model under the risk‐neutral probability is then a GARCH‐type model if the variable is a measurable function of the past of . This generally does not hold because the relation

11.48

entails that the past of is included in the past of ε _t , but not the reverse.

If the relation (11.48) is invertible, in the sense that there exists a measurable function f such that , model (11.47) is of the GARCH type, but the volatility can take a very complicated form as a function of the . Specifically, if the volatility under the historic probability is that of a classical GARCH(1, 1), we have

Finally, using π _{t, T} , the forward risk‐neutral probability for the expiry date T , the price of a derivative is given by the formula

11.49

or, under the historic probability, in view of the temporal coherence restriction ( 11.40),

11.50

It is important to note that, with the affine exponential specification of the SDF, the volatilities coincide with the two probability measures. This will not be the case for other SDFs (Exercise 11.11).

Numerical Pricing of Option Prices

Explicit computation of the expectation involved in Formula (11.49) is not possible, but the expectation can be evaluated by simulation. Note that, under π _{t, T} , S _T , and S _t are linked by the formula

where . At time t , suppose that an estimate of the coefficients of model (11.51) is available, obtained from observations S ₁, …, S _t of the underlying asset. Simulated values of S _T , and thus simulated values of Z _T , for i = 1, …, N , are obtained by simulating, at step i , T − t independent realisations of the and by setting

where the , are recursively computed from

taking, for instance, as initial value , the volatility estimated from the initial GARCH model (this volatility being computed recursively, and the effect of the initialisation being negligible for t large enough under stationarity assumptions). This choice can be justified by noting that for SDFs of the form (11.45), the volatilities coincide under the historic and risk‐neutral probabilities. Finally, a simulation of the derivative price is obtained by taking

The previous approach can obviously be extended to more general GARCH models, with larger orders and/or different volatility specifications. It can also be extended to other SDFs.

Empirical studies show that, comparing the computed prices with actual prices observed on the market, GARCH option pricing provides much better results than the classical Black–Scholes approach (see, for instance, Sabbatini and Linton 1998; Härdle and Hafner 2000).

To conclude this section, we observe that the formula providing the theoretical option prices can also be used to estimate the parameters of the underlying process, using observed options (see, for instance, Hsieh and Ritchken 2005).

Definition

VaR is concerned with the possible loss of a portfolio in a given time horizon. A natural risk measure is the maximum possible loss. However, in most models, the support of the loss distribution is unbounded so that the maximum loss is infinite. The concept of VaR replaces the maximum loss by a maximum loss which is not exceeded with a given (high) probability.

VaR should be computed using the predictive distribution of future losses, that is the conditional distribution of future losses using the current information. However, for horizons h > 1, this conditional distribution may be hard to obtain.

To be more specific, consider a portfolio whose value at time t is a random variable denoted V _t . At horizon h , the loss is denoted

The distribution of L _{t, t + h} is called the loss distribution (conditional or not). This distribution is used to compute the regulatory capital which allows certain risks to be covered, but not all of them. In general, V _t is specified as a function of d unobservable risk factors.

The distribution of −L _t,t + h conditional on the available information at time t is called the profit and loss (P&L) distribution.

The determination of reserves depends on

• the portfolio,
• the available information at time t and the horizon h , ⁸
• a level α ∈ (0, 1) characterising the acceptable risk. ⁹

Denote by R _{t, h} (α) the level of the reserves. Including these reserves, which are not subject to remuneration, the value of the portfolio at time t + h becomes V _t + h  + R _{t, h} (α). The capital used to support risk, the VaR, also includes the current portfolio value,

and satisfies

where P _t is the probability conditional on the information available at time t . ¹⁰ VaR can thus be interpreted as the capital exposed to risk in the event of bankruptcy. Equivalently,

11.53

In probabilistic terms, VaR _{t, h} (α) is thus simply the (1 − α)‐quantile of the conditional loss distribution. If, for instance, for a confidence level 99% and a horizon of 10 days, the VaR of a portfolio is €5000, this means that, if the composition of the portfolio does not change, there is a probability of 1% that the potential loss over 10 days will be larger than €5000.

In particular, it is obvious that VaR _{t, h} (α) is a decreasing function of α .

From Definition (11.53), computing a VaR simply reduces to determining a quantile of the conditional loss distribution. Figure 11.1 compares the VaR of three distributions, with the same variance but different tail thicknesses. The thickest tail, proportional to 1/x ⁴ , is that of the Student t distribution with 3 degrees of freedom, here denoted as ; the thinnest tail, proportional to , is that of the Gaussian ; and the double exponential ℰ possesses a tail of intermediate size, proportional to . For some very small level α , the VaRs are ranked in the order suggested by the thickness of the tails: . However, Figure  11.1b shows that this ranking does not hold for the standard levels α = 1% or  5%.

VaR and Conditional Moments

Let us introduce the first two moments of L _{t, t + h} conditional on the information available at time t :

Suppose that

11.54

where is a random variable with cumulative distribution function F _h . Denote by the quantile function of the variable , defined as the generalised inverse of F _h :

If F _h is continuous and strictly increasing, we simply have , where is the ordinary inverse of F _h . In view of the relations ( 11.53) and (11.54), it follows that

Consequently,

11.55

VaR can thus be decomposed into an ‘expected loss’ m _{t, t + h} , the conditional mean of the loss, and an ‘unexpected loss’ σ _{t, t + h} F ^←(1 − α), also called economic capital.

The apparent simplicity of formula (11.55) masks difficulties in (i) deriving the first conditional moments for a given model, and (ii) determining the law F _h , supposedly independent of t , of the standardised loss at horizon h .

Consider the price of a portfolio, defined as a combination of the prices of d assets, p _t  = a ^′ P _t , where a, P _t  ∈ ℝ ^d . Introducing the price variations ΔP _t  = P _t  − P _t − 1 , we have

The term structure of the VaR, that is its evolution as a function of the horizon, can be analysed in different cases.

It is often more convenient to work with the log‐returns r _t  = Δ₁ log p _t , assumed to be stationary, than with the price variations. Letting q _t (h, α) be the α ‐quantile of the conditional distribution of the future returns r _t + 1 + … + r _t + h , we obtain (Exercise 11.15)

11.60

Non Subadditivity of VaR

VaR is often criticised for not satisfying, for any distribution of the price variations, the ‘subadditivity’ property. Subadditivity means that the VaR for two portfolios after they have been merged should be no greater than the sum of their VaRs before they were merged. However, this property does not hold: if L ₁ and L ₂ are two loss variables, we do not necessarily have, in obvious notation,

11.61

The lack of subadditivity of the VaR is often interpreted negatively as this does not encourage diversification. Note however that, as underlined by the previous example, the distribution of an average of non integrable iid variables can be more dispersed than the common distribution of the individual variables. In such an extreme situation, diversification does not reduce risk, and any good risk measure should be overadditive, as it is the case for the VaR.

Volatility and Moments

In the Markowitz (1952) portfolio theory, the variance is used as a risk measure. It might then seem natural, in a dynamic framework, to use the volatility as a risk measure. However, volatility does not take into account the signs of the differences from the conditional mean. More importantly, this measure does not satisfy some ‘coherency’ properties, as will be seen later (translation invariance, subadditivity).

Expected Shortfall

The expected shortfall (ES), or anticipated loss, is the standard risk measure used in insurance since Solvency II. This risk measure is closely related to VaR, but avoids certain of its conceptual difficulties. It is more sensitive than VaR to the shape of the conditional loss distribution in the tail of the distribution. In contrast to VaR, it is informative about the expected loss when a big loss occurs.

Let L _{t, t + h} be such that . In this section, the conditional distribution of L _{t, t + h} is assumed to have a continuous and strictly increasing cdf. The ES at level α , also referred to as Tailvar, is defined as the conditional expectation of the loss given that the loss exceeds the VaR:

11.62

We have

Now P _t [L _{t, t + h}  > VaR _{t, h} (α)] = 1 − P _t [L _{t, t + h}  ≤ VaR _{t, h} (α)] = 1 − (1 − α) = α , where the last but one equality follows from the continuity of the cdf at VaR _{t, h} (α). Thus

11.63

The following characterisation also holds (Exercise 11.16):

11.64

ES thus can be interpreted, for a given level α , as the mean of the VaR over all levels u ≤ α. Obviously, ES _{t, h} (α) ≥ VaR _{t, h} (α).

Note that the integral representation makes ES _{t, h} (α) a continuous function of α , whatever the distribution (continuous or not) of the loss variable. VaR does not satisfy this property (for loss variables which have a zero mass over certain intervals).

More generally, we have under assumption ( 11.54), in view of Expressions (11.64) and ( 11.55),

11.65

Distortion Risk Measures

Continue to assume that the cdf F of the loss distribution is continuous and strictly increasing. For notational simplicity, we omit the indices t and h . From Definition ( 11.64), the ES is written as

where the term can be interpreted as the density of the uniform distribution over [0, α]. More generally, a distortion risk measure (DRM) is defined as a number

where G is a cdf on [0,1], called distortion function, and F is the loss distribution. The introduction of a probability distribution on the confidence levels is often interpreted in terms of optimism or pessimism. If G admits a density g which is increasing on [0,1], that is, if G is convex, the weight of the quantile F ⁻¹(1 − u) increases with u : large risks receive small weights with this choice of G . Conversely, if g decreases, those large risks receive the bigger weights.

VaR at level α is a DRM, obtained by taking for G the Dirac mass at α . As we have seen, the ES corresponds to the constant density g on [0, α]: it is simply an average over all levels below α.

A family of DRMs is obtained by parameterising the distortion measure as

where the parameter p reflects the confidence level, that is the degree of optimism in the face of risk.

Coherent Risk Measures

In response to criticisms of VaR, several notions of coherent risk measures have been introduced. One of the proposed definitions is the following.

This definition has the following immediate consequences:

1. ρ(0) = 0, using the homogeneity property with L = 0. More generally, ρ(c) = c for all constants c (if a loss of amount c is certain, a cash amount c should be added to the portfolio).
2. If L ≥ 0, then ρ(L) ≥ 0. If a loss is certain, an amount of capital must be added to the position.
3. ρ(L − ρ(L)) = 0, that is the deterministic amount ρ(L) cancels the risk of L .

These requirements, which are controversial (see Example 11.10), are not satisfied for most risk measures used in finance. The variance, or more generally any risk measure based on the centred moments of the loss distribution, does not satisfy the monotonicity property, for instance. The expectation can be seen as a coherent, but uninteresting, risk measure. VaR satisfies all conditions except subadditivity: we have seen that this property holds for (dependent or independent) Gaussian variables, but not for general variables. ES is a coherent risk measure in the sense of Definition 11.2 (Exercise 11.17). It can be shown (see Wang and Dhaene 1998) that DRMs with G concave satisfy the subadditivity requirement. Note that, because finite expectation is required, ES and DRMs do not apply to fat tailed distributions such as the Pareto distributions of Example 11.10, for which overaddidivity is desirable.

Unconditional VaR

The simplest estimation method is based on the K last returns at horizon h , that is, r _{t + h − i} (h) =log(p _{t + h − i} /p _t − i ), for i = h…, h + K − 1. These K returns are viewed as scenarios for future returns. The non‐parametric historical VaR is simply obtained by replacing q _t (h, α) in Relation (11.60) by the empirical α ‐quantile of the last K returns. Typical values are K = 250 and α = 1%, which means that the third worst return is used as the empirical quantile. A parametric version is obtained by fitting a particular distribution to the returns, for example a Gaussian which amounts to replacing q _t (h, α) by , where and are the estimated mean and standard deviation. Apart from the (somewhat unrealistic) case where the returns are iid, these methods have little theoretical justification.

RiskMetrics Model

A popular estimation method for the conditional VaR relies on the RiskMetrics model. This model is defined by the equations

11.66

where λ ∈ ]0, 1[ is a smoothing parameter, for which, according to RiskMetrics (Longerstaey 1996), a reasonable choice is λ = 0.94 for daily series. Thus, is simply the prediction of obtained by simple exponential smoothing. This model can also be viewed as an IGARCH(1, 1) without intercept. It is worth noting that no non‐degenerate solution (r _t ) _{t ∈ ℤ} to Model (11.66) exists (Exercise 11.18). Thus Model (11.66) is not a realistic data generating process for any usual financial series. This model can, however, be used as a simple tool for VaR computation. From Relation ( 11.60), we get

Let Ω _t denote the information generated by ε _t , ε _t − 1, …, ε₁ . Choosing an arbitrary initial value to , we obtain and

for i ≥ 2. It follows that . Note however that the conditional distribution of r _t + 1 + ⋯ + r _t + h is not exactly (Exercise 11.19). Many practitioners, however, systematically use the erroneous formula

11.67

GARCH‐Based Estimation

Of course, one can use more sophisticated GARCH‐type models, rather than the degenerate version of RiskMetrics. To estimate VaR _t (1, α), it suffices to use Relation ( 11.60) and to estimate q _t (1, α) by , where is the conditional variance estimated by a GARCH‐type model (for instance, an EGARCH or TGARCH to account for the leverage effect; see Chapter 4), and is an estimate of the distribution of the normalised residuals. It is, however, important to note that, even for a simple Gaussian GARCH(1, 1), there is no explicit available formula for computing q _t (h, α) when h > 1. Apart from the case h = 1, simulations are required to evaluate this quantile (but, as can be seen from Exercise 11.19, this should also be the case with the RiskMetrics method). The following procedure may then be suggested:

1. Fit a model, for instance a GARCH(1, 1), on the observed returns r _t  = ε _t , t = 1, …, n , and deduce the estimated volatility for t = 1, …, n + 1.
2. Simulate a large number N of scenarios for ε _n + 1, …, ε _n + h by iterating, independently for i = 1, …, N , the following three steps:
   • (b1) simulate the values iid with the distribution ;
   • (b2) set and ;
   • (b3) for k = 2, …, h , set and .
3. Determine the empirical quantile of simulations , i = 1, …, N .

The distribution can be obtained parametrically or non‐parametrically. A simple non‐parametric method involves taking for the empirical distribution of the standardised residuals , which amounts to taking, in step (b1), a bootstrap sample of the standardised residuals.

Assessment of the Estimated VaR (Backtesting)

The Basel accords allow financial institutions to develop their own internal procedures to evaluate their techniques for risk measurement. The term ‘backtesting’ refers to procedures comparing, on a test (out‐of‐sample) period, the observed violations of the VaR (or any other risk measure), the latter being computed from a model estimated on an earlier period (in‐sample).

To fix ideas, define the variables corresponding to the violations of VaR (‘hit variables’)

Ideally, we should have

that is, a correct proportion of effective losses which violate the estimated VaRs, with a minimal average cost.

Numerical Illustration

Consider a portfolio constituted solely by the CAC 40 index, over the period from 1 March 1990 to 23 April 2007. We use the first 2202 daily returns, corresponding to the period from 2 March 1990 to 30 December 1998, to estimate the volatility using different methods. To fix ideas, suppose that on 30 December 1998, the value of the portfolio was, in French Francs, the equivalent of €1 million. For the second period, from 4 January 1999 to 23 April 2007 (2120 values), we estimated VaR at horizon h = 1 and level α = 1% using four methods.

The first method (historical) is based on the empirical quantiles of the last 250 returns. The second method is RiskMetrics. The initial value for was chosen equal to the average of the squared last 250 returns of the period from 2 March 1990 to 30 December 1998, and we took λ = 0.94. The third method (GARCH‐ ) relies on a GARCH(1, 1) model with Gaussian innovations. With this method we set , the 1% quantile of the distribution. The last method (GARCH‐NP) estimates volatility using a GARCH(1, 1) model, and approximates by the empirical 1% quantile of the standardised residuals. For the last two methods, we estimated a GARCH(1, 1) on the first period, and kept this GARCH model for all VaR estimations of the second period. The estimated VaR and the effective losses were compared for the 2120 data of the second period.

Table 11.1 and Figure 11.2 do not allow us to draw definitive conclusions, but the historical method appears to be outperformed by the NP‐GARCH method. On this example, the only method which adequately controls the level 1% is the NP‐GARCH, which is not surprising since the empirical distribution of the standardised residuals is very far from Gaussian.

	Historic	Risk metrics	GARCH‐	GARCH‐NP
Average estimated VaR (€)	38 323	32 235	31 950	35 059
Number of losses > VaR	29	37	37	21

On the 2120 values, the VaR at the 1% level should only be violated 2120 × 1%  = 21.2 times on average.

11.1 (Linear SDE)

Consider the linear SDE ( 11.6). Letting denote the solution obtained for ω = 0, what is the equation satisfied by ?

Hint: the following result, which is a consequence of the multidimensional Itô formula, can be used. If is a two‐dimensional process such that, for a real Brownian motion (W _t ),

under standard assumptions, then

Deduce the solution of Eq. ( 11.6) and verify that if ω ≥ 0, x ₀ ≥ 0 and (ω, x ₀) ≠ (0, 0), then this solution will remain strictly positive.