Role of quantitative analytics
Classification of pricing models
The quantitative analytics team (also known as ‘quants’) is the crucial part of the investment bank front office. They are highly educated individuals who typically hold PhDs in maths and physics, whose job is the creation of complex mathematical models used for pricing existing and innovative financial instruments. Their expertise is best utilised in derivatives markets, as they operate with many unknown variables. They tend to specialise in a particular class of products, so the teams are typically sub-divided into interest rate derivatives, equity and FX derivatives, commodity derivatives and credit derivatives. As the interest rate market is the biggest by volume and offers the widest range of derivatives products, the majority of quantitative analytics work is done in this field, which is reflected in the number of quants. Their work is usually complemented by the support of the quant IT team who ‘translate’ the mathematical models into computer software to be incorporated into a wide range of IT systems used by the bank.
There are numerous quantitative models used to price financial instruments. Whilst there is a market convention when it comes to the main concepts and techniques used to price different product classes, most investment banks use proprietary models or bespoke modifications to standard methods to price their products. This chapter offers a classification of the most popular, and publicly known, models used in derivatives pricing. For each model or technique only a brief description, with the pros and cons, is given. This was done deliberately, as these techniques are very complex, each requiring advanced knowledge of calculus and an entire book devoted to them. Hence this chapter aims just to give an overview of various pricing techniques, as a glimpse into the complexity involved in derivatives valuation.
The main division of existing pricing models is into:
The numerical models can be further sub-divided by their use into:
Whilst the analytical models can be:
The main numerical methods currently in use in quantitative analysis are: Monte Carlo simulation, binomial and trinomial trees, finite difference methods and interest rate trees.
Monte Carlo simulation
Trees (binomial, trinomial)
Finite difference methods
Interest rate trees
The main analytical quantitative methods currently in use are: simple pricing equations (money market products, bonds, swaps etc.); vanilla product models (Black–Scholes, Black models); models for exotic products (models of the entire yield curve) – further subdivided into equilibrium models and no-arbitrage models.
Simple pricing equations (money market products, bonds, swaps etc.)
Vanilla product models (Black–Scholes, Black models)
Exotic products (models of the entire yield curve):
Equilibrium models
No-arbitrage model
Monte Carlo simulation generates random samples of possible behaviours of a variable. Many samples are generated and an average is calculated as a most likely outcome. For example, Monte Carlo calculation of the constant π can be done by drawing a circle inside a square and randomly positioning dots inside the whole area of the square. Since the area of a square is well known (A = a2) and the area of the circle inside it is a2 π/4, π can be calculated from the ratio of dots that fall in and out of the circle. Similarly for the calculation of a payoff of a certain financial product that depends on the evolution of the entire yield curve, simulating different paths that the curve can take will provide an average (and most likely) payoff.
The Monte Carlo technique is useful for pricing derivatives where the payoff is dependent on the history of the underlying variable or where there are several underlying variables. It can accommodate complex payoffs and complex stochastic processes. It is numerically very efficient, as the time taken for the simulation increases approximately linearly with the number of variables. Its drawback is that it can be used for European style derivatives only, i.e. no early exercise is allowed.
Binomial and trinomial trees are typically used to price options on an underlying variable (e.g. stock). They rely on the assumption that the stock price movements consist of very small binary steps in very short time intervals. The time to maturity is divided into small intervals and at each interval the stock price can only go up or down in by a predetermined amount. The probability of up and down movements is assumed. Since payoff at maturity is known, it can be calculated at each branch of the tree. Working backwards through the nodes enables the calculation of the option price today.
Trees are useful for pricing products where decisions can be made at each step (e.g. American options) as well as for pricing dividend paying stocks. Options on indices, currencies and futures contracts can also be valued using this technique. Variables (interest rates, volatility) can be varied between the tree nodes. Binomial and trinomial trees can also be used in conjunction with control variate techniques. For example, the European option price can be calculated using the Black–Scholes formula and compared with the price obtained from the binomial tree. Assuming that this is the error that the tree introduces into pricing, the price of an American style option (obtained using the tree) can be adjusted by this amount. This approach can be easily extended to non-recombining trees, trees with more than up/down movements (e.g. trinomial trees that have a mid-path). However, tree building is time-consuming and computationally extensive. Furthermore, its main shortcoming is that it is not suitable for derivatives dependent on past history.
Finite difference methods are similar to trees, whereby the calculation works backwards from the terminal value to the start date. They are used to solve a differential equation that a variable satisfies when the equation does not have an analytical solution or is very difficult to solve. The equation is converted into a set of finite difference equations and those are solved iteratively. For example, a partial differential equation (containing derivatives of any order) in two variables can be represented by a two-dimensional mesh with one variable on the x-axis and another on the y-axis. Each (x, y) point represents one state of the world, with grid boundaries representing the final values. Akin to tree techniques, the value of the derivative is known at expiry (e.g. the value of a call option with strike 20 is 5 when the stock price at expiry is 25), hence it is used as boundary condition. First, second and higher order derivatives can also be represented by differences in grid positions. In this way a single differential equation is represented by a different equation at every grid point. The set of equations is solved to yield the price of a derivative at inception.
Finite difference methods are used for the same types of derivative pricing problems as trees. Hence, they can price both American and European style options, but do not easily handle derivatives with a payoff dependent on historical values. Finite difference methods can also be combined with control variate techniques for improved accuracy. Furthermore, risk parameters (delta, gamma) can be calculated directly from the grid, making this approach very useful in risk evaluation and management. These techniques can be computationally extensive when used for problems involving several state variables.
The principle of interest rate tree construction is the same as for stock price trees (described earlier). Whereas in stock price trees the discounting rate is typically constant between the nodes, in interest rate trees the rate varies from node to node. It is often more convenient to use trinomial trees (rather than binomial ones) as they give more freedom. For example, trinomial trees can model mean reversion.
As with all similar techniques, interest rate trees are used for pricing products where decisions can be made at each step (e.g. American options) as well as for pricing dividend-paying stocks, options on indices, currencies and futures contracts. Variables can be varied between the tree nodes. Binomial and trinomial trees can also be used in conjunction with control variate techniques. As with all tree-based techniques, this approach is computationally extensive. The main advantage is that it can be used to represent many yield curve models. Given the flexibility of the method, it can be used to fit any term structure.
All of the above methods are reliable and tractable. Which one will be used depends mainly on the characteristics of the derivative being valued and the accuracy required. Monte Carlo simulation works forward from the beginning of the contract to maturity. It cannot be used for derivatives requiring knowledge of history. It is more efficient than tree and grid methods.
On the other hand, tree and finite difference methods work from the expiry backwards in order to evaluate the security in question. Computationally very demanding, these methods can accommodate early exercise. Interest rate trees are effectively just a sub-class of a standard tree, but give a flexibility that accommodates changes of discount rates between the nodes. As they can fit any yield curve term structure required, they can implement many of the analytical models.
This class of methods includes pricing products that have known payments, no optionality or other special features. Typically these products are:
These products all have known payment dates (simple deposits and zero coupon bonds typically have only one payment at maturity) and the payment values are either fixed or linked to the yield curve (floating rate). There is no uncertainty over whether the payments will take place, how much will be paid (we are not certain of the yield curve in the future, but current forward rates are the best estimate of the future cashflows). This is why there are no specific models to price these products. The equations are simple streams of future cashflows discounted to today using discount factors calculated from today’s yield curve. They have all been extensively covered in previous chapters.
The above products have some features that require closer attention. For example:
In summary, pricing the products with the known number, timing and often size of the cashflows reduces to calculating their present value in order to price the products at inception. At some future date the value of the position is calculated in the same way, taking into account all the outstanding payments. These products use yield curves (built from the most liquid products in the market) to project and discount the future cashflows. The discounting and projection index can be, but are not required to be, the same.
This class of models includes, for example, the Black–Scholes (used to price stock options) and Black models (used to price interest rate derivatives), both described in Chapter 9. These models are quick and robust but are not flexible enough to accommodate pricing more exotic products.
Much quicker than any of the numerical methods, the B–S model uses a single equation to price call and put options on stocks. The model assumptions are:
The B–S model has been extended (with minor modifications) to cover dividend-paying stocks (known dividend model), American style options (pseudo American model), options on indices, currency options (Garman–Kohlhagen model), options on futures (Black model), American options with only one dividend (Roll–Geske–Whaley model) etc.
As mentioned above, the Black model is an extension of the B–S model, hence the same assumptions apply. In terms of implementation and the computational speed, it is the same as B–S. It uses the assumption of the B–S model that the security will be held until its maturity, thus the evolution of interest rates during the life of the security is irrelevant and all that matters is its value at maturity. Hence rates are treated as if they are tradable securities.
These models are attempting to accommodate different features of the products emerging in OTC markets. Their aim is to model the evolution of the entire yield curve (rather than working with the terminal values of interest rates). They can be broadly classified as: equilibrium models and no-arbitrage models.
Equilibrium models are also called short rate models as they describe behaviours of economic variables in terms of an instantaneous short-term rate r (the rate that prevails from one moment to the next). As this rate is not a tradable quantity it does not describe the real world. The models based on the short rate assume that derivative prices depend only on the process followed by r in a risk-neutral world (where the positions are perfectly hedged to create zero profit/loss regardless of market moves). Once the process for r is fully defined, it is assumed to implicitly define the initial term structure and its future evolution. The main disadvantage of the equilibrium model is that the initial yield curve term structure is the output of the model, rather than the input to it. Hence calibrating the model outputs to the available market data can be an issue.
In one-factor models the process for the instantaneous short-term rate r has only one source of uncertainty. The drift and the standard deviation are taken to be functions of r, but independent of time. One-factor models imply that, over any short time interval, all rates move in the same direction, albeit by potentially different amounts. This feature enables modelling of a very rich pattern of term structures, but does not allow for curve shape inversion (where long-term rates are lower than the short-term ones).
Examples of one-factor models are:
Here the process for the instantaneous short-term rate r is assumed to have two sources of uncertainty. The drift and the standard deviation are now taken to be functions of both r and time. This allows modelling term structures with an even richer pattern than in one-factor models. Given the two-factor approach, one factor can drive the curve level, whilst the second factor governs its ‘tilt’. Thus the points along the curve can move in opposite directions, allowing for curve inversion. Examples of two-factor models are:
The disadvantage of all equilibrium models is that they do not automatically fit the market-driven yield curve term structure. However, careful parameter selection enables this class of models to approximately fit most of the term structures encountered in practice (albeit with significant errors). Thus traders and market practitioners are reluctant to rely on these models when pricing derivatives securities. This has led to the emergence of the no-arbitrage models described in the following section.
No-arbitrage models implicitly fit today’s term structure. They take today’s term structure for granted and use it as an input to define the process of its evolution over time. Different no-arbitrage models use one of the three distinct (but equivalent) approaches:
Models of bond prices and forward rates (the first two approaches) are generally non-Markov* and have to be solved numerically using Monte Carlo simulation or a non-recombining tree. However, the choice of volatilities used in the model is left to the practitioner, with only one condition – the volatility has to approach zero at maturity.
Models of short rates are usually Markov and analytically tractable. However, unlike the approach above, in this case the practitioner does not have complete freedom of choice of volatility. The initial volatility is typically consistent with the modelled value, but the future model-implied volatility might be inconsistent with the market data.
An example of a non-Markov model based on forward rates is Heath-Jarrow and Morton. Its main characteristics are:
Most Markov models are developed in terms of short rate r, assuming the drift to be a function of time (thus these models are typically consistent with the initial term structure). This makes them a logical extension of the equilibrium models described earlier. The Markov property also makes it possible to use recombining trees.
As these models are explicit functions of short rate r (not observable in practice), they tend to be consistent with only the short end, rather than the entire yield curve.
Some examples of Markov models are:
In summary, short rate models are simple, easy to implement and fast to run. However, neither the short rate nor mean reversion and short rate volatility are directly observable in the market. Thus the models suffer from a general lack of transparency and a difficulty in incorporating market changes.
The BGM (Brace–Gaterek–Musiela) model is an arbitrage-free model. It allows the arbitrage-free evolution of discrete points of a yield curve described either in terms of forward rates or swap rates. Unlike other models described earlier, it does not rely on the dynamics of unobservable or pseudo-observable quantities (e.g. instantaneous short rate, instantaneous forward rates, variance of the short rate etc.). Instead, it directly models market-observable variables, such as Libor forward and swap rates and their volatilities. Due to its direct link to market parameters, it is often called a ‘market model’. It implicitly allows for straightforward calibration to market rates/prices, whilst the forward and swap rate volatilities can be directly derived from the model, as they are directly used for its creation.
For the derivation of no-arbitrage conditions, BGM approach requires the following:
The above was a brief summary of methods and models used by quantitative analytics to price a wide range of derivatives, with the emphasis on interest rate derivatives. Modelling interest rates is equally important for other classes of securities; as funding, cashflow projections and a host of other parameters are ultimately affected by the yield curve evolution. However, pricing credit and commodity derivatives is typically done using product-specific models. Due to the range and complexity of these derivatives, their pricing models are outside the scope of this book.
* Markov process is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future. This implies that all the history is contained in the present state and is irrelevant for the future evolution of the process.
* These are non-stationary models (several parameters are functions of time).
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