Derivative Security Pricing

This chapter outlines the paradigm problem of option pricing and motivates key concepts and techniques that we will develop in Part I when the risk-free rate is deterministic.

This chapter gives an intuitive appreciation and review of many important aspects of the stochastic processes that have been used to model asset price processes. We will be interested in a probabilistic description of the time evolution of asset prices. After imposing some structure on the stochastic process for the return on the asset, this chapter introduces Markov processes, time evolution of conditional probabilities, continuous sample paths, and the Fokker–Planck and Kolmogorov equations.

This chapter uses the concepts developed in Chap. 2 to illustrate the problem of option pricing as a discounted expected option payoff. By assuming that investors are risk neutral and using the Kolmogorov equation for the conditional probability, we demonstrate how the Black–Scholes option formula can be arrived. We also illustrate how the option price can be viewed in a quite natural way as a martingale and the Feynman–Kac formula, two very important concepts of continuous time finance.

To develop the hedging argument of Black and Scholes, this chapter introduces stochastic differential equations to model the evolution of the price path itself and the statistical properties of small price changes over small changes in time. We then consider the stochastic differential equations for the Wiener process, Ornstein–Uhlenbeck process and Poisson process and examine the autocovariance behaviour of the Wiener process. Furthermore we introduce stochastic integrals to define the stochastic differential equations.

Many of the calculations of derivative security pricing involve formal manipulations of stochastic differential equations and stochastic integrals. This chapter derives those that are most frequently used. We also consider transformation of correlated Wiener processes to uncorrelated Wiener processes for higher dimensional stochastic differential equations.

This chapter introduces Ito’s lemma, which is one of the most important tools of stochastic analysis in finance. It relates the change in the price of the derivative security to the change in the price of the underlying asset. Applications of Ito’s lemma to geometric Brownian motion asset price process, the Ornstein–Uhlenbeck process, and Brownian bridge process are discussed in detail. Extension and applications of Ito’s lemma in several variables are also included.

This chapter develops a continuous hedging argument for derivative security pricing. Following fairly closely the original Black and Scholes (1973) article, we make use of Ito’s lemma to derive the expression for the option value and exploit the idea of creating a hedged position by going long in one security, say the stock, and short in the other security, the option. Alternative hedging portfolios based on Merton’s approach and self financing strategy approach are also introduced.

The martingale approach is widely used in the literature on contingent claim analysis. Following the definition of a martingale process, we give some examples, including the Wiener process, stochastic integral, and exponential martingale. We then present the Girsanov’s theorem on a change of measure. As an application, we derive the Black–Scholes formula under risk neutral measure. A brief discussion on the pricing kernel representation and the Feynman–Kac formula is also included.

The Partial Differential Equation (PDE) Approach is one of the techniques in solving the pricing equations for financial instruments. The solution technique of the PDE approach is the Fourier transform, which reduces the problem of solving the PDE to one of solving an ordinary differential equation (ODE). The Fourier transform provides quite a general framework for solving the PDEs of financial instruments when the underlying asset follows a jump-diffusion process and also when we deal with American options. This chapter illustrates that in the case of geometric Brownian motion, the ODE determining the transform can be solved explicitly. It shows how the PDE approach is related to pricing derivatives in terms of integration and expectations under the risk-neutral measure.

This chapter extends the hedging argument developed in Chap. 7 and the martingale approach developed in Chap. 8 by allowing derivative securities to depend on multiple sources of risks and multiple underlying factors, some are tradable and some are not tradable. It provides a general PDE and martingale approaches to pricing derivative securities.

This chapter applies the general pricing framework developed in Chap. 10 to some standard one factor examples including stock options, currency options, futures options and a two factor model of exchange option.

This chapter considers jump-diffusion processes to allow for price fluctuations to have two components, one consisting of the usual increments of a Wiener process, the second allows for “large” jumps from time-to-time. We introduce Poisson jump process with either absolute or proportional jump sizes through the stochastic integrals and provide solutions when both the stock price and Poisson jump size are log-normal. We also extend Ito’s lemma for the jump-diffusion processes.

This chapter extends the hedging argument of option pricing developed for continuous diffusion processes previously to the situations when the underlying asset price is driven by the jump-diffusion stochastic differential equations. By constructing hedging portfolios and employing the capital asset pricing model, we provide an option pricing integro-partial differential equations and a general solution. We also examine alternative ways to construct the hedging portfolio and to price option when the jump sizes are fixed.

In this chapter we consider the solution of the integro-partial differential equation that determines derivative security prices when the underlying asset price is driven by a jump-diffusion process. We take the analysis as far as we can for the case of a European option with a general pay-off and the jump-size distribution is left unspecified. We obtain specific results in the case of a European call option and when the jump size distribution is log-normal. We illustrate two approaches to the problem. The first is the Fourier transform technique that we have used in the case that the underlying asset follows a diffusion process. The second is the direct approach using the expectation operator expression that follows from the martingale representation. We also show how these two approaches are connected.

Empirical studies show that the volatility of asset returns are not constant and the returns are more peaked around the mean and have fatter tails than implied by the normal distribution. These empirical observations have led to models in which the volatility of returns follows a diffusion process. In this chapter, we introduce some stochastic volatility models and consider option prices under stochastic volatility. In particular, we consider the solutions of the option pricing when volatility follows a mean-reverting diffusion process. We also introduce the Heston model, one of the most popular stochastic volatility models.

To understand the problems and techniques of pricing the American feature of an option, this chapter introduces the American option pricing problem from the conventional approach based on compound options and the free boundary value problem which can be solved by using either the Fourier transform technique or a simple approximation procedure. The framework developed is readily extended to other option pricing problems.

This chapter presents the binomial tree approach to the option pricing problem. We first illustrate the basic ideas of option pricing by considering the one-period binomial tree model and then extend to a multi-period binomial tree model. We then show that, by taking limits in an appropriate way, the binomial expression for the option price converges to the Black–Scholes option price and pricing equation. Alternatively, the continuous time model can be discretised in a way that yields the same expressions as obtained by the binomial tree approach.

It is commonly observed across many underlying assets that the implied volatility of the Black Scholes model varies across exercise price and time-to-maturity and has a pattern known as the volatility smile. In this chapter, we first address the volatility smile using the stochastic volatility models which may underestimate the size of the smile. We then develop an approach to calibrate the smile by choosing the volatility function as a deterministic function of the underlying asset price and time so as to fit the model option price to the observed volatility smile.

The discussion in Chaps. 12 and 15 considered a relaxation of one of the key assumptions of the Black–Scholes framework, namely that the asset price changes follow a geometric Brownian motion. Another crucial assumption is the assumption of a constant interest rate over the life of the option. In this chapter we consider the specific case of stock options and retain all the assumptions of the original Black–Scholes model, except that we now allow interest rates to vary stochastically. Along the lines of Merton (Bell J Econ Manag Sci 4:141–183, 1973b), we develop the appropriate hedging argument to derive the stock option pricing partial differential equation and provide the technical details of its solution.

Many computational applications of derivative pricing models such as the determination of derivative prices by simulation or the estimation of derivative pricing models can be significantly simplified by a change of numeraire. In this chapter we discuss the main idea behind the change of numeraire technique and the formation of equivalent probability measures under which options can be priced. In addition, the connection of the associated numeraires via the Radon–Nikodym derivative are presented. We also consider an application of the technique for the option pricing models with stochastic interest rate discussed in Chap. 19 and an extension of the technique to accommodate multiple sources of risk in the dynamics of the underlying assets is also considered.

There are a number of instruments in interest rate markets that are equivalent to an option on an interest rate or an option on a bond. In this chapter we focus on the interest rate caps, which are call options on an interest rate. We show that they can be interpreted as a put option on a bond. The problem of pricing such bonds, and hence the interest rate cap, shall motivate much of the discussion in subsequent chapters. In the last section we briefly discuss the issues associated with the interest rate option problem that distinguish it from the option pricing problem in a world of deterministic interest rates.

In this chapter, we establish the fundamental relationships between interest rates, bond prices and forward rates. We further discuss the modelling of interest rates and analyse typical models for the spot interest rate and the forward rates. As we desire interest rates to be non-negative, we seek stochastic processes with this feature such as the Feller process. Thus we present the motivation of the Feller process and its relevance to the interest rate modelling. We also summarise the main results of Fubini’s theorem, that are very useful for modelling forward rates.

In this chapter we survey models of interest rate derivatives which take the instantaneous spot interest rate as the underlying factor. The continuous hedging argument is extended so as to model the term structure of interest rates and other interest rate derivative securities. This basic approach is due to Vasicek (J Financ Econ 5:177–188, 1977) and hence we shall often refer to it as the Vasicek approach. By specifying different functional forms for the drift, the diffusion and the market price of risk, we develop three well known spot rate models, namely the Vasicek model, the Hull–White model and the Cox–Ingersoll–Ross model. Then we present a general framework for pricing bond options and we apply this framework to obtain closed form solutions for bond options under the specifications of the Hull–White and the Cox–Ingersoll–Ross model. Finally we discuss the calibration of the Hull–White model to the currently observed yield curve.

In this chapter we develop a framework for term structure modelling that allows factors other than the instantaneous spot rate itself to influence the evolution of the term structure of interest rates. The framework allows for multi-factor generalisations of the Hull–White model as well as of the CIR model. First we present a two-factor extension of the Hull–White model. Then we develop a general multi-factor term structure model and the corresponding bond option pricing model. Finally as a specific application, we consider the so called Duffie–Kan affine class of term structure models, which is widely applied in practice.

Interest rate modelling can also be performed by starting from the dynamics of the instantaneous forward rate. As we shall see the dynamics of all other quantities of interest can then be derived from it. This approach has its origin in Ho and Lee (J Finance XLI:1011–1029, 1986) but was most clearly articulated in Heath et al. (Econometrica 60(1):77–105, 1992a), to which we shall subsequently refer as Heath–Jarrow–Morton. In this framework, the condition of no riskless arbitrage results in the drift coefficient of the forward rate dynamics being expressed in terms of the forward rate volatility function. The major weakness in implementing the Heath–Jarrow–Morton approach is that the spot rate dynamics are usually path dependent (non-Markovian). We consider a class of functional forms of the forward rate volatility that allow the model to be reduced to a finite dimensional Markovian system of stochastic differential equations. This class contains some important models considered in the literature.

The modifications to the Heath-Jarrow-Morton framework to cater for market quoted rates such as LIBOR rates were carried out by Brace and Musiela (Math Finance 4(3):259–283, 1994) (henceforth BM). In this chapter, we first describe the BM parameterisation of the Heath–Jarrow–Morton model, and then we outline the choice of volatility functions that produces lognormal dynamics for LIBOR rates. We also discuss the pricing of interest rate caps and swaptions in this framework. In the final section, we summarise the earlier effort to price an interest rate caplet when the forward rate dynamics are Gaussian (i.e. the volatility function is only time dependent).