Risk-Neutral Valuation

The main focus of this book is the pricing of financial assets. Price formation in financial markets may be explained in an absolute manner in terms of fundamentals, as, e.g. in the so-called rational expectation model, or, more modestly, in a relative manner explaining the prices of some assets in terms of other given and observable asset prices. The second approach, which we adopt, is based on the concept of arbitrage. This remarkably simple concept is independent of beliefs and tastes (preferences) of the actors in the financial market. The basic assumption simply states that all participants in the market prefer more to less, and that any increase in consumption opportunities must somehow be paid for. Underlying all arguments is the question: Is it possible for an investor to restructure his current portfolio (the assets currently owned) in such a way that he has to pay less today for his restructured portfolio and still has the same (or a higher) return at a future date? If such an opportunity exists, the arbitrageur can consume the difference today and has gained a free lunch.

No one can predict the future! All that can be done by way of prediction is to use what information is available as well as possible. Our task is to make the best quantitative statements we can about uncertainty — which in the financial context is usually uncertainty about the future. The basic tool to quantify uncertainty is a probability density or distribution. We will assume that most readers will be familiar with such things from an elementary course in probability and statistics; for a clear introduction see, e.g. Grimmett and Welsh (1986), or the first few chapters of Grimmett and Stirzaker (2001); Ross (1997), Resnick (2001), Durrett (1999), Ross (1997), Rosenthal (2000) are also useful.

Access to full, accurate, up-to-date information is clearly essential to anyone actively engaged in financial activity or trading Indeed, information is arguably the most important determinant of success in financial life. Partly for simplicity, partly to reflect the legislation and regulations against insider trading, we shall confine ourselves to the situation where agents take decisions on the basis of information in the public domain, and available to all. We shall further assume that information once known remains known — is not forgotten — and can be accessed in real time.

We will study so-called finite markets — i.e. discrete-time models of financial markets in which all relevant quantities take a finite number of values. Following the approach of Harrison and Pliska (1981) and Taqqu and Willinger (1987), it suffices, to illustrate the ideas, to work with a finite probability space (Ω, F, ℙ), with a finite number |Ω| of points ω, each with positive probability: ℙ({ω}) > 0.

The underlying set-up is as in Chapter 3: we need a complete probability space (Ω, F, ℙ), equipped with a filtration, i.e a nondecreasing family \(\mathbb{F} = {\left( {{F_t}} \right)_{t \geqslant 0}}\) of sub-σ-fields of F: F _s ⊆ F _t ⊆ F for 0 ≤ s < t < ∞. Here, F _t represents the information available at time t, and the filtration \(\mathbb{F}\) represents the information flow evolving with time.

This chapter discusses the general principles of continuous-time financial market models. In the first section we use a rather general model, which will serve also as a reference in the later chapters. A thorough discussion of the benchmark multi-dimensional Black-Scholes model is the topic of the second section. We discuss the valuation of several standard and exotic contingent claims in the continuous-time Black-Scholes model in the third section. After examining the relation between continuous-time and discrete-time models we close with a discussion of futures and currency markets.

We now return to general continuous-time financial market models in the setting of §6.1, i.e. there are d + 1 primary traded assets whose price processes are given by stochastic processes S ₀,..., S _d , which are assumed to be adapted, right-continuous with left-limits (RCLL) and strictly positive semi-martingales on a filtered probability space (Ω, F, ℙ, F) (as usual F = (F _t ) _t≤T ). We assume that the market is free of arbitrage, in the sense that there exist equivalent martingale measures, but it contains non-attainable contingent claims, i.e. there are cash flows that cannot be replicated by self-financing trading strategies. In view of Theorem 6.1.5 this means that we do not have a unique equivalent martingale measure. We try to answer the obvious questions in this setting: how should we price the non-attainable contingent claims, i.e. which of the possible equivalent martingale measures should we pick for our valuation formula based on expectation, and, how can we construct hedging strategies for the non-attainable contingent claims to ‘minimize the risk? We try to answer these two questions in the general setting and then consider a prominent example of an incomplete market, a market with stochastic volatility, in more detail.

In this chapter, we apply the techniques developed in the previous chapters to the fast-growing fixed-income securities market. We mainly focus on the continuous-time model (since the available tools from stochastic calculus allow an elegant presentation) and comment of the discrete-time analogue (Jarrow (1996) gives a splendid account of discrete-time models). As we want to develop a relative pricing theory, based on the no-arbitrage assumption, we will assume prices of some underlying objects as given. In the present context we take zero-coupon bonds as the building blocks of our theory. In doing so we face the additional modelling restriction that the value of a zero-coupon bond at time of maturity is predetermined (= 1). Furthermore, since the entirety of fixed-income securities gives rise to the term-structure of interest rates (sometimes called the yield curve), which describes the relationship between the yield-to-maturity and the maturity of a given fixed-income security, we face the further task of calibrating our model to a whole continuum of initial values (and not just to a vector of prices). A first attempt at explaining the behaviour of the yield curve is in terms of a continuum of spot rates of maturities between τ and T, where τ is the shortest (instantaneous) lending/borrowing period, and T the longest maturity of interest. We model these rates as correlated stochastic variables with the degree of correlation decreasing in terms of the difference in maturity. Discretizing the maturity spectrum, we are tempted to start with a generalized Black-Scholes model (as in §6.2) with W = (W1₁..., W _d) a standard d-dimensional Brownian motion. The degree of (instantaneous) correlation of the different rates can then be described in terms of their covariationt.

Approaches to modelling financial assets subject to credit risk can roughly be divided into two types of models: reduced-form and structural models.