## St VTS on t T

We then solve for V in the region t < T. That is, we solve 'backwards in time', hence the name. If the equation is of forward type we impose an 'initial' condition on t 0, say, and solve in t > 0, in the forward direction. Of course, we can change from backward to forward by the simple change of variables t' t. This is why both types of equation are mathematically equivalent and it is common to transform backward equations into forward equations before any...

## S 0 S E E if S E

Whether S is greater or less than E at expiry the payoff is always the same, namely E. How much would I pay for a portfolio that gives a guaranteed E at t T This is, of course, the same question that we asked in Chapter 1, and the answer is arrived at by discounting the final value of the portfolio. (Note that here we do have to assume the existence of a known risk-free interest rate over the lifetime of the option.) Thus this portfolio is now worth Ee-r(T-i). This equates the return from the...

## It t2S2 rtatSrtPtV 0 616

Now eliminate the coefficients of V and dV OS by choosing 2 Some readers may have already derived these formulae by a different procedure, in Exercise 5 at the end of Chapter 5. and remove the remaining time dependence by setting 7 t j 2 r dr. With these choices 6.16 becomes and has coefficients which are independent of time this equation contains no reference to r or a. If V S,t is any solution of 6.17 , then the corresponding solution of 6.16 , in original variables, is V Let us...

## CTa7 a

Suppose that a2 and r in the Black-Scholes equation are both functions of t, but that r a2 is constant. Derive the Black-Scholes formulae in this case. 5. Suppose that in the Black-Scholes equation, r t and lt J2 t are both non-constant but known functions of t. Show that the following procedure reduces the Black-Scholes equation to the diffusion equation. a Set S Eex, C Ev as before, and put t T t' to get the equation Note that we have not yet scaled time, but merely changed...

## Info

We divide the x, r -plane into a regular finite mesh as usual, and take a finite-difference approximation of the linear complementarity equations 9.1 . As most of this discretisation is a simple extension of the finite-difference formulations given in the previous chapter, we only give a short account of it here. We approximate terms of the form du dr d2u dx2 by finite differences on a regular mesh with step sizes 6r and 6x, and truncating so that x lies between N 6x and N 6x, where N and N are...

## Forward versus Backward

In all the above we have discussed the forward equation with conditions given at r 0. The reader may ask, what is wrong with the equation with the same initial and boundary conditions This equation might, for example, arise if in a forward problem we had replaced r by To r for some constant tq, whereupon du dr becomes du dr. It turns out that this backward problem is ill-posed for most initial and boundary data the solution does not exist at all, and even if it does exist, it is likely to blow...

## Tch

The option value C S, t at the same four times as in Figure 7.8. where o is a 'universal constant' of call option pricing. Beyond this interesting fact, the local analysis is also important for the early stages of a numerical calculation where prices change rapidly the effect of the rapid changes in Sf t is felt throughout the solution region, not just near S S T . In Figures 7.8 and 7.9 we show the values of c x, r in dimensionless variables, and the original C S,t , prior to...

## Introduction

This book is about mathematical models for financial markets, the assets that are traded in them and, especially, financial derivative products such as options and futures. There are many kinds of financial market, but the most important ones for us are Stock markets, such as those in New York, London and Tokyo Bond markets, which deal in government and other bonds Currency markets or foreign exchange markets, where currencies are bought and sold Commodity markets, where physical assets such as...

## Options on Futures

Many options have as their underlying asset not the cash product but rather the corresponding futures contract, which is often more liquid and involves lower transaction costs. Options on futures therefore have a value that depends on F and t, i.e. of the form V F,t . Since we can derive a partial differential equation for V F, t from the ordinary Black-Scholes equation written in terms of S and t via the change of variable rule. That is, we replace S by Fe r T t throughout, and we replace We...

## Call Option Ramp Function

'Underlying securltr Vke t Long dated eapiry nrths Premiums shown are baud on closing o Hit prices Figure 1.1 The traded options section of the Financial Times of 4 February 1993. Figure 1.2 The FT-SE index call option values versus exercise price and the option values at expiry assuming that the index value is then 2872. Figure 1.2 The FT-SE index call option values versus exercise price and the option values at expiry assuming that the index value is then 2872. the longest dated option has a...

## Df iaSdX vSdt a2S2dt doabA

This is Ito's lemma2 relating the small change in a function of a random variable to the small change in the variable itself. Because the order of magnitude of dX is O Vdt , the second derivative of with respect to 5 appears in the expression for df at order dt. The order dt terms play a significant part in our later analyses, and any other choice for the order of dX would not lead to the interesting results we discover. It can be shown that any other order of magnitude for dX leads to...

## VT[dS [dS2 [dS2 [a2S2dX2 a2S2dt

The square root of the variance is the standard deviation, which is thus proportional to o. If we compare two random walks with different values for the parameters jl and lt 7, we see that the one with the larger value of i usually rises more steeply and the one with the larger value of a appears more jagged. Typically, for stocks and indices the value of a is in the range 0.05 to 0.4 the units of a2 are per annum . Government bonds are examples of assets with low volatility, while 'penny...