## Examples

(i) Use of Low Discrepancy Sequences In the last section we made the distinction between the use of Monte Carlo for an occasional pricing and its routine use for calculating on-line prices or regular revaluation of a book. The same remarks apply to quasi-random numbers. There is no doubt that you need to be a lot more careful when setting up low discrepancy sequences. If you are using simulation for once-off calculations, it might be easier just to let your random number Monte Carlo simulations...

## Outside Barrier Options

The barrier options described so far have been European options which are knocked in or knocked out when the price of the underlying variable crosses a barrier. An extension of this is a European option which knocks in when the price of a commodity other than the underlying stock crosses a barrier. For example, an up-and-in call on a stock which knocks in when a foreign exchange rate crosses a barrier. These options are called outside barrier options, as distinct from inside barrier options,...

## Introduction

i Price Volatility Apart from a few stray references, option theory has been developed to this point in the book with the assumption that stock price volatility remains constant. But it is very unlikely that a reader would have got this far without having heard that volatility is not constant. Before plunging into the subject we need to spend a couple of pages both defining the jargon and explaining the market observations which cause us to depart from the previous, well-ordered world of...

## Options On Options Compound Options

i Definitions We will consider an option the compound option on an underlying option. Both the compound and the underlying options can be either put or call options, so that we have four options to consider in all. Half the battle in pricing these of compound underlying option option options is simply getting the notation straight, and this can be summarized as follows UNDERLYING OPTIONS Cu St, X, t Pu St, X, t Stock price at time t and volatility Value at time t of an underlying call put...

## Power Options

i These may take an infinite variety of forms, but the two most common ones encountered in the marketplace have payoffs given by st - X 2 St - X2 - 2X st - X max 0, St - X2 - 2X max 0, st - X so the two options are simply connected by a call option. These options are mostly the domain of leverage junkies and are mathematically rather untidy. Unlike most options we deal with, they are not homogeneous in ST and X. A reflection of this is that the size of the payment depends on the unit of...

## Binomial Model For Two Asset Options

i An extension of the now familiar binomial tree to three dimensions is shown in Figure 12.1. For the sake of simplicity, we use the space variables xt ln Si 1 S01 and yt ln S 2 S02 , rather than working directly with stock prices. This means that the step sizes up down and left right are of constant sizes, rather than proportional to the stock values the first node in the tree has value zero. A tree of this type is described by the basic arithmetic random walk in Appendix A.2. Equation 3.5...

## Local Volatility And The Fokker Planck Equation

In the last section we saw that implied volatilities vary with the strike price and maturity of options. This is tantamount to saying that the Black Scholes model does not quite work. The most straightforward way of getting around this consists of assuming that volatility is a function of both the stock price and of time, which allows us to price options consistently with each other at any given moment in time. This is of course essential if we are ever to use one option to hedge another, or to...

## Binary Digital Options

i Recall the simple derivation of the Black Scholes formula which was given in Section 5.2. In its simplest form, this may be written F St max 0, St - X dST F St St - X dST where F ST is the lognormal probability distribution of ST. The two terms in this equation will now be interpreted separately, rather than together as they were before Reiner and Rubinstein, 1991b . ii Cash or Nothing Option Bet In the term fX F ST X dST, the factor X appears in two un- related roles as a constant...

## Trinomial Trees

i Binomial Tree with Variable Volatility Let us consider a binomial tree of the type studied extensively in Chapter 7. It was assumed that volatility is constant throughout the tree we now relax this idealized assumption, and consider the case where volatility varies both over time and with stock price. This means that the volatility at each node is different and in consequence, the sizes of steps and the transition probabilities are also variable. In each cell of the tree, there are two...

## Shout Options

i Like cliquets, these options are for investors who think that the underlying stock price might peak at some time before maturity. The shout option is usually a call option, but with a difference at any time t before maturity, the holder may shout. The effect of this is that he is guaranteed a minimum payoff of St X, although he will get the payoff of the call option if this is greater than the minimum. ii Payoffs By definition, the final payoff of the option is max 0, St X, ST X . In...

## Forward Contracts

i A forward contract is a contract to buy some security or commodity for a predetermined price, at some time in the future the purchase price remains fixed, whatever happens to the price of the security before maturity. Clearly, the market or spot price and the forward price will tend to converge Figure 1.3 as the maturity date is approached a one-day forward price will be pretty close to the spot price. In the last section we used the example of a forward currency contract this is the...

## Contents

PART 1 ELEMENTS OF OPTION THEORY 1 2.2 Option prices before maturity 16 2.4 Put-call parity for american options 20 2.5 Combinations of options 22 2.6 Combinations before maturity 26 3 Stock Price Distribution 29 3.1 Stock price movements 29 3.2 Properties of stock price distribution 30 3.3 Infinitesimal price movements 33 4 Principles of Option Pricing 35 4.2 Continuous time analysis 38 4.4 Examples of dynamic hedging 46 5 The Black Scholes Model 51 5.2 Derivation of model from expected values...

## Baroneadesi And Whaley Approximation

i This method can be applied to continuous dividend puts and calls. We will restrict our analysis to put options where price divergence between European and American options is greater, but the analysis for calls is exactly analogous Barone-Adesi and Whaley, 1987 . The price of an American put option can be written PA PE lt , where PE is the price of the European option and lt is a premium for the possibility of early exercise. This method seeks a way of calculating lt the Black Scholes model...

## Applications

i European Call Jarrow-Rudd Method u d-1 econsider the tree shown in Figure 7.4. From the specification of the option and equation 7.6 , the following parameters can be calculated Using these u and d factors, we can start filling in the stock prices on the tree shown just above each node . The intermediate values of St are not really necessary for a European option, since the option payoff only depends on the stock price at maturity however, they are shown for ease of understanding. The payoff...

## Black Scholes Equation Revisited

i Figure 6.1 shows the curves for the prices of an American and a European put option, each with strike price X. The American option has an additional constraint PA gt X S0 which does not apply to the European put an in-the-money American put could be exercised at any time, to yield the payoff. In consequence, while the values of the two options might be fairly similar for large values of S0, they might diverge sharply for smaller values of S0, especially below X. Figure 6.1 American vs....