## 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...

## 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...

## 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...

## 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...

## 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...