Derivatives

CHAPTER 22 BOPM: Implementation

Aims

• To show how dynamic delta hedging can be used to price a (two-period) call option, using a portfolio comprising stocks and calls, which is risk-free over small intervals of time.
• To show how dynamic delta hedging is consistent with the no-arbitrage binomial pricing equation – the latter is a backward recursion that can be interpreted using risk-neutral valuation (RNV).
• To replicate the payoff to an option, using stocks and (risk-free) borrowing or lending (e.g. using a bank deposit/loan). This provides an alternative derivation of the binomial formula for options.
• To show that as each time-step in the binomial tree becomes smaller, the tree more closely approximates a continuous time process (Brownian motion) for the stock price – as used in the Black–Scholes approach. Hence, as we increase the number of time periods in the binomial tree, the option price calculated from the BOPM formula converges to the Black–Scholes price.

Using insights from RNV, we price a two-period call option using the BOPM without going through all the details involved in delta hedging and forming a ‘risk-free arbitrage portfolio’ – instead we price the option assuming RNV, using a ‘backward recursion’. This allows us to generalise the BOPM to many periods and to price many different types of option. We show that RNV is consistent with there being no arbitrage opportunities at any node in the binomial tree.

We demonstrate another method of pricing an option using ...