Forwards and Options

In this chapter, we cover forward and option contracts. We introduce generic forward contracts. We move to vanilla options, focusing on options to buy or sell one share, without much loss of generality. We start by briefly defining standard option types—calls vs puts, European vs American vs Bermuda exercise styles, and Asian options. We move to models to value European options when no dividends are due during the option life. Using a simple one-step model of price movements, we explain two approaches to solving option models—the dynamic hedging approach and the risk-neutral probabilities approach. We then introduce the put-call parity relationship. This relationship helps find the price of a European put given that of a similar European call and vice versa. We cover some standard lattice models including the Cox-Ross-Rubinstein binomial model, the equal-probabilities binomial model, and a version of the trinomial model. We introduce the Black-Scholes model as the limit of a binomial model as the number of steps increase. We proceed to adapt the models where possible to the valuation of American options when no dividends are due during the option life. Then we extend the models further where possible to handle fixed dividends that fall due during the option life. We briefly discuss valuation of Bermuda options. We proceed to look at the valuation of employee stock options (ESOs) following the requirements of IFRS 2 Share-based compensation. We briefly discuss the modified Black-Scholes approach allowed under U.S. GAAP, as well as the Hull-White model of ESO valuation, adapted to comply with IFRS 2. We turn to Asian options, showing how they can be valued by Monte Carlo simulation. We close with how to estimate the volatility parameter for option models using approaches based on historical prices as well as one based on the implied volatility from a traded option.