(BQ) Part 2 book "Options futures and other derivatives" has contents: Volatility smiles, basic numerical procedures, exotic options, more on models and numerical procedures, martingales and measures, real options, derivatives mishaps and what we can learn from them,.and other contents. | 20 C H A P T E R Volatility Smiles How close are the market prices of options to those predicted by the Black–Scholes– Merton model? Do traders really use the Black–Scholes–Merton model when determining a price for an option? Are the probability distributions of asset prices really lognormal? This chapter answers these questions. It explains that traders do use the Black– Scholes–Merton model—but not in exactly the way that Black, Scholes, and Merton originally intended. This is because they allow the volatility used to price an option to depend on its strike price and time to maturity. A plot of the implied volatility of an option with a certain life as a function of its strike price is known as a volatility smile. This chapter describes the volatility smiles that traders use in equity and foreign currency markets. It explains the relationship between a volatility smile and the risk-neutral probability distribution being assumed for the future asset price. It also discusses how option traders use volatility surfaces as pricing tools. WHY THE VOLATILITY SMILE IS THE SAME FOR CALLS AND PUTS This section shows that the implied volatility of a European call option is the same as that of a European put option when they have the same strike price and time to maturity. This means that the volatility smile for European calls with a certain maturity is the same as that for European puts with the same maturity. This is a particularly convenient result. It shows that when talking about a volatility smile we do not have to worry about whether the options are calls or puts. As explained in earlier chapters, put–call parity provides a relationship between the prices of European call and put options when they have the same strike price and time to maturity. With a dividend yield on the underlying asset of q, the relationship is p þ S0 eÀqT ¼ c þ KeÀrT ð20:1Þ As usual, c and p are the European call and put price. They have the same strike price, K, and time to maturity,
