Question

Can Black-Scholes PDE describe an option price dynamic with volatility risk? Explain

Can Black-Scholes formula be used in pricing executive stock options? Explain

Answer 1

1. While Black-Scholes model is often used to compute implied volatility, it does not explain the option price dynamic completely. This is well-reflected in a 'volatility smile'. A volatility smile is a geographical pattern of implied volatility for a series of options that has the same expiration date. When plotted against strike prices, these implied volatilities can create a line that slopes upward on either end; hence the term "smile." Volatility smiles should never occur based on standard Black-Scholes option theory, which normally requires a completely flat volatility curve.

The pricing of options is more complicated than the pricing of stocks or commodities, and this is well-reflected in a volatility smile. Three main factors make up an option's value: strike price relative to the underlying asset; the time until expiration, or expiry; and the expected volatility in the underlying asset during the life of the option. Most option valuations rely on the concept of implied volatility, which assumes the same level of volatility exists for all options of the same asset with the same expiry.

Several hypotheses explain the existence of volatility smiles. The simplest and most obvious explanation is that demand is greater for options that are in-the-money or out-of-the-money as opposed to at-the-money options. Others suggest that better-developed options models have led to out-of-the-money options becoming priced more expensively to account for risk of extreme market crashes or black swans. This calls into question any investing strategy that relies too heavily on implied volatility from the Black-Scholes model, particularly with the valuation of downside puts that are far away from the money.

2. Execituve Stock Options (ESOs) are difficult to value using conventional methods of option pricing, which are applicable principally to traded stock options. For example, a traded American call option on a non-dividend paying stock can be valued using the pricing formulas developed for traded European call options using the Black-Scholes model. However, certain critical assumptions needed to derive these option pricing formulas are not met by ESOs. In particular, the Black-Scholes-Merton models assume that the holder can buy and sell short the company’s stock, and that the options themselves can be freely traded. However, ESOs are not transferable and cannot be traded. Further, SEC rules and company policies prevent executives from selling their own company’s stock short. Therefore, valuing an ESO and computing the optimal exercise policy using the Black-Scholes-Merton framework is likely to lead to substantial errors.