Advanced Equity Derivatives: Volatility and Correlation

Chapter 3Implied Distributions

Perhaps the favorite activity of quantitative analysts is to decode market data into information about the future upon which a trader can base his or her decisions. This is the purpose of the implied distribution that translates option prices into probabilities for the underlying stock or stock index to reach certain levels in the future. In this chapter, we derive the implied distribution and show how it may be exploited to price and hedge certain exotic payoffs.

3.1 Butterfly Spreads and the Implied Distribution

Vanilla option prices contain probability information about the market's guess at the future level of the underlying asset S. For example, suppose that Kroger Co. trades at $24 and that one-year calls struck at $24 and $25 trade at $1 and $0.60 respectively. If interest rates are zero, we may then infer that the probability of the terminal spot price S_T in one year to be above $24 must satisfy:

because the digital payoff dominates the call spread as shown in Figure 3.1.

Figure 3.1 The digital payoff dominates the call spread payoff.

Similarly the price of a butterfly spread with strikes $23, $24, and $25 (i.e., long one call struck ...