Equity Options Part 2 - Empirical Evidence

INVESTMENTS Empirical Evidence The Black Scholes Model Stock prices follow a geometric Brownian motion. Translating into discrete time, stock prices follow a Random Walk: rt+∆t = µ · √ � ∆t + σ ∆εt+∆t (1) This model implies that, over time interval ∆t stock returns are normally √ distributed with standard deviation σ ∆t. The market is dynamically complete. Options are redundant, and can be replicated by a dynamic hedging strategy that involves the riskfree account r and the underlying stock S. The Black-Scholes formula for a call option struck at K and expiring at T: C0BS = BS (S0 , K, T, σ, rf ) (2) which depends on today’s stock price So, the volatility of the underlying stock a, and the riskfree rate rf . Implications of the Black & Scholes The Black-Scholes volatilities implied by options with difference strike prices K and maturities T should be the same. They should also remain constant over time. The volatility implied by options should be consistent with the level of volatility observed in the underlying stock market. 0% Figure 2: Which volatility? Implied Volatility 7/15/2001 4/15/2001 1/15/2001 10/15/2000 7/15/2000 4/15/2000 1/15/2000 10/15/1999 7/15/1999 Volatility MA(20) 4/15/1999 1/15/1999 10/15/1998 7/15/1998 4/15/1998 1/15/1998 10/15/1997 7/15/1997 4/15/1997 1/15/1997 10/15/1996 7/15/1996 Implied Volatility S&P500 Index Volatility MA(60) Figure 1: Which Volatility? 45% 40% 35% 30% 25% 20% 15% 10% 5% 7/15/2001 5/15/2001 3/15/2001 1/15/2001 11/15/2000 9/15/2000 7/15/2000 5/15/2000 3/15/2000 1/15/2000 11/15/1999 9/15/1999 7/15/1999 5/15/1999 3/15/1999 1/15/1999 11/15/1998 9/15/1998 7/15/1998 5/15/1998 3/15/1998 1/15/1998 11/15/1997 9/15/1997 7/15/1997 5/15/1997 3/15/1997 1/15/1997 11/15/1996 9/15/1996 7/15/1996 S&P 500 Index Volatility Time Series of Stocks and Options 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% Evidence against the Black & Scholes The Black-Scholes implied volatility does not stay constant over time. The volatility implied by the options market is on average higher than that observed directly from the underlying stock market. On any given day, options (both puts and calls) with different K and T exhibit a pattern of ”smile” or ”smirk”: • OTM puts have higher implied-vol than ATM options and OTM calls. • This ”smile” pattern is more pronounced in short-dated options. • The volatility implied by long-dated options differs from that implied by short-dated options. The Assumption of Constant Volatility The fact that volatility is not constant can be seen from both the underlying stock market and the options market. In fact, volatility is itself a stochastic process. For example, letting Vt = σt2 , a popular stochastic volatility model is: � � � dVt = k V̄ − Vt dt + η Vt dBtv (3) Some empirical ”regularities”: 1. volatility is stochastic (random); 2. volatility is persistent; 3. volatility is mean-reverting; 4. the volatility shocks are negatively correlated with the underlying price shocks. For each regularity, can you provide a relevant empirical evidence from the options market? The Assumption of No Jump The fact that stock prices experience sudden jumps can be seen from both the underlying stock market and the options market (why?). Price is continuous in the Black Scholes: rt+∆t = µ · √ ∆t + σ � ∆εt+∆t (4) √ As we shrink the time interval ∆t to zero, the volatility σ ∆t of the surprise component also shrinks to zero. In 1976, three years after the Black Scholes, Bob Merton proposed a jump extension. rt+∆t = µ · √ � ∆t + σ ∆εt+∆t + Jt+∆t (5) Can you provide the relevant empirical evidence from the options market indicating negative price jumps in the underlying? The Assumption of Complete Market Although it is not an explicit assumption made by the Black Scholes, market is dynamically complete with respect to the underlying stocks and the riskfree account. The intuition is that there is only one source of uncertainty (associated with the Brownian motion), which can be fully reflected by the underlying stock prices. If in addition to the Brownian shocks, there are jump risks in the underlying stock, then the market is no longer complete. Similarly situation arises if the stock volatility has random shocks of its own. In both cases, investors’ aversion to the additional sources of uncertain will be reflected in option prices. Fear of Jumps The options market, especially that of OTM puts, provides a unique opportunity for investors to express their aversion to negative price jumps. The market prices of such OTM puts reflect not only the probability and magnitude of such jumps, but also investors’ aversion to such jumps. This aversion to negative jump risks is closely related to the aversion to negative skewness. Just like backing out Black-Scholes implied volatility, one can back out the probability and magnitude of negative jumps from OTM puts. It is found that option-implied jump risks are much exaggerated in comparison with those observed directly from the spot market (Jun Pan 2001), why? Summary The Black-Scholes model provides a powerful framework for us to formulate option pricing. The Black-Schole formula provides a powerful tool for us to measure the sensitivity of option prices with respect to the market condition. To option traders, such sensitivity measures are as essential, if not more, as breathing air in and out. There is mounting evidence, from both the underlying stock market and the options market, suggesting that the assumptions of the Black Scholes are overly simplified. In this class, we examined the effect of stochastic volatility, price jumps, and fear of jumps.