Academic journal article Journal of Applied Finance

Hedging Strategies for Exploiting Mispriced Options Using the Black-Scholes Model with Excel

Academic journal article Journal of Applied Finance

Hedging Strategies for Exploiting Mispriced Options Using the Black-Scholes Model with Excel

Article excerpt

This paper provides a flexible Excel model that uses the Black-Scholes model to calculate the extent to which European options may be mispriced and what positions to take to profit from this mispricing while at the same time hedging against stock price changes. The model calculates positions that hedge against small stock price changes ("delta hedging") and also positions that hedge against changes in the options' sensitivity to stock price changes ("gamma hedging"). The model can be used to compare the profitability of these different strategies for a range of possible stock prices at the close of the positions. [G13, G32]

The Black-Scholes (1973) option pricing model enables the construction of a portfolio position that lets an investor exploit a mispriced option while still hedging against changes in the price of the underlying stock. Say the investor believes a European call option's current market price implies greater volatility in the stock price than is likely accurate. This in turn implies that the option's price is too high, so the investor can profit by shorting the call and then buying it back when its price has fallen. Other factors, of course, could cause the stock price to rise while the short position is in place, raising the call's price and wiping out the investor's potential profit. To protect against this, the investor can use the underlying stock and other options on that stock to construct a hedged portfolio position.

We provide a flexible Excel model that shows how much an option may be mispriced and what positions to take to profit from this mispricing while hedging against stock price changes. The model provides positions that hedge against small stock price changes and positions that hedge against changes in the options' stock price sensitivity. Investors can use the model to compare the potential profitability of different strategies for a range of possible stock prices at the close of the positions.

I. The Black-Scholes Model and Hedge Portfolios

Given a market price quotation for a European call or put option, plus the values we observe for the remaining variables, we can solve the Black-Scholes formula for the stock price volatility. This implied volatility represents the market's estimate of the volatility of the underlying stock's return over the option's life. If the investor's estimate of the stock's true volatility is different, he or she could speculate on a change in the option's price and at the same time take positions in the underlying stock or other options to hedge against changes in the stock price.

In a delta hedging strategy, for example, an investor believing that an option is overvalued could sell the option and simultaneously purchase a number of the underlying shares equal to the partial derivative of the option price with respect to the stock price. For relatively small changes, the effects of stock price changes on these two positions would be offsetting and protect the investor's profit if the option mispricing is corrected. From Equations (1) and (2), the partial derivative of the option price is N(d^sub 1^) for a European call option and N(d^sub 1^) -1 for a put option. The partial derivative is also called the hedge ratio, or the option delta. Delta hedging a call option position thus consists of buying (selling) N(d^sub 1^) shares for each call option sold (purchased). Since N(d^sub 1^) -1 < 0, delta hedging a put option position consists of buying (selling) 1 - N(d^sub 1^) shares for each1 put option purchased (sold). The resulting portfolios of the option and the stock are referred to as "delta-neutral."

A delta hedging strategy should be reasonably effective if the option mispricing is corrected quickly and the underlying stock price does not have a chance to move very much. If this is not the case, the delta hedge may not be effective, because the option's delta changes as the stock price changes.

Since the gamma of the stock itself is zero, the resulting portfolio will also be gamma-neutral (i. …

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