Option Greeks and Dynamic Hedging

There are five inputs to the BSM model: asset price, exercise price, asset price volatility, time to expiration, and the risk free rate. The relationship between each input (except the exercise price) and the option price is captured by sensitivity factors known as “the Greeks.”

Delta describes the relationship between changes in asset prices and changes in option prices. Delta is also the hedge ratio. Call option deltas are positive because as the underlying asset price increases, call option value also increases.

Conversely, the delta of a put option is negative because the put value falls as the asset price increases. See Figure 29.4.

Figure 29.4: Call and Put Delta: The Relationship Between Option Price and Underlying Asset Price

Deltas for call and put options on dividend paying stocks are given by:

$\delta$ is the continuously compounded dividend yield.

However, because we’re now using a linear relationship to estimate a non linear change, the following relationships are only approximations:

$\Delta C$ and $\Delta P$ are the changes in call and put price.

The approximations are close for small changes in stock price, but the approximation becomes less accurate as the ∆S becomes larger.

Interpreting Delta

The payoff diagrams for a European call option before and at expiration are shown in Figure 29.5. The “at expiration” line represents the call option’s intrinsic value, which is equal to:

Zero when the call option is out of the money. The stock price minus the exercise price when the option is in the money.

Before expiration, the option also has time value, so the prior to expiration curve lies above the at expiration diagram by the amount of the time value.

Figure 29.5: European Call Option Payoff Diagrams

The slope of the “prior to expiration” curve is the change in call price per unit change in stock price. Sound familiar? It should, that’s also the definition of delta.

That means delta is the slope of the prior to expiration curve. The delta of the put option is also the slope of the prior to expiration put curve.

Take a closer look at Figure 29.5. When the call option is deep out of the money, the slope of the at expiration curve is close to zero, which means the call delta will be close to zero. For deep out of the money call options (i.e., when the stock price is low), the option price does not change much for a given change in the underlying stock.

When the call option is in the money (i.e., when the stock price is high), the slope of the at expiration curve is close to 45 degrees, which means the call delta is close to one. In this case, the call option price will change approximately one for one for a given change in the underlying stock price.

The bottom line is that a call option’s delta will increase from 0 to $e^{-\delta T}$ as stock price increases. For a non dividend paying stock, the delta will increase from 0 to 1 as the stock price increases.

For a put option, the put delta is close to zero when the put is out of the money (i.e., when the stock price is high). When the put option is in the money (i.e., when the stock price is close to zero), the put delta is close to − $e^{-\delta T}$.

The bottom line is that a put option’s delta will increase from − $e^{-\delta T}$ to 0 as stock price increases. For a non dividend paying stock, the put delta increases from −1 to 0 as the stock price increases.

Now, let’s consider what happens to delta as the option approaches maturity, assuming that the underlying stock price doesn’t change. The effects on call and put options are different and will depend on whether the options are in or out of the money.

Remember that a call option delta is between 0 and $e^{-\delta T}$. Assuming that the underlying stock price doesn’t change, if the call option is:

Out of the money (the stock price is less than exercise price), the call delta moves closer to 0 as time passes.

In the money (the stock price is greater than exercise price), the call delta moves closer to $e^{-\delta T}$ as time passes.

Remember that a put option delta is between − $e^{-\delta T}$ and 0. If the put option is:

Out of the money (the stock price is greater than exercise price), the put delta moves closer to 0 as time passes.

In the money (the stock price is less than exercise price), the put delta moves closer to − $e^{-\delta T}$ as time passes.

Gamma measures the rate of change in delta as the underlying stock price changes. Gamma captures the curvature of the option value versus stock price relationship. Long positions in calls and puts have positive gammas.

For example, a gamma of 0.04 implies that a $1.00 increase in the price of the underlying stock will cause a call option’s delta to increase by 0.04, making the call option more sensitive to changes in the stock price.

Gamma is highest for at the money options. Deep in the money or deep out of money options have low gamma. Gamma changes with stock price and with time to expiration.

To lower (increase) the overall gamma of a portfolio, one should short (go long) options.

Recall that delta provides an approximation for change in option value in response to a change in the price of underlying. Including gamma in our equation would improve the precision with which change in option value would be estimated (similar to how convexity improves the estimate of change in a bond’s price in response to a change in interest rate).

Call and put options on the same underlying asset with the same exercise price and time to expiration will have equal gammas.

Vega measures the sensitivity of the option price to changes in the volatility of returns on the underlying asset, as shown in Figure 29.6. Both call and put options are more valuable, all else equal, the higher the volatility, so vega is positive for both calls and puts. Volatility is a very important input to any option valuation model (and thus vega is an important sensitivity measure) because option values can be very sensitive to changes in volatility.

Note that vega gets larger as the option gets closer to being at the money.

Figure 29.6: Call and Put Vega: The Relationship Between Option Price and Volatility

Rho measures the sensitivity of the option price to changes in the risk free rate, as shown in Figure 29.7. The price of a European call or put option does not change much if we use different inputs for the risk free rate, so rho is not a very important sensitivity measure. Call options increase in value as the risk free rate increases (the rho of a call option is positive).

You can see this by looking at the BSM model. As the risk free rate increases, the second term decreases, and the value of the call option increases.

Put options decrease in value as the risk free rate increases (the rho of a put option is negative).

Figure 29.7: Call and Put Rho: The Relationship Between Option Price and the Risk Free Rate

Theta measures the sensitivity of option price to the passage of time, as shown in Figure 29.8. As time passes and a call option approaches maturity, its speculative value declines, all else equal. This is called time decay.

That behavior also applies for most put options (though deep in the money put options close to maturity may actually increase in value as time passes). Notice that because it is a measure of time decay, theta is less than zero: as time passes and the option approaches the maturity date, the option value decreases (holding other factors constant). The speed of decline in the option’s value increases as time to expiration decreases (i.e., the absolute value of theta increases as the option nears maturity).

Figure 29.8: Call and Put Theta: The Relationship Between Option Price and the Time to Expiration

Figure 29.9: Direction of European Option Prices for a Change in the Five Black Scholes Merton Model Inputs

Dynamic Hedging

The goal of a delta neutral portfolio (or delta neutral hedge) is to combine a long position in a stock with a short position in a call option so that the value of the portfolio does not change as the stock price changes. The number of calls to sell is equal to:

Delta neutral portfolios can also be constructed using put options. In such a case, a long stock position is combined with a long position in puts. (Remember that put deltas are negative).

Delta neutral portfolios are best illustrated with an example.

If the assumptions of the BSM hold, changes in stock price will be continuous rather than abrupt, and hence there will be no gamma risk. In this context, gamma can be viewed as a measure of how poorly a dynamic hedge will perform when it is not rebalanced in response to a change in the asset price. Gamma risk is therefore the risk that the stock price might abruptly “jump,” leaving an otherwise delta hedged portfolio unhedged.

Consider a delta hedge involving a long position in stock and short position in calls. If the stock price falls abruptly, the loss in the long stock position will not equal the gain in the short call position. This is the gamma risk of the hedge.

Because a stock’s delta is always 1, its gamma is 0. A delta hedged portfolio with a long position in stocks and a short position in calls will have negative net gamma exposure.

Future volatility is one of the inputs in the BSM model. Estimates based on historical volatility are often used because future volatility is not directly observable.

Implied volatility is the standard deviation of continuously compounded asset returns that is “implied” by the market price of the option. Of the five inputs into the BSM model, four are observable: stock price, exercise price, risk free rate, and time to maturity. If we plug these four inputs into the BSM model, we can solve for the value of volatility that makes the BSM model value equal the market price.

Volatility enters the BSM pricing equation in a complex way, and there is no closed form solution for the volatility that will produce a value equal to the market price. Rather, implied volatility must be found by iteration (i.e., trial and error).

Traders often use implied volatilities to gauge market perceptions. For example, implied volatilities in options with different exercise prices on the same underlying may reflect different implied volatilities (a violation of the BSM assumption of constant volatility).

Traders can also use implied volatility as a mechanism to quote option prices. In this way, options with different exercise prices and maturity dates can be quoted using the same unit of measurement.

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