How much will the option value change with a small change in a parameter such as option price or volatility? In this section we examine static risk measures, which capture movements in the option value with a movement in one of the factors, while holding all other factors constant.

Delta

Options prices are extremely dependent on underlying prices, and the price of the underlying is the key determinant of an option price. Delta measures the sensitivity of the option's price to changing underlying prices. Delta is the first derivative of an option's price with respect to a change in the price of the underlying.

This definition is exact.

In fact, the delta can be obtained approximately from the BSM formula.

Delta_c = e^{-δ T}N(d₁), and
Delta_p = -e^{-δ T}N(-d₁)
where δ is the dividend yield.

Note that the deltas are a simple function of N(d₁), and N(d₁) is between 0 and 1.

• The delta of a call option is always positive, as the value of a call increases with an underlying price increase.
  • When the call option is deep-in-the-money, its delta approaches 1.
  • When the call option is deep-out-of-the-money, its delta approaches 0.

• The delta of a put option is always negative, as the value of a put decreases with an underlying price increase.
  • When the put option is deep-in-the-money, its delta approaches -1.
  • When the put option is deep-out-of-the-money, its delta approaches 0.

• When the underlying is near the exercise price (at-the-money), delta is most sensitive to a change in the underlying price.

• For a non-dividend-paying stock, δ is 0, and e^{-δ T} becomes 1. The delta for a call becomes N(d₁) and the delta for a put becomes N(d₁) - 1.

Continue with the example we worked with in the BSM model, where S₀ = $100, X = $100, r(c) = 0.06, T = 1, and σ = 0.1. The call price is $7.46 and the put price is $1.64. The stock does not pay any dividend. The call delta N(d₁) = 0.7422 and the put delta is 0.7422 - 1 = -0.2578. For a $1 change in the price of the stock, we should expect that the:

• change in the call option price = $0.7422, and the new call option price = 7.46 + 0.7422 = $8.20.
• change in the put option price = -$0.2578, and the new put option price = 1.64 - 0.2578 = $1.38.

If we use the BSM model to re-calculate the option price, the new call option price will be $8.21 and the new put option price will be $1.39.

Note that the delta approximation is good but not perfect. For a $10 change in the price of the stock, the call price should change by 0.7422 x 10 = $7.422 and the put option price should change by -0.2578 x 10 = -$2.578. The new call option price would be 7.46 + 7.422 = $14.882 and the new put option price would be 1.64 - 2.578 = -$0.938 (this should not be true as the option price cannot be negative).

If we use the BSM model to re-calculate the option price, the new call option price should be $16.09 and the new put option price should be $0.26. The approximations based on delta are not very accurate. In general, the larger the move in the underlying price, the worse the approximation.

Delta Hedging

Continue with the example above, where S₀ = $100, X = $100, r(c) = 0.06, T = 1, and σ = 0.1. The call price is $7.46 and the call delta N(d₁) = 0.7422. Consider a portfolio, P, of a short position of one call on the stock combined with a long position of 0.7422 units of the stock. The portfolio would have the value: P = -c + N(d₁) S₀ = -7.46 + 0.7422 x 100 = $66.76.

If the stock price were to suddenly change to $101, the portfolio's value would be -8.21 + 0.7422 x 101 = $66.7522.

Thus, the value of the portfolio would change by only $0.0078 for a $1 change in the stock price. If the change in the stock price were infinitesimal, the price of the portfolio would not change at all. If the change in the stock price were larger, the change of the portfolio would be larger, but it would still be quite small relative to the change in the stock price. For example, if the stock price rose from $100 to $110, the portfolio's value would be -16.09 + 0.7422 x 110 = $65.552. In this case, a change of $10 in the stock price caused a change of $1.208 in the value of the portfolio.

A portfolio like the one we are considering is known as a delta-neutral portfolio. It is delta-neutral because an infinitesimal change in the price of the stock does not affect the price of the portfolio. We could also say that the delta of this portfolio is zero: the value of the portfolio is insensitive to the value of the stock.

N_H = - (portfolio delta)/Delta_H where N_H is the number of units of the hedging instrument, and Delta_H is the delta of the hedging instrument.

Investors often use delta to construct hedges to offset the risk they have assumed by buying and selling options. For example, if a dealer sells 1,000 call options discussed above, he would buy 742 shares of the stock to construct a delta-neutral portfolio.

An option is delta-hedged when a position has been taken in the underlying which matches its delta. Such a hedge is only effective instantaneously, because the option's delta is itself altered by changes in the price of the underlying, interest rates, the option's volatility, and time to expiry. A delta hedge must, therefore, be rebalanced continuously to be effective. In fact, delta hedging is often referred to as dynamic hedging.

Gamma

Gamma measures how delta changes with changes in the stock price. In terms of calculus, gamma is the second derivative of the option price with respect to the stock price.

The following exhibit shows the relationship between the option price and the underlying stock price, and the relationship between the gamma and the underlying stock price. (X = 50, r(c) = 0.0488, T = 0.75, σ = 0.35)

Call option values increase as the value of the underlying becomes greater, and put option values decrease. However, the amount of change is not the same in each direction. N(d₁) measures the slope of this line at a given point (it only measures the slope for a very small change in the underlying).

When the underlying changes by more than a very small amount, the curvature of the line comes into play and distorts the relationship between the option price and the underlying price that is explained by delta. The curvature or second-order effect is known as gamma in the options world ("convexity" in the fixed income world). When gamma is large, delta changes rapidly and cannot provide a good approximation of how much the option moves for each unit of movement in the underlying.

A few properties:

• The gammas are the same for a put and call with the same underlying instrument, time to expiration, and strike price.
• Gamma is always non-negative.
• Gamma is large when the option is near-the-money. However, it is not necessarily the largest when the option is at-the-money (in the above graph, when the underlying is at 50), as it is affected by other factors as well. Gamma is small when the option is deep-in-the-money or deep-out-of-the-money.
• Gamma also varies with the time remaining until expiration. For an option that is near-the-money, gamma increases as expiration approaches. However, it is difficult to make solid generalizations about how gamma will change without actually calculating the effects of the passage of the time.

Gamma risk is created when stock prices jump rather than move continuously and smoothly.

Theta

If the stock price and all other parameters of the option pricing model remain constant, the price of options will still change with the passage of time. Theta is the negative of the first derivative of the option price with respect to the time remaining until expiration. There is no uncertainty - the passage of time will happen. This is the fundamental difference of theta from delta and gamma.

THETAc and THETAp are generally less than 0. The tendency for option prices to change merely due to the passage of time is known as time decay. If all parameters remain constant except that the expiration date draws nearer, a call or a put option will have to fall in value, and the theta will be negative. As time goes by, the speed of the option value decline increases. It will be worthless at expiration as S = X. Therefore, a call or a put option will eventually lose its entire value through time decay.

Vega

Volatility is the standard deviation of the continuously compounded return on the stock. It is the only variable that cannot be obtained easily and directly from another source.

Option prices are extremely sensitive to volatility. Vega is the first derivative of an option's price with respect to the future volatility of the underlying stock.

• VEGAc and VEGAp are identical and always positive. This means that if volatility increases, both call and put prices increase.
• Vega tends to be greatest for an option near-the-money. When an option is deep-in-the-money or deep-out-of-the-money, the vega is low and can approach zero.

Rho

The sensitivity of the option price to the risk-free rate is called the rho. Although the BSM model assumes a constant risk-free rate, we can still explore how the option price would differ if the current rate were different.

• Large changes in the interest rate have relatively little impact on the option prices. That is, the price of a European option on an asset is not very sensitive to the risk-free rate.
• For a call option, rho (RHOc) is always positive. For a put option, rho (RHOp ) is always negative.
• RHOc and RHOp change as time passes, with both tending toward zero as expiration approaches. The interest rate affects the price of an option in conjunction with the time remaining until expiration mainly through the time value of money. If little time remains until expiration, the interest rate is relatively unimportant, and the price of the option becomes less sensitive to the interest rate.

Implied Volatility

Volatility is an extremely important variable to price options. We are interested in the future (not past or current) volatility over the life of the option.

There are two basic ways to estimate the volatility.

The historical estimate of the volatility is based on what happened in the past. To get the best estimate, we must use a lot of prices, but that means going back farther in time. The farther back we go, the less current the data become, and the less reliable our estimate of the volatility.

Although historical volatility can be estimated, there is no objective measure of future volatility.

Implied volatility is the BSM model volatility that yields the market option price. There are five inputs in the BSM model, which the model relates to a sixth variable, the option price. With a total of six variables, any five imply a unique value for the sixth. The technique of implied volatility uses known values of five variables to estimate the standard deviation. The estimated standard deviation is an implied volatility because it is the value implied by the other five variables in the model.

Consider the following example of an implied standard deviation based on a call option. We assume that X = $100, S0 = $100, r(c) = 0.1, price of the call option = $5, and the option has 90 days remaining until expiration. To find the implied standard deviation, we need to find the standard deviation that is consistent with these other values. To do this, we can compute the Black-Scholes-Merton model price for alternative standard deviations. We adjust the standard deviation to make the option price converge to its actual price of $5.

In our example we first try σ = 0.1, which gives a call price of $3.41. This price is too low. Next, σ = 0.2 results in a call price of $5.24, which is too high. We keep doing this until we try σ = 0.187, which results in a call price of $5.00.

Therefore, if the option is selling for about $5, we say that the market is pricing it at a volatility of 0.187. This number represents the market's best estimate of true volatility.

The implied volatility is non-stochastic. It also changes through calendar time.

The implied volatility is a source of valuable information on uncertainty in the underlying. It can be interpreted as the market's view of how to value options. Since the implied volatility is often the same order of magnitude across exercise prices and expiration, and thus easy to understand, traders routinely use it as the medium in which to quote options.