Chapter 19: Hedging and the Greeks Flashcards
Provide the definition of delta and describe how the delta changes with the stock price
Generally speaking the delta (∆) is defined as the rate of change in the call price with respect to the underlying asset. In the BSM, it can be shown that delta is equal to ∆_c=N(d_1). For a call option we know that 0 ≤ ∆_c ≤ 1 since the call option carries the possibility of entering into a forward (which has ∆_c=1) of throwing it away, thereby ending up with nothing (∆_c=0). The value of the option increases with the stock price since the likelihood of the option ending up ITM increases. Hence, the delta is non-negative. The rate of which the option changes in value is largest for ATM options. When the option is ATM, even small increases in the stock price will have a large impact on the option value because it increases the probability of the option being ITM. If the option already is deep ITM a small increase in the stock price will have little effect on the option value, because the probability of exercising already is large. Hence, the change in delta will be largest for stock prices close to the strike price.
The variation of delta with stock price is graphically shown below
Provide the definition of gamma and describe how the gamma changes with the stock price
In general, gamma (Γ) is defined as the rate of change in the ∆_c with respect to the underlying asset. The variation in gamma for a call option follows from the explanation above regarding the delta. Since the sensitivity of the call option to the stock price is largest for ATM options, this means that gamma will peak here. Similarly for deep ITM or OTM options, delta is less sensitive to the stock price as changes in the latter will not materially impact whether we end up ITM og OTM.
If gamma is small then delta changes slowly, i.e. we have to change the delta hedge more gradually.
If gamma is large then delta changes quickly, i.e. in order to stay hedged we need to change the portfolio frequently.
The variation of gamma with stock price is shown below
Provide the definition of theta and describe how the theta changes with the stock price
In general, the theta (Θ) is defined as the rate of change in the call price with respect to the passage of time. As we know, the value of an option is positively dependent on time to maturity as the likelihood of the option ending up ITM increases. Therefore, theta is usually negative for an option since, all else being equal, the option tends to become less valuable as time passes (time to maturity decreases, lowering the value of choice). When the stock price is low, the theta is close to zero since the passage of time will have a small impact on whether we will end up ITM. This choice seems rather certain. When the option is ATM, and the stock price is close to the strike price, the theta is negative since the passage of time reduces the value of choice, and the latter is particularly valuable in this situation. When the stock price is high, the passage of time still reduces the value of choice, but not as much as for ATM options since we are more certain that the option will end up ITM.
Provide the definition of vega and describe how the vega changes with the stock price
The vega of a portfolio of derivatives, V, is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset. If vega is highly positive or highly negative, the portfolio’s value is very sensitive to small changes in volatility. If it is close to zero, volatility changes have relatively little impact on the value of the portfolio. A position in the underlying asset has zero vega.
The variation in vega with stock price is shown below
