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Changes in Volatility and Risk-Free Rates

One of the assumptions made by Black-Scholes is that at,T and rt,T must remain constant during the life of the hedging program. In practice, this is far from true as stock price volatility and interest rates change just as frequently as stock prices. As a consequence, for the hedger to be immunized against these movements, the hedger needs to ensure that sensitivities of the option premium to changes in volatility (called vega) and risk-free rate (called rho) need to be also rebalanced. The expressions for vega and rho in the instance of calls and puts on dividend-paying stocks are given as follow:

(7.3a)

(7.3b)

(7.4a)

(7.4b)

At first glance, it may seem counterintuitive to the reader as to how it is possible to assume that σt,T and rt,T are constants on one hand and then calculate both vega and rho as though these variables are in fact random on the other hand. On a closer examination, this is not counterintuitive if one recognizes the fact that the option premiums are correct for a given σt,T and rt,T and that the values of σt,T and rt,T are subject to change. In practice, it is more appropriate to use the relevant distributions for both volatilities and risk-free rates so as to allow for extra variability inherent in these options, which tend to be more pronounced as the maturities get longer. The consequence of doing this is higher option premiums as compared to those obtained using constant σt,T and rt,T – an issue that is discussed further in Chapter 8.

As in the gamma risks, hedging vega risks also involves the use of stock options. Despite this, one cannot use the same set of options to simultaneously mitigate both gamma and vega risks – unless the hedge option is identical to the initial option sold. In this case, all sensitivities, like delta/gamma/vega/rho and so on, of the entire portfolio after hedging would be identically zero and continue to be such as the option life decays and market conditions change. Like delta risks, hedging rho risks can easily be done using interest rate swaps as well as stock options although the use of interest rate swaps tends to be cleaner and cheaper. Thus, to implement delta, gamma, vega, and rho hedging in practice, every time a rebalancing needs to done (i.e., neutralize the net portfolio delta/vega/rho values to zero), the hedger has to transact in a combination of stocks, stock options, and interest rate swaps.

Redoing the example discussed in Table 7.7 (and not taking into consideration any transaction costs), one can see in Table 7.10 how the gamma, vega, and rho change over the option life.

From Table 7.10, the reader can easily observe that I only computed the deltas, gammas, vegas, rhos of an option. In practice, one can mathematically compute higher order derivatives{{Examples of such derivatives are Vanna =

 
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