When evaluating hedging strategies it is also important to consider their robustness to misspecification of stock parameters. In the Black-Scholes world, hedging requires the input of only one unobserved parameter: the volatility. Ahmad and Wilmott (2005) show that the Black-Scholes hedge is quite robust to misspecification of this parameter, and in fact you can profit safely from option misprices even if your estimate of the volatility is not particularly close. The maximal utility and minimal variance hedges both require estimates of five unobserved parameters, while the adjusted Black-Scholes hedge requires only two. For this reason, it is reasonable
to suggest that the profit distributions for the three strategies may be effected differently by parameter misspecification. We study this by generating 10 000 stock trajectory pairs with 500 uniform rebalancing times for each strategy, and each value of a certain parameter. We use the parameters of Table 3.7, unless otherwise specified.
Tables 3.9-3.13 summarize the results on the resulting profit distributions. We can see that the adjusted Black-Scholes hedge profit standard deviation is much more sensitive to misspecification of the η and σ parameters, and of course, does not differ for different values of the other parameters. The two other strategies are less sensitive to misspecification of the volatility parameters, which is the investor’s reward for having to estimate an additional three parameters. These three parame-ters do not seem to have much of an impact on the profit distributions. This should be reassuring to an investor, due to the notorious difficulty in estimating the drift terms. However, it is not entirely clear how to interpret these results. Comparing the mean profits with the standard deviations, it is clear that losses, even significant ones, are entirely possible here. For this reason, it is reasonable to evaluate the robustness of the strategies in terms of the effect on the investor’s terminal utility.
We do this next.
ηh Optimal Minimum Local Variance Black-Scholes
Table 3.9: The distributions of the total profit when hedging the sale of a bundle of 500 put options with the stock and option parameters given in Table 3.7, with different values for η.
σh Optimal Minimal Variance Adjusted Black-Scholes
Table 3.10: The distributions of the total profit when hedging the sale of a bundle of 500 put options with the stock and option parameters given in Table 3.7, with different values for σ.
vh Optimal Minimum Local Variance Adjusted Black-Scholes
Table 3.11: The distributions of the total profit when hedging the sale of a bundle of 500 put options with the stock and option parameters given in Table 3.7, with different values for v.
µh Optimal Minimum Local Variance Adjusted Black-Scholes
Table 3.12: The distributions of the total profit when hedging the sale of a bundle of 500 put options with the stock and option parameters given in Table 3.7, with different values for µ.
ρh Optimal Minimum Local Variance Adjusted Black-Scholes
Table 3.13: The distributions of the total profit when hedging the sale of a bundle of 500 put options with the stock and option parameters given in Table 3.7, with different values for ρ.
3.11 Comparing the Hedging Strategies - Robustness to Parameter Mis-specification - Utility Loss
Our second means of evaluating the results of hedging with different parameter values comes from a utility indifference argument, similar to that used to arrive at the utility indifference price earlier. We borrow this methodology from Larsen and Munk (2012), who introduce the value function:
Jλ(t, w, y)π = −1
γEw,y,texp −γ(WT + λh(eYT)) .
We use the superscript (...)π to designate that this function is to be evaluated using a particular hedging strategy, not necessarily the optimal strategy as in Eqn.(3.2).
We can calculate the loss due to using a suboptimal hedging strategy by finding the amount of money added to the initial wealth when we hedge with a subopti-mal strategy, that leaves us indifferent between this strategy and using the optisubopti-mal hedging strategy. Our method here differs from that of Larsen and Munk (2012) only in that we do not present the loss as a fraction of the initial wealth. This is because with exponential utility functions, the loss is independent of initial wealth.
Definition 4. The utility loss due to a suboptimal strategy to hedge a position with λ options with expiration time T is the L that solves:
Jλ(t, w, y; ) = Jλ(t, w + L, y)π.
Note that the loss is not calculated as a fraction of the initial wealth, as it is in Larsen
and Munk (2012). This is because we are only studying investors with exponential utility functions, and thus the results of solving Eqn.(3.8) will be independent of initial wealth. Our results have shown that an investor who plans to hedge with a correlated stock as a means of securing a profit via volatility arbitrage will have to accept significant risk. For this reason, the risk preferences of the investor are of the utmost importance, which justifies analysing the effect of misspecified hedging parameters on the expected utility of terminal wealth. We take the same stock and option parameters as in the previous section. The results are summarized in Tables 3.15-3.19. Once again, we note the sensitivity of the adjusted Black-Scholes strategy to misspecification of the volatility terms. While one of the advantages of the adjusted Black-Scholes strategy is that there is no need to specify v, µ or ρ, we can see how little the utility loss is due to misspecification of these parameters.
It is also interesting to see how little utility loss there is from overestimating the correlation between the two stocks. It seems to be more important for an investor to choose a strongly correlated stock to hedge with in the first place, than to have an accurate estimate of exactly how correlated they are. These results seem to justify using the two more sophisticated strategies, even if the investor only possesses rough estimates of the three extra parameters. It seems difficult for an investor to justify utilizing the minimal variance hedge it does not result in notably smoother profit trajectories. In addition, it does not incorporate the risk preferences of the investor, which in such a risky environment is a critical flaw. For these reasons, it would
seem the maximal utility strategy should be the preferred method of securing a profit from volatility arbitrage in this incomplete market setting.
Loss Due to Misspecification
η Maximal Utility Minimal Variance Adjusted Black-Scholes
0.05 71.09 71.09 40.55
0.1 33.36 34.26 1.81
0.2 - 0 33.36
0.3 17.14 9.03 185.26
0.4 79.16 52.24 432.51
Table 3.14: Utility loss due to misspecification of η when hedging the sale of 500 European put options.
Loss Due to Misspecification
σ Maximal Utility Minimal Variance Adjusted Black-Scholes
0.05 71.09 56.73 492.99
0.1 - 0 33.36
0.2 35.16 36.95 2.71
0.3 57.63 59.42 25.25
0.4 71.09 72.88 42.35
Table 3.15: Utility loss due to misspecification of σ when hedging the sale of 500 European put options.
Loss Due to Misspecification
v Maximal Utility Minimal Variance Adjusted Black-Scholes
-0.15 14.44 10.83 33.36
-0.05 5.42 2.71 33.36
0.05 - 0 33.36
0.15 1.81 3.61 33.36
0.25 10.83 14.44 33.36
Table 3.16: Utility loss due to misspecification of v when hedging the sale of 500 European put options.
Loss Due to Misspecification
µ Maximal Utility Minimal Variance Adjusted Black-Scholes
-0.15 13.54 16.24 33.36
-0.05 2.71 4.51 33.36
0.05 - 0 33.36
0.15 5.42 3.61 33.36
0.25 16.24 13.54 33.36
Table 3.17: Utility loss due to misspecification of µ when hedging the sale of 500 European put options.
Loss Due to Misspecification
ρ Maximal Utility Minimal Variance Adjusted Black-Scholes
0.65 3.61 3.60 33.36
0.75 21.38 17.42 33.36
0.85 40.34 37.19 33.36
0.95 73.00 62.42 33.36
Table 3.18: Utility loss due to misspecification of ρ when hedging the sale of 500 European put options.
Chapter 4
NON-PROBABILISTIC OPTION PRICING