6 Hedging Performance on Exotic Options
6.1 Minimum Variance Hedging Strategy
The minimum variance hedging strategy involves only the underlying futures contract as the hedging instrument. We consider the minimum variance hedging strategy because the market is not complete in the jump diffusion model and in the stochastic volatility model, and the hedging ratio should reflect the jump risk and the volatility risk.
A minimum variance hedging portfolio consists of one unit of the hedged option and Xs units of the underlying futures, where the hedging ratio X is determined by minimizing s the variance of the hedging portfolio value.
To be more specific, suppose that an option trader writes one unit of option C . If the writer relies on the minimum variance hedging strategy to hedge this option, then the value of the hedging portfolio at time t is:
B S X C
H =− + s + ,
where B=C−XsS is the amount of risk free investment. The hedging portfolio is self-financing, and the change of H from t to t+ can be written as dt
Brdt dS X dC
dH =− + s + .
The total variance of dH is given by
Var(dH)=Var(dC)+Xs2Var(dS) 2− XsCov(dS dC, ). By minimizing Var(dH , the hedging ratio can be solved as: )
Cov( , ) Var( )
s
dS dC
X = dS . (18)
In the Black-Scholes model and the CEV model, the market is complete, and an option can be perfectly hedged by taking positions in the underlying asset and risk free investment. For these two models, the minimum variance hedging strategy is the same as the delta neutral hedging strategy, and the hedging ratio is the delta of the hedged option.
However, in the jump diffusion or the stochastic volatility model, the minimum variance hedging is not perfect in the sense that one cannot perfectly replicate the payoff of an option by only taking positions in the underlying asset and risk free investment.
In the case of the jump diffusion model, the minimum variance hedging ratio is given by (See Appendix C for details)
(
[ (, (1 ))] ( , ))
) ( 2
2 2
t t
j t
t j
S E JC t S J kC t S
V S
S C
X V + −
+ +
∂
∂
= +
σ λ σ
σ , (19)
whereVj =λk2+λ(eδ2 −1)(1+k)2.
Equation (19) shows that if there is no jump risk (i.e.λ =0) the minimum variance hedging is the same as the delta neutral hedging. However, if there is jump risk, its impact will be reflected in the second term of Equation (19).
In the stochastic volatility model, the minimum variance hedging ratio is given by (see Appendix C)
Sv v C S
X C
t S
ρ σ
∂ + ∂
∂
= ∂ . (20)
Equation (20) shows that if the volatility is deterministic or stock returns are uncorrelated with volatility changes then the minimum variance hedging ratio is the same as the delta ratio. As we have mentioned in Section 5, ρ is usually negative in the equity markets, and as a result, minimum variance hedging ratio is usually less than the delta ratio to reflect the impact of volatility changes.
In this study, the options to be hedged are barrier options and compound options on S&P 500 futures. To see the effect of the maturity of an option on the hedging performance, we consider the barrier or compound call options with three different terms to expiration: short-term, medium-term, and long-term.
The hedging procedure is described as follows: At time t , model parameters are estimated by fitting the model to the prices of the traded futures options. The price of the exotic option under consideration, Ct, can then be calculated from the model. To hedge this exotic option, a hedging portfolio is constructed with Xs units of futuresFt, and Ct units in the risk free investments. Since the futures contracts require zero initial cash outlay, the total cost of such a portfolio is zero:
0
0+ =
⋅ +
−
= t s t
t C X C
H .
At time t+∆t, the hedging portfolio is rebalanced. Using model parameters estimated at time t+∆t, the value of the hedging portfolio is given by
) 1 ( )
(F F C r t
X C
Ht+∆t =− t+∆t + s t+∆t − t + t + ∆ .
t
Ht+∆ is referred to as the hedging error over the rebalancing interval t∆ . These steps are repeated up to 7 days before the option’s maturity date. This will give the average dollar hedging errors, average absolute hedging errors for this hedging strategy as described.
This procedure tracks the hedging errors for one realization of the option being hedged. In order to perform empirical analysis, the procedure is repeated for every 7 days in the sample period and each repeat represents a realization of the sample path. Average dollar and absolute errors are calculated for each model through these hedging errors, and the results are reported in Tables 3 and 4.
Table 3 reports the hedging errors when the target options are up-and-out call options. The barrier level of each barrier option is set equal to 1.1 times of the underlying
futures price, while strikes are set equal to 0.94, 0.96, 0.98, 1.00, 1.02, 1.04, and1.06 times of underlying futures price, respectively. The hedging portfolios are rebalanced daily up to 7 days before the maturity date of the hedged options.
Several observations can be made from the average dollar hedging errors in Table 3. First, the Black-Scholes model outperforms the other three alternative option-pricing models for hedging out-of-the-money barrier options for all maturities. The jump diffusion model, on the other hand, performs the best for hedging short-term in-the-money barrier options, and the stochastic volatility model performs the best for hedging long-term in-the-money barrier options. Second, the average hedging errors increase as maturities of barrier options increase for any given model. Based on the average absolute hedging errors, this maturity effect becomes much more significant. Note that the prices of barrier options may decrease as the maturities increase; this result indicates that the hedging performance (relative to the option value) is quite poor for long-term barrier options for any given model. This confirms the result noted by Hull and Suo (2002) that model performance depends on the degree of path dependence of the exotic option. For a barrier option, the probability of hitting the barrier becomes relatively large when the time to maturity increases, and therefore the knockout feature becomes more important. To reconfirm this conclusion, we repeat the above hedging procedure for barrier options whose barriers are closer to the underlying futures price (e.g. the barrier is set equal to 1.05 times of the futures price).8 The hedging errors, as we expected, are larger in this case than their counterparts in Table 3. Based on the sizes of the absolute hedging errors, however, the Black-Scholes model yields the smallest sizes of hedging errors for almost all of the options considered indicating that the hedging performance of the Black-Scholes model is quite stable.
8 The results are not reported here.
Table 4 reports the hedging errors for compound options. The compound option considered is the call on call option. The underlying call option is a futures option with 60 days to expiration. The strike of the underlying call option is set as the underlying futures price. Strikes of the call-on-call options are set equal to 1.5, 6.0, 10.5, 15.0, 19.5, and 24.0, respectively.
Based on the dollar errors and the absolute hedging errors in Table 4, the stochastic volatility model and the jump diffusion model generally perform better than the Black-Scholes model and the CEV model for hedging most of the compound options.
These findings are in line with those in the current literature.
Another observation is that the average dollar and absolute hedging errors relative to the values of the exotic options being hedged for a given model do not change much when the maturities of the target options increase. Intuitively, increasing the maturity of a compound option does not affect the importance of its exotic feature, or the degree of path dependence. As a result, the hedging performance of a compound option changes very mildly when its maturity increases.
For any given model, we can also see that the relative performance for hedging the short-term barrier options is better than that for hedging the short-term compound options, while the performance for hedging the long-term barrier options is poorer than that for hedging the long-term compound options. For the Black-Scholes model, for example, the average dollar hedging errors relative to the target option values are from 0.07% to 1.2% for short-term barrier options, while they are from 0.23% to 1.2% for short-term compound options. The relative hedging errors are from 0.5% to 1.5% for long-term barrier options while they are from 0.23%to 0.56% for long-term compound options. This evidence becomes much pronounced in terms of relative absolute hedging errors.