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Modelling and Hedging Equity Derivatives
Thus for hedging applications that are subject to model risk or basis risk, an IR hedge tends to improve on delta hedging, something that other hedging schemes, such as gamma and vega, do not attain. These latter results do not generally depend on the basis risk present in the hedging applications.
Table 3 compares the six hedging schemes under four scenarios of rolling hedges and several conclusions can be drawn from these investigations. The longer maturity hedging instruments Scenario 4 consistently provide a better hedging performance across all hedging schemes compared to the shorter maturity hedging instruments e.
This is to be expected, since Scenario 1 requires the hedge to be rolled over most frequently, introducing the most basis risk. With the shorter maturity futures contracts, the improvement relative to the unhedged position across the different schemes ranges between With the longer maturity futures contracts, there is a larger improvement across all schemes, ranging between The improvement in the hedging performance also holds over the three subperiods. As expected, over the more volatile periods, the hedges are less effective, yet the magnitude of improvement between Scenarios 4 and 1 is far more substantial compare for example the last two panels of Table 3.
Thus, in volatile market environments, hedges which need to be rolled over less frequently outperform frequently rolling hedges by more than 10 percentage points, while in low volatility environments, this distinction has little impact on the performance of the hedge. As noted in Section 4. Consequently, part of the basis risk is managed by hedging the interest rate risk.
This is in line with what would be expected theoretically, since the difference between futures prices for different maturities is in part due to interest rates though convenience yields would also play a role. Again, this is an indication that the presence of basis risk magnifies the model risk inherent in vega and gamma hedges, thus in some cases the inclusion of a vega or gamma hedge in addition to delta and IR hedges may actually be detrimental to hedge performance.
The extent to which the modelling of stochastic interest rates results in improved hedges is investigated next, even if interest rate risk is not hedged. Table 4 displays the percentage improvement of hedging errors over the unhedged positions between models with stochastic interest rates and deterministic interest rates, for a range of hedging schemes and under the four scenarios of rolling hedges.
Delta, vega, and gamma hedges are considered, and the option position is hedged using hedge ratios from stochastic interest rate and deterministic interest model specifications. Figure 4 compares the RMSEs of portfolios assuming stochastic interest rates and deterministic interest rates, under Scenarios 1 and 4 representing scenarios with high and low basis risk, respectively. These plots compare the average RMSEs of hedge portfolios constructed assuming stochastic interest rates and deterministic interest rates, under Scenario 1 top and Scenario 4 bottom [Color figure can be viewed at wileyonlinelibrary.
One sees that using hedge ratios from the stochastic interest rate model consistently improves hedging performance compared to the deterministic interest rate model for all hedging schemes and scenarios, even when interest rate risk is not hedged. When considering the improvement over the entire period, from July to October see top panel of Table 4 , there are marginal differences between hedging based on stochastic and deterministic interest rate models. However, when comparing the improvement over the three subperiods, there are noticeable differences.
As expected, hedging with shorter maturity contracts Scenarios 1, 2, and 3 yields less efficient hedges, predominantly due to the additional basis risk involved, but the improvement resulting from modelling stochastic interest rates is more marked. Thus using hedge ratios from stochastic interest rate models in periods of stable market conditions in particular low interest rate volatility does not provide any advantage, but, importantly, when interest rate volatility is high, it is better to construct hedges based on a model incorporating stochastic interest rates—even if that interest rate risk is not hedged.
As expected theoretically, using hedging instruments with maturities that are closer to the maturity of the option reduces the hedging error. Consequently, part of the basis risk is managed via the hedging of interest rate risk. When hedging is carried out with shorter maturity hedging instruments and over periods of high interest rate volatility, adding an IR hedge to a delta hedge improves hedging performance, while adding a gamma or vega hedge to the delta hedge can make the hedge less effective.
This is a key conclusion: When we have more exposure to model risk due to turbulent market conditions and basis risk due to a mismatch between the maturity of the option to be hedged and the hedge instruments , an IR hedge combined with a delta hedge can outperform all other hedging schemes. Thus, when interest rate volatility is high, there is an advantage in modelling interest rate risk when determining any hedges, even if one chooses not to hedge this risk.
The current interest rate environment in most developed economies is extremely benign both in terms of interest rate levels and volatility by historical standards, but one should not expect this situation to persist indefinitely. Australian Research Council financial support DP is gratefully acknowledged. The usual disclaimer applies. Let be the index of the Monte Carlo simulation with 1, iterations i.
We note that the 4th of July, is a public holiday Independence Day in the United States, so represents the trading day of the 5th of July, Using these samples and following the steps outlined in Section 3 , we can construct a hedging portfolio for each Monte Carlo iteration , with hedge ratios , making explicit the dependence on the iteration. Then the average of the RMSE over 1, simulations for each hedging scheme is computed, for each of the scenarios.
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Modelling and Hedging Equity Derivatives | eBay
Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. This volatility process evolves as 1. Then, the change in the price of the target option is determined by the difference 8.
Thus, the change in the price of the futures options due to a shock in the volatility can be determined by In this case, the forward interest rate curve is expressed as For times to maturity of up to 3 years, we use June and December contracts, while for times to maturity of more than 3 years, only December contracts are sufficiently liquid. Each scenario uses three futures contracts with consecutive maturities, and the scenarios differ by the time to maturity of the contract used. The shortest times to maturity in Scenarios 1, 2, 3, and 4 are 2, 14, 26, and 38 months, respectively.
For instance, in Scenario 4 and assuming delta hedging, a call option on futures is hedged on the 3 July by using three futures contracts with maturities of December , December , and December June contracts are not included here because on 3 July , futures with a maturity of June are not available. Learn on your own schedule, where and when you can, on your computer or tablet.
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