 # TESTING HEDGING STRATEGIES

My discussion thus far focused on replicating risks arising from the options sold. In doing this, I briefly touched on the tradeoff between transaction costs FIGURE 7.1 Impact of Trading Frequency on Standardized Hedge P&L Standard Deviation

and hedging errors. Succinctly stated, the more frequent the hedging, the less volatile – albeit higher – the hedging costs. On the contrary, the less frequent the hedging, the more volatile – albeit lower – the hedging costs. In this section, I will discuss a couple of practical ways to evaluate the effectiveness of hedging strategies.

Assuming that there are no transaction costs, one can rerun the deltahedging analysis presented in Tables 7.7 and 7.8 to arrive at Figure 7.1.

As can be seen from Figure 7.1, the bigger the time between rebalancing, the greater the standard deviation in the simulated standardized hedge P&L. As a consequence, one would like to rebalance as frequently as possible so as to reduce the noise in the simulated standardized hedge P&L. Doing this will be ineffective if there were transaction costs associated with the hedging program. Rerunning the analysis that resulted in Figure 7.1 across varying transaction costs yields the results in Table 7.11.

TABLE 7.11 Effect of Transaction Costs on Rebalancing Intervals

 Rebalancing Interval (days) Transaction Costs 0% 2% 4% Average Std Deviation Average Std Deviation Average Std Deviation 1 0.00 0.04 1.30 0.45 2.59 0.89 2 0.00 0.05 0.95 0.32 1.91 0.64 3 0.00 0.06 0.80 0.27 1.61 0.53 4 0.00 0.07 0.72 0.25 1.43 0.47 5 0.00 0.08 0.66 0.23 1.31 0.43 6 0.00 0.09 0.61 0.22 1.22 0.39 7 0.00 0.09 0.58 0.20 1.15 0.37 8 0.00 0.10 0.55 0.20 1.10 0.36 9 0.00 0.10 0.53 0.20 1.05 0.34 10 0.00 0.11 0.51 0.20 1.01 0.34 11 0.00 0.12 0.49 0.20 0.97 0.32 12 0.00 0.12 0.48 0.20 0.95 0.32 13 0.00 0.12 0.46 0.20 0.92 0.31 14 0.00 0.13 0.45 0.20 0.90 0.31 15 0.00 0.13 0.44 0.19 0.88 0.30 20 0.00 0.15 0.40 0.20 0.80 0.29 30 0.01 0.18 0.36 0.22 0.72 0.28 40 0.01 0.21 0.33 0.24 0.66 0.29 50 0.00 0.23 0.31 0.25 0.62 0.30 60 0.01 0.25 0.30 0.27 0.60 0.31 70 0.00 0.27 0.29 0.29 0.57 0.32 80 0.00 0.29 0.28 0.30 0.55 0.34

As can be seen from Table 7.11, when there are no transaction costs, the average hedge P&L associated with the delta-hedging program using 2,500 paths is 0 (as expected). Consistent with Figure 7.1, the standard deviation of the standardized hedge P&L increased from 0.04 to 0.29 in the absence of transaction costs. As the transaction costs increased, the average standardized hedge P&L also increased by roughly the same amount (i.e., a two-fold increase in transaction costs resulted in approximately a two-fold increase in the average standardized hedge P&L). Not surprisingly the standard deviation of these standardized hedge P&L also increased as transaction costs increased – illustrating the fact that the more frequent the rebalancing, the bigger the increase in standard deviation. Based on the analysis done, for a given transaction-cost level, the hedger can now decide (using the table) on how much fluctuation the hedger can stomach in the realized hedge

P&L – in the process making a decision on how frequently the hedger has to rebalance. It is important for the reader to understand that for a given transaction cost, it is not unusual to have two hedgers selling identical options and choosing different rebalancing frequencies simply due to the fact that each of them would have different appetites for risks given their level of capital, risk limits, etc.

In practice, instead of waiting once every day, or 10 days, or 50 days, and so on to rebalance their positions, practitioners tend to work with the notion of a delta-trading limit. More precisely, with this approach, hedgers only neutralize their deltas if the absolute value of the total net portfolio delta exceeds a certain pre-defined level. In this way, one can avoid hedging costs due to rollercoaster market movements (i.e., an upward movement followed by a downward movement or a downward movement followed by an upward movement) as long as the trading limits are not breached during such rollercoaster rides. Thus, a delta-trading limit of 0 would be akin to rebalancing continuously (as this would mean that any slippages in the portfolio delta need to be rebalanced immediately). Similarly, a very large delta-trading limit would imply an infrequent rebalancing of deltas. In taking this approach, one can replace all references to rebalancing frequency with portfolio-delta limits. One can also extend this philosophy to implement portfolio gamma, vega, and rho trading limits when hedging gamma, vega, and rho risks respectively.

In discussing the implementation of delta hedging and the impact of transaction costs on standard deviation associated with standardized hedge P&L, I pointed out that for the hedger to decide how best to run a delta hedging program, the hedger must decide how much standard deviation in standardized hedge P&L he/she is prepared to stomach for a given level of transaction costs – which is done in the guise of a delta-trading limit (or time interval between rebalancing periods or both). Once the delta-trading limit and the frequency of delta monitoring has been established, the hedger can now more readily compare different strategies by looking at both the average standardized hedge costs and the standard deviation of the simulated standardized hedge P&L. To understand this better, suppose for a given transaction-cost level and a specified delta-trading limit frequency of delta monitoring, the distribution of the hedge P&L associated with the use of stocks produced an average and standard deviation of 1.00 and 0.40 respectively. If the distribution of hedge P&L using another hedging strategy (e.g., using at-the-money options) with the same trading limits and frequency of

TABLE 7.12 Compa rison of Hedging Strategies for a Given Trading Limit Constraint

 Statistics from Delta Hedging Profit Distribution At-the-Money Options Standard Deviation Average 0.5 1 1.5 0.2 Better than using stocks. Better than using stocks. 0.4 Better than using stocks. No advantage over using stocks. Worse than using stocks. 0.6 Worse than using stocks. Worse than using stocks.

monitoring constraints produced an average and standard deviation of 1.00 and 0.2 respectively, then it is obvious that the strategy based on using at- the-money options is more effective than the one based on stocks. Table 7.12 better summarizes all possible scenarios and instances when one strategy dominates the other.

From Table 7.12, one can make the following four observations:

1. While the comments given for seven of the nine scenarios are intuitively reasonable, nothing can be concluded in the two instances (that are shaded in grey). In such an instance, one would need to look at other metrics to arrive at a decision.

2. Since I only ran 2,500 paths to produce the distribution of simulated hedge P&L for both the strategies, some allowance has to be made to allow for simulation errors that possibly arise from not running sufficient paths or biasedness in the sampling of the random numbers – as discussed in Chapter 4. This is despite the fact that the same stock price paths are used to compare the stock-based strategy against an at-the- money-option strategy. As a consequence, the effectiveness of one strategy over the other cannot be established with certainty if the average and the standard deviation associated with the hedge-P&L distributions are close to each other. To overcome this impasse, one can either run more simulations, or use different decision metrics, or do a combination of both.

3. I have kept the stock price volatility and risk-free rate constant thus far. In practice, these variables have their own distribution (and sometimes can even be correlated to stock price movements). To do a more rigorous analysis of the hedge-P&L distribution, one must test these strategies against movements in volatilities and risk-free rates so as to ensure that the results obtained are a more realistic indication of what can be achieved in practice – especially when vega and rho risks turn out to be the biggest risk drivers in the option sold.

4. In addition to simulating stock prices into the future, another sensibility check that is usually carried out in practice is the experience check. With this type of check, instead of solely relying on simulated stock prices, one would additionally use real-world scenarios (which may contain historical stock prices, risk-free rates, volatilities) to test the effectiveness of the hedging strategies. In doing so, one can get a better sense of how well or badly the strategies perform under market scenarios that are deemed catastrophic.

•  To arrive at data supporting the chart, I computed the hedge P&L using the delta- hedging example in Tables 7.7 and 7.8, standardized it (by dividing with option premium), and then calculated the standard deviation of the standardized-hedge P&L obtained by using 2,500 runs. To avoid any error introduced by the generation of random numbers, I used the same set of random numbers so as to ensure the same stock price path scenarios.
•  To put this table together, I assumed that transaction costs of 0, 2, and 4 percent of the stock price were incurred when buying and selling the shares during the deltahedging activity. For example, when the transaction cost of 2 percent was used, I simulated a mid-market stock price S and then assumed a stock price of 1.025 when purchasing shares and a stock price of 0.985 when selling the shares – all consistent with the mechanics presented in Table 7.9.
•  When the delta trading limit is breached, in order to neutralize the deltas, I have assumed that the hedger unwinds the existing option positions and then puts on new at-the-money option positions.
•  Examples of these metrics are the coefficient of variations (which is computed by dividing the standard deviation by the mean) or conditional tail expectations (which is computed by calculating the worst-case scenarios and taking the average of the selected worse-case scenarios).