E ff ect of Delta Limit

Recall that the Delta limit,Nlimwas imposed on the initial hedging option employed in Portfolio

II. This was put in place to insure that the hedge ratio would not “blow up” thus requiring that the investor take an infinite position in an option. We are interested to determine what effect, if any, the magnitude of this Delta limit has on the mean profit/loss of our trading strategies and on the percentage of time this limit is actually breached over the course of trading.

0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 ∆₂ Limit

Mean % of Trading Days with Limit Breaches

Figure 6.3: Effect of∆2limit on mean % of trading days with limit breaches.µ=0%, all other

parameters as given in Table 6.1.

It was found that the overall percentage of limit breaches was independent of both stock price drift and on which option we initially hedged with. We only show results for one choice

0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 ∆₃ Limit

Mean % of Trading Days with Limit Breaches

Figure 6.4: Effect of∆3limit on mean % of trading days with limit breaches.µ=0%, all other

parameters as given in Table 6.1.

of stock drift,µ=0%, however similar results follow for other market conditions. Figures 6.3 and 6.4 show that when the limit is a fraction of an option, the limit is breached between 50% and 90% of the time. There is an exponential decay that can be observed in the percentage of limit breaches as the Delta limit increases. No observable change is observed for values above

Nlim = 4 as limit breaches become non-existent. It should be noted that almost identical results

are observed for when we start hedging Portfolio II with eitherCi

2(S,t) orC

i

3(S,t).

Finally, we investigated the effect that the Delta limit has on the mean profit/loss of Portfolio II. The same effects for increases in Delta limit were observed for all market conditions. Figure 6.5 illustrates the effect that the Delta limit has on the mean profit/loss of Portfolio I when we choose to hedge withCi

2(S,t) such thatK1 <K2. The mean profit/loss and corresponding 95%

confidence intervals decrease with respect to the ∆2 limit. As the hedge ratio limit increases,

the mean profit/loss decreases by a magnitude of a few tenths of cents. This decrease appears to flatten out for larger values of the limit. When∆2limit take on the value of zero, financially

this means that we will only be hedging with our additional hedging option Ci₃(S,t). Figure 6.5 indicates that hedging with our additional hedging option produces maximum portfolio performance.

Similarly, Figure 6.6 shows the effect that the Delta limit has on the mean profit/loss of Portfolio II when we choose to hedge withCi₃(S,t) such that K3 < K1. Increases in the Delta

limit are shown to cause increases in our mean profit/loss. This difference in profit/loss is only a few cents in comparison and this begins to flatten out over time. There is a peak in the mean profit/loss which occurs between Nlim = 1 andNlim = 2. This suggests that for this

portfolio, there is an optimal limit to impose to maximize our portfolio performance. When the limit takes on the value of zero, this is a scenario in which we would only be hedging with

6.5. HedgingStrategiesAnalysis 113 0 1 2 3 4 5 6 7 8 9 10 −0.01 −0.005 0 0.005 0.01 0.015 0.02 ∆₂ Limit Profit/Loss ($) Mean 95% Confidence Interval

Figure 6.5: Effect of∆2 limit on profit/loss of Portfolio II. µ = 0%, all other parameters as

given in Table 6.1.

our additional option,Ci

2(S,t). Figure 6.6 indicates that this scenario would lead to our worst

portfolio performance for all values of the hedge ratio limit.

In general, larger values for our Delta limit will allow us to take larger positions in our first hedging option. If this limit is breached, we must introduce a new hedging option which will incur additional costs. For the portfolio starting withCi₂(S,t), when the limit is breached, we must also start hedging withC₃i(S,t). Recall that it was assumed that K3 < K1 < K2. All else

being equal, it turns out thatCi

3(S,t) > C

i

2(S,t). However, given that all hedging options are

in-the-money, it will always be cheaper at maturity to exerciseCi₃(S,t) to buy one share of the stock for the low price ofK3. Therefore keeping some dynamic position in this option over the

life of the portfolio is cheaper in the long run, rather than trying to set up a large position in it closer to maturity. Thus increasing the hedging ratio limit such that we will use our original hedging optionCi₂(S,t) will result in a lower terminal portfolio profit. On the other hand, for the portfolio in which we begin hedging with Ci

3(S,t), our initial hedging option is the more

expensive option of the two available to us. However since this option is more likely to finish in-the-money, making it the most beneficial hedging instrument in our portfolio, it is advisable to maintain a dynamic position in it for longer through a larger hedging limit. If our hedging limit is too low, our dynamic position would be in the other optionCi₂(S,t), which due to the set-up of the portfolio with respect to the strike prices, is less likely to finish in-the-money. This makes it a less desirable hedging option in the long run since we want an option in our portfolio which enables us, if necessary, to buy one share of the stock at maturity for the cheapest price

0 1 2 3 4 5 6 7 8 9 10 −0.01 −0.005 0 0.005 0.01 0.015 0.02 ∆₃ Limit Profit/Loss ($) Mean 95% Confidence Interval

Figure 6.6: Effect of∆3 limit on profit/loss of Portfolio II. µ = 0%, all other parameters as

given in Table 6.1.

in order to close out our short position.

Using the results illustrated in Figures 6.3 and 6.5 together, we can observe that it is most profitable to have a hedge limit as low as possible forCi

2(S,t), in order to ensure that we will

hedge with our option that has a higher probability of finishing the money. On the other hand, the results of Figures 6.4 and 6.6, indicate that there is an optimal Delta limit which would maximize our profit. This limit would ensure that we always have a position in our original option Ci

3(S,t), whether dynamic or static, which has a higher probability of being in-the-

money at maturity. The additional option would provide some added value in hedging against the inherent risks, while minimizing trading costs associated with our higher valued, original hedging option.