hi i am looking at question 4.10(iii) from the Q&A and wanted to check that my understanding of the solution is correct. The question says a trader has a short position in 100 Calls and 100 Puts on future contracts. The delta of her position is +10 and she has therefore sold short 10 future contracts, to delta hedge her position. In part (iii) you are asked to describe the effect on the trader's profit and loss of the market falling sharply in the next few days. The solution describes how his "portfolio" will probably expected to fall in value, since the sharp fall would be in excess of the volatility assumed in the market prices. This seems a little odd to me. She is delta hedged, so the fall will not result in direct losses. To suggest the market is underestimating volatility seems rather speculative, I would imagine that post LTCM, traders are using rather more sophisticated valuation methods... Finally, in part (iv), you are asked if your solution to (iii) would differ in the fall occured a week before expiry. This states the final loss will be greater because the fall would directly affect the intrinsic value. BUT are we not forgetting Mrs Trader is delta hedged?? She should not experience a loss, so long as she dynamically hedges throughout these turbulant "few days". Or am I missing something here? Thx.
ST6 Q&A 4.10(iii) I'll explain the ActEd solution, but apologies in advance if I'm looking at a different version of the question - the trader in my version is male, but apart from this your comments seem to tie up with the question in front of me. Incidentally, the question comes from the original specimen exam for CiD, and the profession's solution is consistent with ours (if that gives you any comfort). There are 2 points to make here: 1. The question tells us the trader has hedged his position at the current market level. This suggests a one-off action with no implication that an ongoing dynamic hedging strategy is in place. So the traders' total (hedged) portfolio value is well protected against small, immediate changes in the FTSE 100. However, we now have sharp falls over a period of a few days, so the value of the portfolio will be affected by both gamma and theta. Shorting 200 options will have caused gamma to be fairly large and negative and the portfolio value is likely to fall regardless of the direction of the large movement in the underlying (see Chapter 8, page 6). 2. We don't need to assume the trader used a poor (unsophisticated) estimate of volatility. It could have been based on very sound techniques and endorsed by every other trader in the market - but even if the estimate is the genuinely real best theoretical estimate, it will still be too low half the time and too high half the time. This time, we're saying it turned out to be too low, given the benefit of hindsight, but the trader wasn't to know there'd be sharp market movements in the next few days. It just happens sometimes. Aha, that's the key. Yes, continuous dynamic hedging would theoretically protect the trader from losses, but that's not the situation described here. The hedge described in the question is a static (one-off) hedge. So the protection is lessened the moment the FTSE 100 moves away from its level at the time of the hedge. I hope that helps - it sounds like it's just a misinterpretation of the question rather than a fundamental misunderstanding of the course material. So you just need to remember in future to look to see if we're dealing with a static or dynamic hedging strategy. Best wishes for your continuing studies.
Mike, thanks for clearing that up. Your comments have raised some other questions for me, which I will outline here. 1) How can we quantify whether volatility has been over- or under-estimated after a shock event? In my limited view, I would say that volatility is a parameter of my model for stock prices and I should determine whether it has been correctly estimated by considering if observed values of stock prices lie reasonably close to the mean. E.g. say I assume log-normality: then post-event, I might consider if whether for my assumed value of sigma, the observed value of the stock prices, lies within a 95% confidence interval for S_t. If it does, the I would conclude that I did not under-estimate volatility under my model of stock prices. 2) If using the method in 1) I conclude my estimate of volatility lies within the range I would expect to see in normal circumstances, does this mean I am safe using this for delta hedging? I suspect that it would depend whether my model of option prices is consistent with the market's view of prices. E.g. even if I am 99% sure of my volatility estimate, I could still face model error, and thus I would not really be hedged - e.g. market prices of options could move differently to how my model predicted (perhaps due to irrational investors, or other events). 3) Let's suppose my model is good and my volatility estimate is good. This should mean my static delta hedge will protect me for a short time period - but how short? If prices in a day are steady and then a shock event occurs the price will jump very quickly. Perhaps in 10 minutes we may see a 20% change in the value of the underlying stock. At what point does the hedge run out? My feeling is that we are protected for as long as delta stays constant. So if we have a low value of Gamma, then our static hedge will protect the portfolio for some time. Am I right to say that in an ideal portfolio we would hedge gamma and delta to zero, and we could sit tight without rehedging through a shock event, and make no loss? Or would we find higher derivatives of the option price with respect to the stock price, e.g. cause gamma to change during this turbulent period, thus requiring dynamic hedging? I know I am asking a bit of detail here, but I have read Hull, Wilmott and Baxter and these do not really address the practical side of the subject. Thanks, Gareth
Hello Gareth, Delta hedging is undertaken to protect you from small movements in the price of the underlying, while gamma hedging is done to protect you from large movements in the value of the underlying. To delta hedge is it necessary to simply take the necessary position in the underlying (which is well documented in the notes), however the second derivative of the value underlying with respect to the value of the underlying is zero (also true for forwards/futures). Thus taking a position in the underlying (or forwards/futures) will not affect the gamma of the portfolio. If you do not understand this, consider holding the underlying as the only security in your portfolio. Your portfolio has a delta of +1 as the value of your portfolio increases by one times whatever the change in the value of the underlying is. However, no matter how much the value of the underlying changes, it is still only going to give you a delta of +1, so its gamma must be zero as delta does not change with the price of the underlying. It is theoretically possible to be both delta and gamma neutral using options (or other derivative that is not linearly dependent on the underlying asset) however traders in the real world do not often aim for this. If you want to protect against small movements in the value of the underlying you delta hedge, to protect against large movements, it is necessary to gamma hedge. However you must consider that in practice it will not always feasible to find non-linear derivatives with the required liquidity, price, availability and maturity date to make a large portfolio gamma neutral. As it states in Hull, most traders will attempt to keep zero, or near zero, delta portfolios, however they will only monitor gamma and take corrective action if things get out of control. Regards other points you have made, you are delta hedged as long as the delta of the portfolio is zero, and you cease to be delta hedged when the delta of the portfolio is no longer zero. I do not see the advantage of trying to quantify the length of time your portfolio will continue to have a delta of zero. Theoretically, the time will depend on the gamma and on sufficient movements in the price of the underlying causing the delta to change. About volatility, do not forget that most pricing models, and thus calculation of delta and gamma, assume constant volatility. Best wishes, Brian
