I've got a question on Delta Hedging:How can one make money on continuously hedging the Delta of an option? I've "googled" this (i know, not a very sophisticated approach) and found this:"In fact, the amount by which a hedge has to be adjusted to stay delta neutral is related to gamma, the second derivative of the portfolio value with respect to the price of the asset in question. For example, if a position is 'long gamma', i.e., has a positive gamma, an increase in the asset price will lead to a positive delta, and one will need to sell some of the asset to 'flatten' the delta. Similarly, a decrease in asset price will cause one to buy more of the asset. From this it is intuitively clear that a high volatility of the underlying asset will lead to trading profits."To me this isn't intuitive. What's the downside? how can I lose money? ...don't get it...It seems quite conceptual and I would really appreciate it if someone could explain this to me.Thank You!

From a really simple level, you become long gamma from say, buying a vanilla option - you paid a premium to do this (unless you know somewhere that sells options for free - let me know!). Think about paying this premium in terms of buying vol - the vol you buy at (implied vol) is the 'expected' vol over the life of the option. Theoretically, if this turns out to be true, then through continuously delta hedging (buying low, selling high) you should make back the premium you paid at the outset. If realised vol turns out to be a lot higher than the implied vol you bought the option at, then you should be able to make back more than your initial premium. So gamma is lilke trading the realised vol. But of course, there's a little more than that involved in the real world... remember, the essence of hedging is to minimise p/l variance..

in addition to PS1980's point that the p/l of delta hedging and option largely comes from the actual, experienced vol of the asset vs the implied vol at which you bought the option, there are other, more mundane but very important factors.....first, it is impossible to trade ANY assect continuously.... besides the physical logisticcal issue, even the most liquid markets (G7 currencies and USD and EUR govt bonds and swaps) will have moments of illiquidity i which and event causes gappping moves.... so even if the experienced vol is exactly equal to the purchased implied vol, realistic market conditions can have a big influence on whether you make or lose $$$$

Thanks, you all, that has helped a lot.Let's say though, that I continuously hedge my delta through the life of the option, from what I understood, I should be making back the premium that I paid for it. if that is correct, than my option would be a "free" option, because at the end it didn't cost me anything, otehr than opportunity cost...

well let me reiterate that "continuous hegding" is a futile concept to consider as it cannot be done......but "free option" generally refers to a very very rare case in which one obtains an option without paying ANY premium..... in your example the trade is just a "push" as your delta hedging profit exactly offsets the premium, leavinging you no better off.....

QuoteOriginally posted by: PS1980...remember, the essence of hedging is to minimise p/l variance..Hmm. I disagree. In fact, I would say that a "real" trader is someone with very large PnL swings. Show me a guy with low variance of returns, and I will show you a guy who is on thin ice.

Does anyone have a simulation (in Excel) for the P&L of a put option?Thanks.

It's fairly straightforward to put one together. For guidance, see Hull, Table 14.2, or see Wilmott's book.

I'm not sure exactly what you mean but here's an approach.I assume you have already written an option pricer. If not, you have to start with that. I'm going to assume you've made it a formula so BS(S,E,v,r,t,P) gives the price of a put on an underlying with spot price S, exercise price E, volatility v, risk-free interest rate r, time to expiry t and "P" tells me it's a put. Delta(S,E,v,r,t,P) gives me the delta (expressed as a negative number for a put).Next you want to simulate a series of future price movements of the underlying. You might or might not want to assume that your pricing implied volatility was correct; or that the price follows the Black-Scholes assumptions. Assuming you want to see performance if the assumptions are correct (in which case, the P&L results from the fact that you hedge at discrete intervals rather than continuously), you could"Put 0 in cell A1, A1 + d in A2, where d is the time interval between hedge rebalancing, and copy it down until the value in the cell equals t. Or you could use nonconstant time intervals.Put S in cell b1, then in B2 S*exp(r*(A2-A1) + v*(A2-A1)^0.5*normsinv(rand())). Copy that down to the end.Put BS(S,E,v,r,t,P) in C1 (I assume you sold the put). In C2, put C1 + Delta(B1,E,v,r,t,P)*(C2 - C1) and copy that down to the end. This assumes you do a naive hedge and don't update your parameters during the life of the option.The last number in C is you P&L from selling the put and hedging it. Subtract Max(0,Bn - E) for the payout you have to make on the put you sold. The result is your profit (loss if negative) from selling the put and hedging it.

Hi vienneseblues,I once did an exercise like the one Aaron mentioned. This is what I found - might be wrong ofcourse but interested to find out.In the case of +ve gamma on a long call:if the underlying price goes up the option price goes up by more than the constant delta factor (that little bit more is the second order gamma effect) in order to stay delta neutral you need to sell some underlying (you have too much of it according to your own delta hedging strategy)if the underlying price goes down the option price doesn't go down quite as much as the delta factor would suggest. Again according to your strategy you need to buy more underlying. This is a nice situation for a trader. In order to maintain delta neutrality they have to sell after the underlying after price rises and buy just after its fallen! They make money on both rebalancings. Thats due to those "little bit extras" that the option price is "off by" due to +ve gamma. Those "extra bits" are bigger the further the underlying price moves in either direction. Therefore you can make lots of money if there is a lot of volatility. I suppose this is just a long winded way of saying the same thing as your original quote from google. The downside with options is time decay. From the day you have them in your book the damn things start decaying in value ( all other things being constant ). +ve gamma is your compensation for having time decay. I think.

That's correct. If the actual volatility measured at the actual rebalancing points is equal to the volatility used to price the option, the hedger will break even. If the volatility is higher, the option holder who hedges will make money. If the volatility is lower, the option seller who hedges will make money.

Suppose you start with an arbitrary spread of several short and long legs on the same underlying. Can one give a precise mathematical statement of the conditions needed for continuous delta hedging to guaranteee a profit at first expiration of an option in the spread? If not, for what spreads can we give such a condition? Obviously, I am asking a mathematical rather than a practical question. Thanks.

The Black-Scholes assumptions are sufficient to guarantee a profit, assuming you enter into the positions at better than Black-Scholes prices. You can weaken the assumptions considerably, especially if you build in a lot of profit at the start and also if you have a position in which is gamma and theta neutral.

Aaron, you said, in effect, below that the option seller/hedger makes money if the IV of the option, at his/her opening transaction, is greater than the volatility at the rebalancing points and the option buyer/hedger makes money if the IV at his/her opening transaction is lower than the volatility at the rebalancing points. I don't see then, what you can say about an arbitrary spread with both long and short positions in it. Please explicitly give the condition that it will be profitable and a reference. I can't find anything about the mathematics of spreads in options textbooks. Thanks.

I do not seem to understand hedging completely... In the book I am reading, it says, "Note that the pricing formula provides the derivatives firm with an estimate of the expected hedging costs but it does not take the desired orofit margin into account. Profit has to be added separately"In continuous time (not trans. costs) hedging costs are equal to the price of a derivative...so what's the point of hedging if it costs as much as the derivative's price. Also, how do you "add the profit separately"?which book explains it the best??