Hi all, I have another questions on option greeks. I find out that some put options' gamma is negative. Here is how I define gamma:Gamma = (Delta at S*(1.01) - Delta at S*0.99)/2 . i.e. the average change of my delta per 1% change in index. Delta at S is calculated by (Option Price at S*1.01 - Option price at S*0.99)/2. I do capture a vol surface so any change in stock price would result in a slight change in vol. I have checked the calculation a lot of times and doesn't seem like I make a mistake. It seems like the vol surface impact is quite large which dominates the decrease in delta when stock prices decrease. If I replace the whole vol surface by a flat vol surface, the negative gamma goes away. Could anyone tell me what does negative gamma mean? Any flaw in my methodology? Thanks!

Check out the difference between a total derivative and a partial derivative.

Substituting in your definition,Gamma = P(1.0201*S)/4 - P(0.9999*S)/2 + P(0.9801*S)/4where P is the price of the put option when the underlying equals the argument.Assume first that the underlying at expiry can only be either a or b, and let K be the strike price of the put, a >= K >= b, and ignore dividends and interest. The put is then the same as -(K - b) / (a - b) shares of stock plus (K - b)*a / (a - b) cash.P(X) = (a - X)*(K - b) / (a - b)Substituting that into your Gamma formula gives:Gamma = -(K - b)*S*0.00005 / (a - b) <= 0So gamma is always negative (you say positive, because you're really defining Delta as the opposite of your definition). This result will be true as long as you have a binomial model, that is as long as the price can move to only two possible other prices at any given time. In the continuous time case, this means you have a local volatility, including the special case of constant volatility, model.To get a positive gamma by this definition, you need to have three potential outcomes, which corresponds in the continuous case to a stochastic volatility model, or any other model that cannot be described by local volatility. Suppose, for example, that from either 1.01*S or 0.9801*S the underlying price either doubles or goes to zero, both of which are absorbing states. From 0.9999*S, the price either goes to 1.01*S or 0.9801*S immediately, or stays at 0.9999*S forever.Now P(1.0201*S) = P(0.9801*S) = K/2, while P(0.9999*S) is somewhere between K/2 and Max(K - 0.9999*S,0), call it K/2 - c. Now your definition of gamma reduces to c/2, which is positive. The key here is that the average volatility is higher at 1.0201*S and 0.9801*S than at 0.9999*S.