Greeks and options hedging

Why is it that theta is sometimes taken as the proxy for gamma of the underlying asset in options hedging?

• For a delta neutral portfolio, the following equation holds $\Theta+\frac{1}{2} \sigma^2 S^2 \Gamma = r \Pi$. From this if know Theta you can calculate Gamma (or vice versa). – Alex C Jan 30 at 21:34

I can argue your case as follows, consider a portfolio such that The value of $$\Pi$$ of a portfolio satisfies the differential equation given by: $$\frac{\delta \Pi}{\delta t}+rS\frac{\delta \Pi}{\delta S}+\frac{1}{2}\sigma^{2}S^{2}\frac{\delta^{2}\Pi}{\delta S^{2}}=r\Pi$$ From the differential equation, $$\Theta=\frac{\delta \Pi}{\delta t}$$ $$\Delta=\frac{\delta \Pi}{\delta S}$$ $$\Gamma=\frac{\delta^{2} \Pi}{\delta S^{2}}$$ substituting the above to our differential equation we shall have: $$\Theta + rS\Delta+\frac{1}{2}\sigma^{2}S^{2}\Gamma=r\Pi$$ We know that for a delta-neutral portfolio, $$\Delta=0$$, thus we can write the equatio as $$\Theta+\frac{1}{2}\sigma^{2}S^{2}\Gamma=r\Pi$$ From the last equation, we note that when Gamma is large and positive, the theta of the portfolio tends to be large and negative, this explains why theta can be regarded as gamma proxy strictly in delta-neutral portfolio not all scenarios.

• This is the correct answer. Theta and Gamma are related only for delta neutral portfolios absent interest rates, dividends or repo. – Ezy Feb 5 at 8:55

I dont think that people would usually use one as the substitute for the other, as:

$$\theta/\Gamma=-\frac{S^{2}\sigma^{2}}{2}$$

which is arrived upon by neglecting the terms of the formula for $$\theta$$, which are preceded by the interest rate $$r$$. I think the background to your question stems from the fact that option market practitioners will consider theta and gamma as essentially the same thing - decay ($$\theta$$) occurs, where there is convexity ($$\Gamma$$).

• Option market practitionners do not consider gamma and thera « essentially the same thing » – Ezy Feb 5 at 8:37
• ok - meant to say decay is a consequence of convexity – ZRH Feb 5 at 8:46
• Deep in-the-money european call is still convex but has positive theta quant.stackexchange.com/questions/42611/… – Ezy Feb 5 at 8:52
• I’m an options market practitioner. I don’t care about the $rPI$ term because I run a funding neutral portfolio, so that term cancels versus the interest I have to pay to run my books. – dm63 Mar 31 at 11:41