Why Delta-Neutral Option Portfolios Link Theta and Gamma

# Greeks and options hedging


# Greeks and options hedging












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

## Answer by Nick Mugisha (score 4)

https://quant.stackexchange.com/a/43825

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.

## Answer by ZRH (score 2)

https://quant.stackexchange.com/a/43805

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 ($\Gamma$).