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12

Gamma is the second partial derivative of the change in the price of the option wrt to the change in the underlying. Said another way, it is the change in delta. If you write down the Black-Scholes pricing formula, you's see the gamma term: $$...\frac{1}{2}\frac{\partial^2C}{\partial C^2}(\Delta S)^2...$$ Notice that the $\Delta S$ (change in stock price) ...


3

At a high level, just look at the delta. If it's so close to zero that it won't shift the price of the option by a penny, then you could say, "the option no long responds to the price of the underlying" Any price it has more or less a function of theta and vega only. In practice however, depending on the model you use, delta has some volatility input. To ...


2

The problem with your formula is the equation sign $=$. The second order finite difference is only an approximation to the true gamma: $$ f^{\prime \prime}(x) \approx \frac{f(x+h)-2f(x)+f(x-h)}{h^2}. $$ $h$ can not be a result. Ideally, it should be small (whatever that means), so your original choice of $1\text{bp}$ seems appropriate for this ...


2

I think what you are missing is simply the Vega-Gamma relation in the Black-Scholes model. Namely: $$ Vega = \frac{\partial v}{\partial \sigma} = \sigma(T-t)S^2 \frac{\partial^2 v}{\partial S^2} = \sigma \tau S^2 \Gamma $$ Plugging this into your coverage error, you get its expression in terms of the Vega which is the most natural measurement of your ...


1

I'd say that the shock size depends on the situation/asset. If your model produces somewhat noisy PVs, it is advisable to use a slightly larger $h$ to avoid numerical issues. You may also want to base your decision on empirical hedging performance. This may or may not help, but most bond index providers (Citi/Barclays) use a shock size of 25bp when reporting ...


1

I, personally, like to see gamma as change in dollar delta per percent (most systems have it as "GammaP"). This way, it's much easier to think about you delta position as the market is moving around. The number above is the BS gamma which is an unscaled 2nd derivative of delta. You need to rescale it to get gammap (delta change per percent). In general, it ...


1

Yes, if the two rates belong to two different currencies having different yield curves. Or in fact any two indices (bases) having different yield curves. e.g OIS vs LIBOR or LIBOR vs UST etc.



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