Premium-Adjusted Delta and Gamma for Quanto Options

# Price adjustment of Black-Scholes delta and gamma for a quanto option


# Price adjustment of Black-Scholes delta and gamma for a quanto option












A quanto option is a derivative with the underlying and strike price denominated in one currency, but the instrument itself is settled in another currency. This has consequences for the calculation of the greeks.

The `BS delta` measures the rate of change of the option price relative to the change of underlying price. `BS gamma` measures the rate of change of BS delta relative to the change of underlying price

A price-adjusted delta (`PA delta`) measures the rate of change of the option price (in settlement currency) relative to the percentage change of the underlying price. `PA gamma` measures the rate of change of PA delta relative to the percentage change of the underlying price

According to this link the difference between the `PA delta` and `BS delta` is the price of the option (in BTC). My interpretation (with interest rate=0, USD and BTC as currencies):

Is it also possible to determine the difference between `BS gamma` and `PA gamma`?

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/53858

Note that \begin{align*} Call_{\rm BTC}=\frac{1}{S}Call_{\rm USD}. \end{align*} The premium adjusted delta $Delta_{PA}$ is defined as the change of $Call_{\rm BTC}$ with respect to the change of the spot in BTC, that is, \begin{align*} Delta_{PA} &= \lim_{\Delta S\rightarrow 0}\frac{\Delta Call_{\rm BTC}}{\frac{\Delta S}{S}}\\ &=\lim_{\Delta S\rightarrow 0}\frac{\Delta Call_{\rm USD}}{\Delta S} - \frac{1}{S}Call_{\rm USD}\\ &=Delta_{BS} - Call_{\rm BTC}. \end{align*} The premium adjusted gamma $Gamma_{PA}$ is defined as the change of $Delta_{PA}$ with respect to the change of the spot, that is, \begin{align*} Gamma_{PA} &= \lim_{\Delta S\rightarrow 0}\frac{\Delta Delta_{PA}}{\Delta S}\\ &=Gamma_{BS} +\frac{1}{S^2}Call_{\rm USD}-\frac{1}{S} \lim_{\Delta S\rightarrow 0}\frac{\Delta Call_{\rm USD}}{\Delta S}\\ &=Gamma_{BS} + \frac{1}{S}Call_{\rm BTC} -\frac{1}{S}Delta_{BS}\\ &=Gamma_{BS} - \frac{1}{S}Delta_{PA}. \end{align*} See also Page 19 of this paper; however, there are some typos there.

The purpose of such definitions are to maintain the units. For example, the delta is always in units of BTC, while the gamma is in units of $({\rm BTC} \times {\rm BTC})/{\rm USD}$.