Compo Feature in Asian Option on Futures

I'm pricing an Asian option on futures using Turnbull–Wakeman (other suggestions welcome) where the average is defined as $A _ { t _ { 1 } , t _ { n } } ^ { A , f } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } f _ { t _ { i } , \vec { T } \left( t _ { i } \right) }$ over a strip of futures where each is described by the SDE $$d f _ { t , T _ { k } } = \sigma _ { k } f _ { t , T _ { k } } d W _ { t } ^ { k }$$ with correlation structure $$\left\langle d W _ { t } ^ { k } , d W _ { t } ^ { l } \right\rangle =\rho _ { k l } d t$$.

The first moment is given by $$\mathbf { E } ^ { d } \left[ A _ { t _ { 1 } , t _ { n } } ^ { A , f } \right] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } f _ { 0 , \overline { T } \left( t _ { i } \right) }$$ and the second $$\mathbf { E } ^ { d } \left[ \left[ A _ { t _ { 1 } , t _ { n } } ^ { A , f } \right] ^ { 2 } \right] = \mathbf { E } ^ { d } \left[ \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } f _ { t _ { i } , \vec { T } \left( t _ { i } \right)} \right) \left( \frac { 1 } { n } \sum _ { j = 1 } ^ { n } f _ { t _ { p } , \vec { T } \left( t _ { j } \right)} \right) \right]$$ $$= \frac { 1 } { n ^ { 2 } } \sum _ { i , j = 1 } ^ { n } f _ { 0 , \overline { T } \left( t _ { i } \right) } f _ { 0 , \vec { T } \left( t _ { j } \right) } \exp \left( \rho _ { m ( i ) , m ( j ) } \sigma _ { m ( i ) } \sigma _ { m ( j ) } t _ { m ( j ) } t _ { \min ( \{ i , j \} ) } \right)$$.

The Black-76 model can then be used with volatility

$$\sigma _ { A } = \sqrt{\frac { 1 } { t _ { n } }[ln\frac{\mathbf { E } ^ { d } \left[ \left[ A _ { t , t _ { n } } ^ { A , t _ { n } } \right] ^ { 2 } \right]}{\left( \mathbf { E } ^ { d } \left[ \left[ A _ { t _ { 1 } , t _ { n } } ^ { A f } \right] \right] \right) ^ { 2 }}]}$$

and an initial futures price $$f _ { 0 , T } = \mathbf { E } ^ { d } \left[ \left[ A _ { t _ { 1 } , t _ { n } } ^ { A , f } \right] \right] = A _ { 0 ; \left\{ t _ { 1 } , t _ { n } \right\} } ^ { A ; f }$$.

My question is how I can tweak this to become a compo option. For example if the underlying is in USD, how could I adjust the procedure to price the option and find the optimal hedge in EUR?

I believe I need to synthesise the underlying to account for the exchange rate dynamics and adjust the volatility input to the Black model with the correlation between the underlying and the exchange rate. I am unsure how to go about this however, any help would be greatly appreciated.

• In the constant volatility case, this can be done similarly based on the dynamics for $X_t f_{t, T_k}$, where $X_t$ is the exchange rate. – Gordon Aug 9 '18 at 14:31
• Given the dynamics for $X_t$ and the respective correlation, you can do as you already have. Of course, with the consideration skew, this can be tricky. – Gordon Aug 14 '18 at 14:59
• @Gordon: Would you care to take a look at my question quant.stackexchange.com/q/41286/6686? Thank you. – Hans Aug 16 '18 at 1:31
• @Hans: I will have a look. But you may start with Chapter 16 of the book Introduces Quantitative Finance. Given all other parameters, it can be estimated like we estimate $\theta(t)$ in the Hull-White model from the yield curve. – Gordon Aug 16 '18 at 13:53