# Hedging an option on a non-traded asset in BS world

I have given the following task given.

Suppose you are in a Black-Scholes World where you have the standard assets $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ $$dB_t = r B_t dt$$ and now you also have a non-financial asset $$dY_t =\alpha dt + \beta dW_t$$ where the Wiener process $$dW_t$$ drives both S and Y.

Give the pricing function at time t=0 for an option with the payout $$X=S_T * (Y_T)^2$$

My approach would have been to take $$S_T$$ as a numeraire in order to get:

$$\Pi_o[X] = S_0 E^{Q^S} \left[ S_T \frac{(Y_T)^2}{S_T} \right] = S_0 E^{Q^S} \left[ (Y_T)^2 \right]$$

Then I have to derive the $$Q^S$$ dynamics of $$Y_t$$. First transforming $$S_t$$ and $$Y_t$$ to the risk neutral measure $$Q$$ gives me

$$dS_t = r S_t dt + \sigma S_t dW_t^{Q}$$

$$dY_t = \left\lbrace \alpha + \beta \left( - \frac{ \mu - r }{\sigma} \right) \right\rbrace dt + \beta dW_t^{Q}$$ Then going from $$Q$$ to $$Q^S$$ gives me $$dS_t = (r * \sigma^2) S_t dt + \sigma S_t dW_t^{S}$$ $$dY_t = \left\lbrace \alpha + \beta \left( - \frac{ \mu - r }{\sigma} + \sigma \right) \right\rbrace dt + \beta dW_t^{S}$$

Now I have the dynamics of $$Y_t$$ under $$Q^S$$ but I don't know how to go on since until now I have completely ignored that $$Y_t$$ is a non-traded asset, thus I can't use the BS replication argument...

• I'd first write everything under the usual risk-neutral measure, and then compare $d \ln S_t$ to $dY_t$. I think that will then lead to a semi-static replication formula using Carr-Madan. – Frido Rolloos Aug 23 '19 at 11:40

For the next steps, you need to use $$dY$$ is place of $$dW$$ everywhere in the expression of $$dS$$. In fact when you replicate the payoff of a vanilla option using the underlying, it is simply because the correlation between the underlying and its tradable counterparty is equal to 1. By chance in your case the correlation between a simple transformation of $$dS$$ and $$dY$$ is 1 too, thus the dimension of your underlyings is not 2 but 1: all can be expressed as a function of $$dW$$ and $$dY$$ only.