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

Question

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...

Answer 1

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.