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Pricing a Payoff Linked to a Non-Traded Asset in Black–Scholes

Article Quant Q&A · Author: Methamortix
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Summary

The document asks how to price a claim paying the stock price multiplied by the square of a non-traded variable at expiration. The setup specifies a Black–Scholes stock and bond, plus a non-financial process driven by the same Brownian motion as the stock. The author proposes using the stock as numeraire, changes measures, and derives candidate dynamics for the non-traded variable, but is unsure how to proceed because the usual replication argument appears unavailable.

The response focuses on the shared source of randomness: it argues that the stock and the non-traded variable are perfectly correlated in this setup, so the effective risk dimension is one and the stock’s movement can be expressed using the other process. This is offered as the route to continue the calculation. The answer does not derive the pricing expectation or give a final price, and the post leaves measure-change details and assumptions about the non-traded process’s market price of risk unexplored.

Key ideas

  • The proposed payoff depends on both a traded stock and the square of a non-traded variable.
  • The stock and non-traded variable are driven by the same Brownian motion in the stated model.
  • The response treats their shared randomness as leaving one effective risk dimension.
  • The author proposes a stock-numeraire change but does not complete the pricing calculation.
  • The response gives a direction for further derivation rather than a final price or full set of assumptions.

Tags

OptionsDerivatives pricingRisk management
Full text
# Hedging an option on a non-traded asset in BS world


# 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...

## Answer by lehalle (score 3, accepted)

https://quant.stackexchange.com/a/47349

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.