Pricing a Call on the Ratio of Two Correlated Stocks
Summary
The document poses a risk-neutral pricing problem for a claim paying the positive part of the difference between two stocks at maturity. It rewrites the payoff as the second stock multiplied by a call payoff on the ratio of the first stock to the second, suggesting a change of numeraire that uses the second stock as the pricing asset. The question derives the ratio dynamics with Itô’s formula and asks how correlated Brownian shocks and the money market account enter the calculation.
This is a problem statement rather than a complete solution: it provides stochastic dynamics for both stocks, their shared Brownian drivers, and a proposed measure change, but gives no final valuation formula or numerical evidence. The key modeling lesson is that a ratio’s drift depends on covariance terms, while choosing an appropriate numeraire can simplify pricing. A valid derivation still needs consistent assumptions about tradability, dividends, and the measure under which the ratio is valued.
Key ideas
- The payoff can be expressed as the second stock times a call payoff on the stock ratio.
- The ratio’s dynamics include variance and covariance contributions from Itô’s formula.
- A change of numeraire may simplify the valuation of a claim expressed relative to a traded asset.
- The document raises the pricing question but does not provide a completed solution.
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# Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$
# Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$
Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$.There is also a money market account $dB_t=rB_tdt$. The task is to compute the price at time $t$ of a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$ using matringale-based pricing. My idea was to decompose $(S_{1T}-S_{2T})^+=S_{2T}\left(\frac{S_{1T}}{S_{2T}}-1\right)^+$ and define a process $Z_t=\frac{S_{1t}}{S_{2t}}$. Thus, the price at time $t$ should be $S_{2t}$ times the price of a call option on $Z_t$ with strike $K=1$. So, using Ito's formula: $$dZ_t=\frac{\partial Z}{\partial t}dt+\frac{\partial Z}{\partial S_1}dS_{1t}+\frac{\partial Z}{\partial S_2}dS_{2t}+\frac{1}{2}\frac{\partial^2 Z}{\partial S^2_1}(dS_{1t})^2+\frac{1}{2}\frac{\partial^2 Z}{\partial S^2_2}(dS_{2t})^2+\frac{\partial^2 Z}{\partial S_1 \partial S_2}dS_{1t}dS_{2t}=$$ $$=\frac{1}{S_{2t}}dS_{1t}-\frac{S_{1t}}{S^2_{2t}}dS_{2t}+\frac{S_{1t}}{S^3_{2t}}(dS_{2t})^2-\frac{1}{S^2_{2t}}dS_{1t}dS_{2t}=$$ $$=Z_t\left((\mu_1-\mu_2)dt+(\sigma_{11}-\sigma_{21})dW_{1t}+(\sigma_{12}-\sigma_{22})dW_{2t}+(\sigma_{21}^2+\sigma_{22}^2+2\rho\sigma_{21}\sigma_{22})dt-(\sigma_{11}\sigma_{21}+\rho\sigma_{11}\sigma_{22}+\rho\sigma_{12}\sigma_{21}+\sigma_{12}\sigma_{22})dt\right)$$ However, nothing cancels out, so it leads me believe that I am on the wrong track. Furthermore, I am not using the money market account given in the task this way. Is there a neat trick for solving this task? Do not really know how to go from there. I understand that I can change the measure $d\hat{W}_{it}=dW_{it}+\lambda_idt$ for $i\in{1;2}$ and choose the lambdas to make the drift term of both stocks equal to $r$, but I do not see how this helps to solve the problem. I also see that I can rewrite the system using matrix notation, but, yet again, I do not see how this helps. The problems that I have seen before only had one Brownian motion per equation, so I am confused. Any help is appreciated.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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