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Suppose I have a model with 2 primary assets, a stock $S$ and a short rate.

The stock will be driven by a Brownian motion $W_1$. The short rate will be random and will be driven by a Brownian motion $W_2$. These Brownian motions may or may not be correlated. I want to price a European option call option on $S$.

I want to compute the measure with respect to having either of the assets as numeraire. However, I have two driving Brownian motions and only 2 assets. I need a third asset for the market to be complete.

In the Black-Scholes model with random interest rates, what is usually taken as this third primary asset?

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  • $\begingroup$ The zero coupon bond. $\endgroup$
    – user34971
    Commented Jan 1, 2022 at 8:59
  • $\begingroup$ @FridoRolloos You can price a zero-coupon bond using the formula $B(t, T) = \widetilde{\mathbb{E}}\Big(e^{\int_{t}^{T} -r(t) \mathrm{d}t} | \mathcal{F}(t)\Big)$. How do you use the prices of zero-coupon bonds to compute the market prices of risks for $W_1(t)$ and $W_2(t)$? $\endgroup$
    – user60304
    Commented Jan 1, 2022 at 10:15
  • $\begingroup$ Why would you want to know the market prices of risks? You are given that stocks are traded and zero coupon bonds are traded. To price options on $S$ with stochastic interest rates you need to do a change of numeraire and use the zero-coupon as numeraire. $\endgroup$
    – user34971
    Commented Jan 1, 2022 at 11:57
  • $\begingroup$ @FridoRolloos Pricing a call option is just an example. I'm computing the market prices of risks to understand how to write $S(t)$ under the equivalent measure with the zero-coupon as numeraire. Some sources have work-arounds that use the forward price, but dealing with market prices of risks is more intuitive and more generalizable. (Besides, I'm not convinced that there isn't some circular reasoning with the forward price argument.) $\endgroup$
    – user60304
    Commented Jan 1, 2022 at 13:20

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