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Changing to a Bond Numeraire for Stock Option Pricing

Article Quant Q&A · Author: Jan Stuller
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Summary

The document asks how to price a stock call under stochastic interest rates using a maturity-matched bond as numeraire. It starts from the change-of-numeraire relation and rewrites the discounted option value as the bond price times an expectation under the bond measure. The remaining challenge is to determine how the stock process changes under that measure.

The question proposes applying Cameron–Martin–Girsanov, but observes that the density appears to involve the integrated short rate rather than a stochastic integral against Brownian motion. It does not provide a derivation or resolution. The discussion is therefore a useful statement of the modeling problem, not a completed pricing method; applying the theorem requires a suitable bond price process and its dynamics, which are not supplied here.

Key ideas

  • A bond numeraire can express an option price as a bond price times a payoff expectation under the bond measure.
  • The stock’s dynamics must be transformed when moving from the bank account measure to the bond measure.
  • The proposed density is not directly in the familiar Brownian exponential form used for Girsanov.
  • The document leaves the measure change unresolved and provides no completed pricing formula.

Tags

OptionsFixed incomeDerivatives pricingStatistics
Full text
# Stock price under Bond numeraire


# Stock price under Bond numeraire












The Radon-Nikodym derivative going from the bank-acount Numeraire $N(t)$ to the bond numeraire $P(t,T)$ is:

$$\frac{dP}{dN}(T|\mathcal{F}_t)=\frac{1}{N(T)P(t,T)}$$

Suppose I now want to price an option on stock $S_t$, but under non-deterministic rates, i.e. $N(t)=e^{\int_0^tr(t)dt}$. I am interested in $C(t_0,T)=\mathbb{E}_{t_0}^N\left[e^{-\int_0^Tr(t)dt}(S_T-K)^+\right]$. Suppose I want to use the Radon-Nikodym from above, I could do the following:

$$C(t_0,T)=P(t_0,T)\mathbb{E}_{t_0}^N\left[P(t_0,T)^{-1}e^{-\int_0^Tr(t)dt}(S_T-K)^+\right]=P(t_0,T)\mathbb{E}_{t_0}^P\left[(S_T-K)^+\right]$$

But now I'd need to figure out the process $S_t$ under $P(t,T)$. Supposing that under $N(t)$, $S_t$ follows the process:

$$S_t=S_0e^{\int_0^tr(t)dt-0.5 \sigma^2 t +\sigma W_t}$$

How can I use the Radon-Nikodym:

$$\frac{1}{N(t) P(t,T)}=\frac{1}{e^{\int_0^tr(t)dt} \mathbb{E}^N[e^{\int_0^tr(t)dt}]}$$

To apply the Cameron-Martin-Girsanov Theorem to the process for $S_t$ under $N_t$?

Here is my attempt:

We want Radon-Nikodym that looks like $exp\left\{\int_0^T \mu (t) dW_t-0.5 \int_0^T \mu^2 (t) dt\right\}$. By CMG Theroem, applying this Radon-Nikodym to a Brownian $W_t$ will result in a new measure under which the same Brownian motion will acquire a drift $\int_0^T \mu(t) dt$.

Instead, we have:

$$exp\left\{-\int_0^Tr(t)dt\right\}*constant$$

Is there a way to proceed further?

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.