Risk-Neutral Measures in a One-Period Market Model

Summary

The document sets up a one-period market with a risk-free asset earning rate r and a risky asset whose price has a strictly positive density on the positive real numbers. It asks how to find a risk-neutral probability measure, given the asset’s current price and its future price distribution. The author correctly frames the pricing condition as requiring the expected future risky-asset price under the risk-neutral measure to equal its current price grown at the risk-free rate.

The proposed route is to express the risk-neutral expectation using a Radon–Nikodym derivative relative to the real-world measure. However, the question does not provide a candidate derivative or a solution. In a model with a continuously distributed single risky payoff, the pricing condition alone generally does not identify a unique equivalent measure; further choices or constraints are needed to specify one. The setup therefore illustrates the risk-neutral pricing condition and the challenge of determining a measure, rather than giving a complete construction.

Key ideas

• The one-period model contains a risk-free asset and a risky asset with a positive continuous density.
• A risk-neutral measure must make the discounted risky-asset price have the appropriate expectation.
• The Radon–Nikodym derivative expresses expectations under the proposed measure using the original measure.
• The pricing condition alone generally leaves multiple possible measures in this continuous-state setup.

Tags

# Given the density function of $S^{1}$ in one-period model, find the risk-neutral measure


# Given the density function of $S^{1}$ in one-period model, find the risk-neutral measure












Consider the one period market model $\left(\overline{\pi},\overline{S}\right)$ consisting of a risk-free asset $\left(\pi^{0},S^{0}\right)=(1,1+r)$ and a risky $\left(\pi^{1},S^{1}\right)$

Let $ r > -1$ and $\pi^{1}>0$. $S^{1}$ has strictly positive density function $f: (0,\infty)\to \left(0,\infty\right)$, i.e. $P(S^{1}\leq x)=\int_{0}^{x}f(y)\mathrm{d}y$ for $x > 0$.

Find the risk-neutral measure $\mathbb Q$.

My idea:

I think we need to characterize $\mathbb Q$ in terms of its Radon-Nikodym derivative with respect to $\mathbb P$, i.e. $\frac{\mathrm{d}\mathbb Q}{\mathrm{d}\mathbb P}$. By definition of being risk-neutral, $\mathbb Q$ has to satisfy:

$$(1+r)\pi^{1}=E_{\mathbb Q}[S^{1}]$$

By the Radon-Nikodym derivative approach, we get

$$(1+r)\pi^{1}=E_{\mathbb Q}[S^{1}]=E_{\mathbb P}\left[S^{1}\frac{d\mathbb Q}{d\mathbb P}\right]=\int_{\Omega}S^{1}(\omega)\frac{\mathrm{d}\mathbb Q}{\mathrm{d}\mathbb P}(\omega)\mathbb P(\mathrm{d}\omega)=\int _{0}^{\infty}\left(S^{1}\circ f\right)(y)\left(\frac{d\mathbb Q}{d\mathbb P}\circ f\right)(y)f(y)\mathrm{d}y$$

I have no idea how to proceed to get an expression of $\frac{\mathrm{d}\mathbb Q}{\mathrm{d}\mathbb P}$. Any ideas? Am I on the right track?

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