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I have to deal with this one-factor model:

\begin{equation*} \begin{cases} dS_t = \alpha \bigl(\mu - \log(S_t) \bigr)S_t \, dt + \sigma S_{t} \, dW_t \, , t \geq 0,\\ S|_{t=0} = S_0 > 0, \end{cases} \end{equation*}

which gives me the following PDE for an European Call option:

\begin{equation*} \begin{cases} \frac{\partial V}{\partial t} + \Bigl [ \alpha \Bigl(\mu - \frac{\lambda}{\alpha} - \log (S) \Bigr) S \Bigr ] \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0, \\ V(S,T) = (S -K)^+, \end{cases} \end{equation*}

where the parameter $\lambda$ represents the risk premium. Solving numerically the PDE, if I increase $\lambda$ (usually I take positive values) then the price of the option decreases. Is it possible? And what is the economic interpretation of this phenomenon? Thanks in advance.

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1 Answer 1

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I assume $\alpha>0$.

Let $V^\lambda$ be the solution of : \begin{equation*} \begin{cases} \frac{\partial V^\lambda}{\partial t} + \Bigl [ \alpha \Bigl(\mu - \frac{\lambda}{\alpha} - \log (S) \Bigr) S \Bigr ] \frac{\partial V^\lambda}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V^\lambda}{\partial S^2} - rV^\lambda = 0, \\ V^\lambda(S,T) = (S -K)^+, \end{cases} \end{equation*}

then you want to prove :

$$\lambda<\lambda' \Rightarrow V^\lambda \geq V^{\lambda'}$$

  1. Use Feyman Kac to prove that, under $\mathbb{P}$, by denoting : $$dX^\lambda_t = \alpha(\mu-\frac{\lambda}{\alpha}-X^\lambda_t)dt-\frac{\sigma^2}{2}dt + \sigma dW_t$$

$$V^\lambda(t,S) = \mathbb{E}\left[ \left.e^{-r(T-t)}\left(e^{X^\lambda_{T}}-K\right)^+ \right| X^\lambda_t = \ln S \right]$$

  1. Prove using Ito's lemma that: $$X^{\lambda_1}_T - X^{\lambda_2}_T = e^{-\alpha(T-t)}(X^{\lambda_1}_t-X^{\lambda_2}_t)+(1-e^{-\alpha(T-t)})(\frac{\lambda_2}{\alpha}-\frac{\lambda_1}{\alpha})$$

  2. Using that $x\to (e^{x}-K)^+$ is increasing, prove that:

$$ \lambda_1 < \lambda_2 \Rightarrow V^{\lambda_1}(t,S)\geq V^{\lambda_2}(t,S)$$

Economic intuition

$\lambda \uparrow \Rightarrow \mu-\frac{\lambda}{\alpha} \downarrow \Rightarrow$ drift of asset is lower, so lower up-trend, and since call option is increasing with the price of the asset, you get lower values.

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    $\begingroup$ +1 nice answer. For point 2, it may help to notice that the SDE satisfied by $X_t = \ln (S_t) $ corresponds to the Vacisek model. And by "using Ito's lemma", I guess MJ73550 meant "by solving the SDE". $\endgroup$
    – Quantuple
    Jul 15, 2016 at 22:04
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    $\begingroup$ Thanks for your answers. But reasoning on pure economic arguments, why risk premium should be inversely related to asset price? If risk premium is higher this means that investors are more risk averse, then they should buy the asset causing an increase in the price. What is wrong? $\endgroup$
    – derik
    Jul 21, 2016 at 16:23

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