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So there is the following PDE given:

$\frac{\partial}{\partial t}f(t,x) + rx\frac{\partial}{\partial x}f(t,x)+\frac{\sigma^2 x^2}{2}\frac{{\partial}^2}{\partial x^2}f(t,x) = rf(t,x)$

With boundary condition $f(T,x) = x^{\frac{2r}{\sigma^2}}$

Here $r$ and $\sigma$ are positive constants.

From what I have learned, I the solution is from the boundary condition $f(t,x) =e^{-r(T-t)}E[x^{\frac{2r}{\sigma^2}}]$

So first I look fro the stochastic representation which I find as:

$dX(t) = rdt + \sigma dW(t)$ with X(t) = x

The solution is:

$X(T) = x + r(T-t) + \sigma(W(T)-W(t))$

This is normally distributed with mean $x + r(T-t)$ and variance $\sigma \sqrt{T-t}$

Now from boundary condition I have $f(t,x) =e^{-r(T-t)}E[(x + r(T-t) + \sigma(W(T)-W(t)))^{\frac{2r}{\sigma^2}}]$

However I don't know if this method is correct. If it is correct, how should I calculate this expectation? By the way, I take this expectation under $Q$ martingale measure.

Thanks

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Define $$ f(x,t):=\mathbb E[e^{-r(T-t)}X_T^{\frac{2r}{\sigma^2}} \vert X_t=x]\,, $$

where the Ito-process, $\text dX_u = rX_u\text du + \sigma X_u\text dW_u\ $ (with $X_t=x$), and the Weiner process, $W_u$, have been defined with respect to some suitable underlying filtered probability space (with filtration $\{\mathcal F_t\}_{t\geqslant 0}$). Now, by solving the SDE defining the process $X_u$, we obtain $$ X_u=xe^{(r-\frac{1}{2}\sigma^2)(u-t)+\sigma (W_{u}-W_{t})}\ \ (\text{for }u\geqslant t). $$

Thus, by considering conditional expectations with respect to the sigma-algebra $\mathcal F_t$, $$ \mathbb E[e^{-r(T-t)}X_T^{\frac{2r}{\sigma^2}} \vert \mathcal F_t] = e^{-r(T-t)}\mathbb E[X_T^{\frac{2r}{\sigma^2}} \vert \mathcal F_t] = e^{-r(T-t)}\mathbb E[(xe^{(r-\frac{1}{2}\sigma^2)(T-t)+\sigma (W_{T}-W_{t})})^{\frac{2r}{\sigma^2}} \vert \mathcal F_t] = e^{-r(T-t)}x^{\frac{2r}{\sigma^2}}e^{\frac{2r}{\sigma^2}(r-\frac{1}{2}\sigma^2)(T-t)}\mathbb E[(e^{\sigma (W_{T}-W_{t})})^{\frac{2r}{\sigma^2}} \vert \mathcal F_t] = e^{-r(T-t)}x^{\frac{2r}{\sigma^2}}e^{\frac{2r}{\sigma^2}(r-\frac{1}{2}\sigma^2)(T-t)}\mathbb E[e^{\frac{2r}{\sigma} (W_{T}-W_{t})}]\,, $$

where $\mathbb E[e^{\frac{2r}{\sigma} (W_{T}-W_{t})}\vert \mathcal F_t] = \mathbb E[e^{\frac{2r}{\sigma} (W_{T}-W_{t})}]$ since $(W_{T}-W_{t})$ is independent of $\mathcal F_t$. So,

$$ f(x,t) = x^{\frac{2r}{\sigma^2}}e^{-2r(T-t)}e^{2\frac{r^2}{\sigma^2}(T-t)}\mathbb E[e^{\frac{2r}{\sigma} (W_{T}-W_{t})}]\,. $$

We are almost done; we simply need to evaluate $\mathbb E[e^{\frac{2r}{\sigma} (W_{T}-W_{t})}]$ using the fact that $W_{T}-W_{t}\sim\mathcal N(0,T-t)$. Note that, $$ \begin{eqnarray*} \mathbb E[e^{\frac{2r}{\sigma} (W_{T}-W_{t})}] &=& \frac{1}{\sqrt{2\pi(T-t)}}\int_{-\infty}^{\infty}e^{\frac{2r}{\sigma} w-\frac{1}{2}w^2/(T-t)}\text dw \\ && \\& = &\ldots \\ && \\ & = & e^{2\frac{r^2}{\sigma^2}(T-t)}\,. \end{eqnarray*} $$

Therefore,

$$ f(x,t) = x^{\frac{2r}{\sigma^2}}e^{-2r(T-t)}e^{4\frac{r^2}{\sigma^2}(T-t)}\,. $$


If you are curious, you can convince yourself that the solution above is correct by showing that it satisfies the given PDE: simply compute each of the partial derivatives and plug them in the PDE.

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  • $\begingroup$ Hi, thanks. How did you come up with the Ito process? Why use exactly that one? $\endgroup$ – Elekko Mar 16 '15 at 18:24
  • $\begingroup$ @Elekko, That is the form of Ito-process required by the Feynman-Kac formula (see en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula). Using the statement of the formula on that webpage, simply identify $\mu(x,t)$ and $\sigma^2(x,t)$ for your PDE and use these in the Ito-process on that webpage. $\endgroup$ – ki3i Mar 16 '15 at 19:26

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