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I have a spot price process for $T>t$ : $S_T=K*C$

With K scalar and C:

$C=\exp{\left[\int_t^T e^{-\alpha(T-s)} (\int_t^s e^{-\beta(s-r)}dW_r^Uds)+\int_t^T e^{-\alpha(T-s)}dW_s^V\right]}$

With:

$=\rho dt$

$\alpha,\beta$ constants.

I'm trying to express $\Bbb{E}_t(S_T|\mathcal F_t)$ to evaluate the forward price but i'm strugling with this double integral.

I tried to use info in Math Stackexchange Post which is quite similar for the first part of C.

Thanks

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    What does $A.B.C.D$ mean? – 2017-02-08
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    This is the product of function of t , where C only is stochastic. So one may consider A,B and D as constants. – 2017-02-08
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    The other two things out front can be considered constants too, right? (conditional on $\mathcal F_t$) – 2017-02-08
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    Yes, Its really expectation of C that I'm trying to solve, so you think I should edit the question and remove everything but C ? – 2017-02-08
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    Yeah, it would probably be a good idea to remove as much clutter as possible You might not be getting any bites since the expressions look so complicated. (Also would probably be good to use $\exp(.)$ rather than $e^.$ so the expression in the exponent is more readable .) – 2017-02-08
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    Done, I changed as you suggested – 2017-02-08

1 Answers 1

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So for the first part of C , all I'm doing is following the computation of Gordon in the link and replacing :

$(t-1) \rightarrow{t}, t \rightarrow{T}, 0 \rightarrow{t} , s\rightarrow{s}$

So I end up with:

\begin{align*} \ln{C} &= \int_{t}^{T} \int_{t}^{s}e^{-\alpha(T-s)-\beta(s-r)}dW(r) \,ds +... \\ &= \int_{t}^{t} \int_{t}^{T}e^{-\alpha(T-s)-\beta(s-r)}ds\,dW(r) + \int_{t}^{T} \int_{r}^{T}e^{-\alpha(T-s)-\beta(s-r)}ds\,dW(r)+...\\ &= 0 + \frac{e^{-\alpha T}}{\alpha-\beta}\int_{t}^T e^{\beta r} \left[e^{(\alpha -\beta)T}-e^{(\alpha -\beta)r} \right] dW(r)+...\\ &= 0 + \frac{e^{-\beta T}}{\alpha -\beta}\int_{t}^T e^{\beta r} dW(r) - \frac{e^{-\alpha T}}{\alpha -\beta}\int_{t}^T e^{\alpha r} dW(r)+... \end{align*}

From there its not too complicated to compute variance of the process, but I have no idea if this work like that as I dont fully understand the decomposition in the first place.

EDIT: This is working, tested with some closed form solutions on special cases, limit cases and Monte Carlo simulation.