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for a stochastic model \begin{equation} dS_{t} = rS_{t}dt + \sigma_1 dW_{1t} + \sigma_2dW_{2t}\\ dW_{1t} dW_{2t} = 0 \end{equation} under the risk neutral measure, I need to calculate the value of a call option, \begin{equation} C_{t} = E\big( (S_{T} - K)_{+}\big| F_t\big) \end{equation} where $T$ is the maturity and $K$ is a s trike of the call option.

When calcultating this, I encountered an integral that is hard for me to solve as below

\begin{equation} \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}N(ax + b) dx \end{equation} where $N(x)$ is the cumulative normal distribution function.

Is there any way to calculate this?

Not only a perfect solution but also a hint will be appreciated.

Hoseong Ryu
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  • You can first note that the sum of Wiener processes is a Wiener process. Hence this is a special case of the Vasicek model. You can find the derivations for call option pricing under this model online. – fes May 14 '21 at 08:28
  • This is essentially an integral of an error function and Gaussian. There is a solution eg here – Sal May 14 '21 at 14:23

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