# Risk-neutral price of $H=e^{X_T^1+X_T^3}$

Let $$B=(B_t^1,B_t^2,B_t^3)$$ a $$\mathbb R^3$$-valued Brownian motion. Let $$r_t$$ (risk free rate) be bounded and deterministic. Let consider the DISCOUNTED market $$d\overline X_t^1=\frac52dt+2dB_t^1-dB_t^2-dB_t^3$$ $$d\overline X_t^2=7dt+2dB_t^1+2dB_t^2-10dB_t^3$$ $$d\overline X_t^3=\frac72dt+4dB_t^1-3dB_t^2+dB_t^3$$ I have already found that the market is arbitrage free.

I would like to find the risk-neutral price of the following claim:$$H=e^{X_T^1+ X_T^3}$$ (note: here the $$X_T^1,X_T^3$$ are not discounted) but i'm stack. Any help please?

• Are there 3 different $X_t$ processes? In your notation it seems the same process. – Daneel Olivaw Jan 26 at 0:30
• Also what are the $B$s at the end of each equation? – Daneel Olivaw Jan 26 at 0:32
• @DaneelOlivaw yes sorry, now I edit it – Buddy_ Jan 26 at 8:02
• @DaneelOlivaw don't you have any suggestion? – Buddy_ Jan 26 at 15:57
• why not to use montecarlo to compute the price?? – Valometrics.com Jan 26 at 20:43