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remia89s
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Posts: 2
Joined: June 9th, 2012, 9:33 am

### convex payoff, convex price

Hello everyone!I have a question that I'm sure someone can help me with.Suppose that you have a convex function $\Phi(x)$ and a stochastic process $(X_t)_{t\geq 0}$ that is a martingale: $\mathbb{E}[X_T|X_t]=X_t$.Define $p(x)=\mathbb{E}[\Phi(X_T)|X_t=x]$. I don't know how to show that the function $p(x)$ is convex.Basically this problem is equivalent to say that the price of a convex european payoff is a convex function. I'm pretty sure that this is true, but I don't know how to prove it!Thank you for your help.

Alan
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Joined: December 19th, 2001, 4:01 am
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### convex payoff, convex price

I'm not sure how general that is. It's relatively easy to prove for a simple diffusion. Take $dX_t = a(X_t) dW_t$. Changing your notation somewhat, the value function $V(t,x) = E_{t,x}[\Phi(X_T)]$ satisfies $-V_t = \frac{1}{2} a^2(x) V_{xx}$.Let $H(t,x) \equiv V_{xx}$. Differentiate the value function pde twice to get a pde for H. This second pde has a probabilistic representation for itssolution and the convexity follows readily from that. See Appendix 2.3 of 'Option Valuation under Stochastic Volatility' for details anda generalization to stochastic volatility processes.
Last edited by Alan on July 7th, 2015, 10:00 pm, edited 1 time in total.

savr
Posts: 48
Joined: January 21st, 2013, 3:28 pm

### convex payoff, convex price

Section 4 of this paper has a counterexample.