I have a contingent claim and I want to find out what is the best structure to meet the continent claim, how to price it and how to hedge it. I am looking more for a qualitative answer.
Suppose I want to best replicate this claim H$H$:
Given a stock S_t$S_t$, exp = 1$\text{exp} = 1$ (yrs), I need a payoff H$H$ in which,
Conditional on (S_exp / S_t) <= 0.8$S_\text{exp} / S_0 \leq 0.8$, i.e the stock price decreased 20% 1$20\%$ one year from now relative to the current price, then H = max(0, V_t - 0.17)$H = \max{(0, V_\text{exp} - 0.17)}$, where V_t$V_\text{exp}$ is the realized volatility 1-yearone year from now. If the stock price did not meet the first criteria, the payout is just zero.
I decided to to use a stochastic vol process. I found the parameters of the stochastic vol process by running Monte Carlo simulations and simulating stock paths, and trying to find the parameters such that I am able to best fit the market prices.
An important assumption is that I can only trade the stock and options on the stock. I cannot trade volatility. Clearly, the market is incomplete because I have two uncertainties (Brownian motion in the stock and in the stochastic volatility). I am having difficulty deciding what is the best structure to best fulfill this contingent claim and yet be able to sufficiently hedge it using stocks and options.
