# How would you hedge this structure?

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 $H$:

Given a stock $S_t$, $\text{exp} = 1$ (yrs), I need a payoff $H$ in which,

Conditional on $S_\text{exp} / S_0 \leq 0.8$, i.e the stock price decreased $20\%$ one year from now relative to the current price, then $H = \max{(0, V_\text{exp} - 0.17)}$, where $V_\text{exp}$ is the realized volatility one 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.

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## 1 Answer

So just to clear the payoff, it's an option on realized volatility (not variance) conditional on the stock? Are you sure it's not a conditional variance swap or a knock-in variance swap?

(a) I hope you are doing it in some sort of index, cause I'd hate to hedge this in single stock. (b) In an index this would be very costly (the skew would make the probability pretty rich. (c) No model properly replicates the volatility dynamics, you are going to have be super-conservative about your hedging assumptions.

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Sorry if I wasn't clear. It is basically a conditional option on realized volatility. And the condition is that the drawdown from the max stock price over the year at the end of the year is greater than 10%. Note that the condition is on stock price and the payoff is on the realized vol so its tricky. Yes it is on some index and not a single name stock. Can you explain point b)? Yea I am making a huge assumption about the volatility dynamics – inquisitive Sep 21 '12 at 1:18

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on the point (b) simple local volatility expectation is approximately 0.5*ATM + 0.5*Barrier. I assume this is on S&P (the only other index I'd trade it on would be Stoxx) and it is sufficiently long-dated, say 1y. One year sk10 is approximately 3.5, so you gain 3.5 vols in addition to ATM which is over 20. So, even without any vol of vol, your option is already ITM. Since you also going to assume high vol of vol at lowet strike, the option will come in very very rich. Just my intuition as an ex-index-exotics-trader. What sort of client is this for anyway, a hedge fund? – Strange Sep 21 '12 at 1:31
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