Hedging a Stock-Conditional Realized Volatility Payoff in an Incomplete Market
Summary
The document poses a hedging problem for a one-year payoff that activates only if a stock falls at least 20% from its initial level. If that condition is met, the holder receives the positive excess of realized volatility over a threshold; otherwise the payoff is zero. The proposed approach calibrates a stochastic-volatility process by Monte Carlo simulation to fit observed market option prices.
Because the payoff depends on both the stock path and volatility, while trading is restricted to the stock and its options, the market is incomplete. The response first questions whether the intended contract is an option on realized volatility or instead a conditional variance or knock-in variance swap. It cautions that volatility dynamics are not reliably replicated by a model, that downside skew can make an index version expensive, and that hedging assumptions should therefore be conservative. The brief exchange offers no replication portfolio, pricing formula, or quantitative evidence, so it raises modeling and risk concerns without resolving the hedge design.
Key ideas
- The payoff combines a downside stock condition with a thresholded realized-volatility payment.
- Stock and options alone may not span both price and stochastic-volatility risks.
- Clarify whether the intended exposure is to realized volatility or to conditional variance.
- Model-based replication is limited by uncertainty in volatility dynamics.
- Downside skew may raise the cost of an index-linked version, calling for conservative hedging assumptions.
Tags
Full text
# How would you hedge this structure?
# 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.
## Answer by Strange (score 1)
https://quant.stackexchange.com/a/4166
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.