Estimating CVaR Under Physical Heston Dynamics

Summary

The document considers whether simulated returns from a Heston model calibrated to S&P 500 option prices can estimate 99% conditional value at risk (CVaR). It identifies a key issue: option prices generally imply risk-neutral dynamics, while risk management measures such as VaR and CVaR are usually intended to describe outcomes under the real-world, or physical, probability measure.

The response recommends estimating Heston parameters from historical data, for example with maximum likelihood, to model physical-measure returns before simulating losses and calculating tail averages. It explains why risk-neutral simulations may not represent the historical profit-and-loss changes relevant to risk limits. The discussion is brief and gives no implementation details, validation results, or treatment of estimation uncertainty; fitting a physical model alone does not establish that its tail forecasts are accurate.

Key ideas

• Risk measures intended to describe actual profit-and-loss movements are generally estimated under the physical measure.
• Option prices are associated with risk-neutral dynamics, which can differ from historical dynamics.
• Historical data and maximum likelihood estimation can be used to fit a physical-measure Heston model.
• Simulated lower-tail returns can then be used to estimate CVaR, subject to model and parameter uncertainty.

Tags

# Why Can I not estimate a CVAR from Heston Model


# Why Can I not estimate a CVAR from Heston Model












I fit the parameters of Heston model, using option data for SPX. Now I have the process S and P 500 is expected to follow. I make 100,000 simulations of this process and then calculate the expected return. The average of lowest 1% return is 99%-CVAR.

Why does the method not work. I get that I can not calibrate r the drift of S&P 500 because of risk neutral measure thing. But say I somehow use MLE or MM to get estimates for the drift. Will this method work.

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/42324

Given the main uses of the VaR relate to risk management such as limit management, and measurement of P&L volatility, it is usually calculated under the physical/real world measure. Reason being that the risk measure are normally used to predict or explain the P&L movements from one day to another, which one can relate to their historical movements. Risk neutral dynamics are usually very different than the historical measure. So that’s the reason physical measure is more widely used in risk management. For Heston, you can use use maximum likelihood approach to estimate the parameters from historical data. Here is a good reference:

https://www.princeton.edu/~yacine/stochvol.pdf

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