Modeling Implied Volatility Changes in Delta–Gamma–Vega VaR

Summary

The document asks how to model changes in implied volatility when approximating an option portfolio’s profit and loss with delta, gamma, and vega terms. It distinguishes historical VaR, which can reprice a portfolio across observed market scenarios without specifying factor distributions, from Monte Carlo VaR, which requires distribution assumptions for market factors. For simulation, the answer suggests treating implied volatility as an observable factor with its own historical changes and correlations.

Volatility is structured across option moneyness and expiry, and sometimes across the tenor of the underlying, forming a volatility surface or cube. One possible simplifying assumption is that changes at each point are normally distributed with zero mean, with volatilities and cross-point correlations estimated from history. These are modeling choices, not universal properties: the note does not specify a calibration window, test distributional fit, or address nonlinear surface dynamics. Historical repricing also depends on having relevant market scenarios and a way to revalue the portfolio under them.

Key ideas

• Historical VaR can reprice positions under observed market scenarios without assuming factor distributions.
• Monte Carlo VaR requires assumptions about the distributions of market factors.
• Implied volatility can be treated as a market factor with historical observations.
• Volatility changes may be modeled across moneyness and expiry, and sometimes underlying tenor.
• Historical data can be used to estimate volatility-change variability and correlations across surface points.

Tags

# delta-gamma-vega VaR approximation: how to calculate the delta volatility?


# delta-gamma-vega VaR approximation: how to calculate the delta volatility?












For an option with price C, the P&L, with respect to changes of the underlying asset price S and volatility σ, is given by

P&L=δΔS+12γ(ΔS)2+νΔσ,

where δ, γ, and ν are respectively the delta, gamma, and vega greeks.

My question is: while we are able to calculate ΔS by using the spot price of underlying S and by assuming a normal distribution of its returns, what about Δσ ?

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/74537

If you use historical VaR, i.e. reprice the portfolio under many historical market move scenarios, then you need not assume anything about the distributions of the market factors. But you need distribution assumptions to use Monte Carlo simulation, rather than history.

For the underlying, you may want to assume normal or lognormal.

The implied volatility is just another market factor, observed in the market, for which you can get history. It has structure: moneyness and expiry of the option (implied volatility surface) and for some underlyings also the tenor of the underlying (implied volatility cube). You can assume that the changes in each point of the vol surface are normally distributed, with mean 0, and, from the history, calculate the historical volatility of implied volatility, as well as the historical correlarions between the points on the vol surface.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.