Projecting Forward Implied Volatility with Forward Variance and Heston

Summary

The document proposes estimating the implied volatility for a one-year option starting one year in the future from the current one-year and two-year at-the-money implied volatilities. It uses a forward-variance relation: the future period’s variance is inferred by subtracting the near-term variance from the longer-term variance, then converting that variance to volatility. This gives a way to derive a single forward volatility estimate from the two maturities.

For generating a distribution of possible future volatilities, the answer suggests calibrating a stochastic volatility model, with Heston offered as an example. Calibration is described as fitting model parameters to observed European call and put prices, after which the model can be simulated. The response does not supply calibration details, assumptions, or evidence, and its claim that the relation is always valid is too broad without conditions such as compatible forward variance inputs and conventions. The suggested workflow therefore needs careful specification and validation for the target option and market.

Key ideas

• Forward variance can be inferred from longer- and shorter-maturity implied variances.
• The resulting forward variance can be converted into a volatility for the future period.
• A stochastic volatility model can be calibrated to option prices and simulated to generate possible volatility outcomes.
• The forward-volatility relation requires compatible maturity and market conventions; the document gives no calibration implementation.

Tags

# Answer by Valometrics.com (score 1)


# How to project 1 Year ATM Implied volatility for SPX 500 1Year from now? Final goal is to calculate 1 Year Call prices on SPX 500 1 year from now?












I have the historical data for 1Year ATM Implied Volatility on SPX 500. I want to simulate the 1 year call option prices 1 year from now. What methods and approaches do I need to use? (Heston,GARCH, Black-Scholes etc...)

## Answer by Valometrics.com (score 1)

https://quant.stackexchange.com/a/51084

The best solution is to compute the implied volatility for a call that matures in two years then the implied volatility for one year call one year from now will be equal to:

$$\sqrt{2*vol^2_{2y}-vol^2_{1y}}$$

You can find this formula in the wikipedia article about forward volatility:

Forward volatility

Now in order to generate many volatilities, the only solution is to use a stochastic volatility model (I have a preference for Heston model) to generate 2 years IV and one year IV then use the formula above which is always valid. To do that, you need to estimate heston model parameters which uses as inputs european calls and puts prices.

The calibration procedure consists on minimising the distance between market options prices and prices given using the parameters of Heston model. You can find the calibration algorithm in the following article:

Heston calibration

Once this is done, you generate as many volatilities as you want by simulating the heston equation.

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