Why Historical GARCH Volatility May Misprice Call Options

Summary

The document describes an attempt to price equity calls by calibrating a GARCH(1,1) model to historical data with maximum likelihood, then simulating the stock under a risk-neutral diffusion using forecasted conditional volatility. The proposed pricing method takes the discounted expected payoff, estimated with Monte Carlo simulation. The author reports that it substantially overprices calls, while using unconditional long-run variance produces more acceptable prices, though that assumes constant volatility.

The discussion raises a key modeling issue: fitting volatility dynamics to historical returns does not by itself specify their risk-neutral behavior for option valuation. It mentions Duan’s work but provides no answer, calibration results, or comparison against market quotes. The described outcomes are the author's observations, not evidence that the approach or constant-variance alternative is generally reliable; the text leaves the source of the pricing discrepancy unresolved.

Key ideas

• The author fits a GARCH(1,1) model to historical equity data using maximum likelihood.
• The proposed simulation uses forecasted conditional volatility and discounted expected payoffs to price calls.
• The author reports substantial overpricing with forecast volatility and more acceptable prices using long-run variance.
• Historical volatility calibration alone does not settle how volatility should behave under risk-neutral pricing.
• The document raises the modeling question but does not resolve it or provide comparative evidence.

Tags

# GARCH option pricing


# GARCH option pricing












I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity through MLE. Now I want to price some call options for that particular equity for that I had tried to use the following risk neutral SDE: $$dS_t = rS_t + \sigma(t)S_tdB_t$$ It can be checked that the discounted stock price process $e^{(-rt)}S_t$ is a martingale(Source - https://math.stackexchange.com/questions/3266010/conditional-expectation-of-exponential-brownian-motion?noredirect=1&lq=1). So as long as the discounted stock price process is a martingale the price of options is simply the given by the expectation of the discounted payoffs(Monte Carlo approach for pricing). $\sigma(t)$ is the forecasted Garch volatility that I have used for simulating the price process but this approach is overpricing the call options very much. I am aware of the work done by Duan 1995 but I am curious to know what may have gone wrong in this approach.

Further instead of $\sigma(t)$ I also used the long term variance or the unconditional variance then in that case the option prices that I am getting is good enough but for this we are assuming that the volatility is constant which I don't want.

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