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Published on: September 19, 2012
Analytic Approximation for Bachelier Option Prices and Applications.
1Department of Economics and Business, Barcelona School of Economics, Universitat Pompeu Fabra, Ramón Trias Fargas 25-27, 08005 Barcelona, Spain.
Entropy (Basel, Switzerland)
|June 26, 2026
Summary
In the Bachelier model, option pricing for out-of-the-money and in-the-money options is derived using volatility expansions. This method is applied to reduce Monte Carlo simulation variance in correlated asset price scenarios.
Area of Science:
- Quantitative Finance
- Financial Mathematics
- Stochastic Calculus
Background:
- The Bachelier model assumes asset prices and volatilities are uncorrelated.
- At-the-money implied volatility equals the fair value of a volatility swap under this assumption.
Purpose of the Study:
- To develop an analytical pricing method for out-of-the-money (OTM) and in-the-money (ITM) options.
- To apply this method as a variance reduction technique in Monte Carlo simulations for correlated asset prices.
Main Methods:
- Utilizing Itô calculus and Taylor expansions.
- Expressing option prices as a series expansion based on moneyness.
- Relating expansion coefficients to powers of future mean volatility.
Main Results:
- Derived an expansion for OTM and ITM option prices in the Bachelier model.
- Demonstrated the coefficients' dependence on future mean volatility.
- Successfully employed the derived prices as a control variate to decrease Monte Carlo variance.
Conclusions:
- The study provides a novel analytical approach to option pricing under specific model assumptions.
- The developed method offers an effective variance reduction strategy for complex financial simulations.
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