Numerical Approximation of the Implied Volatility Under Arithmetic Brownian Motion

Abstract

We provide an accurate approximation method for inverting an option price to the implied volatility under arithmetic Brownian motion, which is widely quoted in Fixed Income markets. The maximum error in the volatility is in the order of 10⁻¹⁰ of the given option price and much smaller for the near-the-money options. Thus our approximation can be used as an exact solution without further refinements of iterative methods.

Key Words:

• Normal implied volatility
• basis point volatility
• arithmetic Brownian motion
• rational approximation
• closed form approximation

Acknowledgements

This work was supported by a BK21 project of the Department of Mathematical Science, KAIST.