In this article we apply the ADE method to a number of partial differential equations in option pricing using one-factor models (Black–Scholes, local volatility, uncertain volatility). We first give an introduction to ADE. We discuss the stability, accuracy, and performance of ADE for a generic one-factor partial differential equation. Of particular importance is how we transform a problem on an unbounded domain to one on a bounded domain, thus avoiding complex mathematical techniques to find the optimal truncated boundary and the determination of the corresponding numerical boundary conditions.
The second part of the article examines a number of applications. We show that oscillation-free Greeks make ADE a suitable candidate for uncertain volatility models. We also apply ADE to the problem of calibration of local volatility using the Kolmogorov forward equation and show that the performance and the accuracy obtained make ADE the method of choice instead of the popular Crank–Nicolson method.
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