Abstract

Computing implied volatility from observed option prices is a frequent and challenging task in finance, even more in the presence of dividends. In this work, we employ a data-driven machine learning approach to determine the Black–Scholes implied volatility, including European-style and American-style options. The inverse function of the pricing model is approximated by an artificial neural network, which decouples the offline (training) and online (prediction) phases and eliminates the need for an iterative process to solve a minimization problem. Meanwhile, two challenging issues are tackled to improve accuracy and robustness, i.e., steep gradients of the volatility with respect to the option price and irregular early-exercise domains for American options. It is shown that deep neural networks can be used as an efficient numerical technique to compute implied volatility from European/American options. An extended version of this work can be found in .

Highlights

  • Computing implied volatility from observed option prices is a frequent and challenging task in finance, even more in the presence of dividends

  • Given an observed market option price V mkt (European or American), the Black–Scholes implied volatility σ∗ is defined by BS(σ∗ ; S0, K, τ, r, q, α) = V mkt

  • In order to avoid an iterative algorithm, we provide a data-driven approach for directly approximating the inverse function of (3) via neural networks

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Summary

Problem Formulation

Where r and q are the risk-less interest rate and continuous dividend yield, respectively. For American options, the original Black–Scholes equation becomes a variational inequality,. We can employ a numerical method, here the COS method [2], to solve the American pricing model. The European/American Black–Scholes solution is denoted by Veu/am = BSeu/am (σ, S0 , K, τ, r, q, α). Given an observed market option price V mkt (European or American), the Black–Scholes implied volatility σ∗ is defined by BS(σ∗ ; S0 , K, τ, r, q, α) = V mkt. There does not exist a closed-form expression of the inverse function for neither American-style or European-style options. There are several root-finding numerical algorithms to solve (4), for example, Newton–Raphson, σk∗+1 = σk∗ −. Some issues are likely to arise when using derivative-based algorithms, see Figure 1

Methodology
ANN for European Implied Volatility
ANN for American Implied Volatility
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