SABR equipped with AI wings

Abstract

This study proposes an artificial neural network (ANN) based option pricing framework under the SABR (stochastic alpha beta rho) and free boundary SABR volatility models. Unlike previous research, we do not directly apply the ANN technique to train and predict option implied volatilities. Instead, we use the technique to train and predict the differences between the implied volatilities calculated by asymptotic approximation and those computed by Monte Carlo simulation. Since the analytical solution for an option written on an underlying asset price in the SABR model is intractable, approximation techniques are used to derive an approximate closed-form solution in practice. However, these approximation formulas worsen as maturity increases and the underlying asset's volatility is high. The accuracy decreases rapidly in wings, which represents deep in-the-money and out-of-the-money cases. By combining the approximation formulas and ANN framework, our new option pricing method offers the following improvements: (1) the training becomes more robust, and the predictions produce more stable and accurate results: (2) it significantly speeds up the training procedure: and (3) the accuracy and efficiency of approximation are improved in the wings without sacrificing performance.

Keywords:

• Artificial neural network
• SABR stochastic volatility model
• Derivatives
• Asymptotic expansion
• Monte Carlo simulation

Acknowledgments

The author is grateful to anonymous referee and managing editor for invaluable comments that improved the original manuscript considerably.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 If the maturity is shorter than 3 years, the accuracy of the approximations seems high enough for practical usage; hence, we omit these cases.

2 We compute M times MC simulations to create training and testing data $N=M\times 21$ and for each MC simulation, W trials are run.

3 We generate K in the range defined by (43(43) $\begin{array}{rl}{K}_1& =max(\mu -2\sqrt{V},0.4f),\end{array}$(43) ) and (44(44) $\begin{array}{rl}{K}_{21}& =min(\mu +2\sqrt{V},2.1f),\end{array}$(44) ) but do not use fixed values because, depending on the combination of $(\alpha ,\text{β},\nu ,\rho )$, volatility of the underlying asset becomes extremely small or large. If the volatility is extremely small, the fixed strike could be unreasonably high or small, which differs from the actual market. On the other hand, if the volatility is large, $K_1$ could be negative; therefore, we set the floor in (43(43) $\begin{array}{rl}{K}_1& =max(\mu -2\sqrt{V},0.4f),\end{array}$(43) ). Similarly, since an extremely high strike is unlikely, we impose a cap in (44(44) $\begin{array}{rl}{K}_{21}& =min(\mu +2\sqrt{V},2.1f),\end{array}$(44) )

4 The computational cost depends on the choice of the hyper-parameters.

5 Here, we listed results used CPU-only TensorFlow with intel(R) Core(TM) i9-10980XE CPU to make it easier to understand. We also examined TensorFlow-gpu with Quadro RTX A6000 but it did not improve computational performance because thanks to the mapping ${\mathcal{M}}_D$, we can reduce the amount of training data, batchsize, and number of hidden layers.

6 We omit T<1 because in these cases the accuracy of the Hagan et al. (2002)'s approximation is high enough for practical usage.

7 We used one PC with an intel(R) Core(TM) i9-10980XE CPU with 18 cores and 36 threads. The test is performed under a multi-thread application running on a multiprocessor with 30 threads and takes thirteen days to compute 5,000 scenarios.

8 We use PRelu as activation function since input parameter could take a negative value.

9 If the asset price follows a CEV process under a certain measure, absorption condition is the only acceptable boundary condition at zero to ensure arbitrage-free condition. This is because, borrowing the words of Chen et al. (2012),

if there were a reflecting boundary at zero, then for an initially worthless portfolio, one would take a long position in the asset once the price zero is reached (which would happen with a strictly positive probability) and sell it immediately when the 0 boundary has reflected the price process to obtain risk-less profit.

10 Note that, Bayer et al. (2016) calibrated their rough Bergomi parameters to the SPX surface to obtain H = 0.05, $\eta =2.3$, and $\rho =-0.9$. Hence, the approximation is accurate only for a short-dated.

Additional information

Funding

This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) JP22K13436.