Can a Machine Correct Option Pricing Models?

Abstract

We introduce a novel two-step approach to predict implied volatility surfaces. Given any fitted parametric option pricing model, we train a feedforward neural network on the model-implied pricing errors to correct for mispricing and boost performance. Using a large dataset of S&P 500 options, we test our nonparametric correction on several parametric models ranging from ad-hoc Black–Scholes to structural stochastic volatility models and demonstrate the boosted performance for each model. Out-of-sample prediction exercises in the cross-section and in the option panel show that machine-corrected models always outperform their respective original ones, often by a large extent. Our method is relatively indiscriminate, bringing pricing errors down to a similar magnitude regardless of the misspecification of the original parametric model. Even so, correcting models that are less misspecified usually leads to additional improvements in performance and also outperforms a neural network fitted directly to the implied volatility surface.

Keywords:

• Boosting
• Deep learning
• Implied volatility
• Model correction
• Stochastic volatility

Disclosure Statement

The authors report there are no competing interests to declare.

$t=1,\dots ,T$$\sigma (t,m,\tau )$${\sigma }_p(t,m,\tau )$${\sigma }_{BS}(t,m,\tau )=a_0$${ϵ}_p(t,m,\tau )=\sigma (t,m,\tau )-{\sigma }_p(t,m,\tau )$$t=1,\dots ,T$$j=1,\dots ,n_t$${\{\sigma (t,m_{j,t},{\tau }_{j,t}), j=1,\dots ,n_t\}}_{t=1}^T$${\widehat{\sigma }}_p(t,m_{j,t},{\tau }_{j,t})$${\widehat{ϵ}}_p(t,m_{j,t},{\tau }_{j,t})=\sigma (t,m_{j,t},{\tau }_{j,t})-{\widehat{\sigma }}_p(t,m_{j,t},{\tau }_{j,t})$$\frac{1}{T}{\sum }_{t=1}^T\frac{1}{n_t}{\sum }_{j=1}^{n_t}{[{\widehat{ϵ}}_p(t,m_{j,t},{\tau }_{j,t})-f(m_{j,t},{\tau }_{j,t})]}^2,$${\widehat{\sigma }}_p(t,m_{j,t},{\tau }_{j,t})+\widehat{f}(m_{j,t},{\tau }_{j,t})$${\mathbf{y}}_t=(y_{1,t},\dots ,y_{q,t})$$\frac{1}{T}{\sum }_{t=1}^T\frac{1}{n_t}{\sum }_{j=1}^{n_t}{[{\widehat{ϵ}}_p(t,m_{j,t},{\tau }_{j,t})-f({\mathbf{y}}_t,m_{j,t},{\tau }_{j,t})]}^2,$${\widehat{\sigma }}_p(t,m_{j,t},{\tau }_{j,t})+\widehat{f}({\mathbf{y}}_t,m_{j,t},{\tau }_{j,t})$${\mathbf{y}}_t$$f(.)$${\mathbf{x}}_{i,t}=(m_{i,t},{\tau }_{i,t})\prime ,f({\mathbf{x}}_{i,t})$${\mathbf{x}}_{i,t}=({\mathbf{y}}_t,m_{i,t},{\tau }_{i,t})\prime ,f({\mathbf{x}}_{i,t})$$f:{\mathbb{R}}^{(q+2)}\to \mathbb{R}$$d_0=q+2$Acknowledgments

We would like to thank the Associate Editor, two anonymous referees and conference participants at Econometric and Big Data Analyses of Global Economy, Financial Markets and Economic Policies for useful comments and suggestions.

Notes

1 The implied volatility surface represents the implied volatility of each option as a function of its moneyness and time to maturity. The implied volatility of a given option is the volatility parameter that makes the Black and Scholes (1973) formula deliver the observed option price.

2 See Glad (1998), Fan and Ullah (1999), and Fan, Wu, and Feng (2009) for the advantages of a parametrically guided nonparametric approach.

3 More recently, Ait-Sahalia, Li, and Li (2021) reverse this logic by linking observed shape characteristics of the implied volatility surface to the coefficients of stochastic volatility models, while Bandi, Fusari, and Renò (2021) expand the characteristic function of the underlying asset process to price short-maturity options and study the relation between equity characteristics and sources of structural risk.

4 Essentially, the formula is obtained by noticing that the option can be dynamically hedged by buying and/or selling the underlying security financing with a risk-free bond.

5 There are several iterative methods available to solve for $C_{\text{BS}}^{-1}$, including the Newton–Raphson method, the bisection method and the Brent method.

6 Dumas, Fleming, and Whaley (1998) choose a quadratic specification due to the parabolic shape of implied volatilities in the cross-section and to favor a parsimonious model.

7 Specifying the regression as a function of the strike price instead of moneyness leads to essentially the same results.

8 Since our main goal is prediction, we do not impose the Feller condition in the estimation.

9 Provided that we allow $f(.)$ to be composed by a constant plus other functions of moneyness and maturity, which is generally the case.

10 Note that, for l = 1, the previous layer l – 1 is the input layer 0, where ${\mathbf{z}}_0={\mathbf{x}}_{i,t}$ and $d_0=2$.

11 In this sense, “deep learning” refers to the use of a deep neural network, where “deep” usually means that $L\ge 2$.

12 We use the implementation of this method available in Matlab, with its default values.

13 See, for instance, Hutchinson, Lo, and Poggio (1994), Rubinstein (1994), Bakshi, Cao, and Chen (1997), Ait-Sahalia and Lo (1998), Dumas, Fleming, and Whaley (1998), Duffie, Pan, and Singleton (2000), Garcia and Gençay (2000), Ait-Sahalia and Duarte (2003), Fan and Mancini (2009), Andersen, Fusari, and Todorov (2015), Carr and Wu (2016), and Ait-Sahalia, Li, and Li (2021).

14 More specifically, observations with zero volume, with price lower than 1/8 or violating the usual no-arbitrage conditions are dropped.

15 For PM settled options, the time to expiration is the number of days between the trade date and the expiration date, while for AM settled options, it is the number of days between the dates less one.

16 For the rare case that there is no such pair of ATM call and put options for a given time to maturity, we use the dividend yield provided by OptionMetrics.

17 See, for instance, Ait-Sahalia and Lo (1998), Fan and Mancini (2009), and Andersen, Fusari, and Todorov (2015).

18 See, among others, Bakshi, Cao, and Chen (1997), Dumas, Fleming, and Whaley (1998), and Fan and Mancini (2009).

19 For instance, Gonçalves and Guidolin (2006) propose a vector autoregressive approach to model the dynamics of the parameters of the AHBS model and predict their values in the future.

20 In unreported results available upon request, we also test Heston using the model-implied expectation at t of future variance at t + h, ${\mathbb{E}}_t[V_{t+h}]=\widehat{\overline{v}}+e^{-\widehat{\kappa }h/252}({\widehat{V}}_t-\widehat{\overline{v}})$, instead of ${\widehat{V}}_t$. We find that the predictive performance improves only marginally, without changing the qualitative nature of our results.

21 As a sanity check, we also conducted our tests with a standalone neural network and the results are identical to those of correcting the BS model. To avoid redundancy and save space, we omit these results.

22 Results for other network architectures and 5- and 21-day ahead predictions are qualitatively similar.

23 We thank an anonymous referee for this suggestion.

24 The in-sample training set consists of 604,749 options over 503 days, while the out-of-sample testing set consists of 662,442 options over 375 days.

25 https://tailindex.com/index.html.

26 That is, we use neural networks with three hidden layers with 32, 16, and 8 neurons, respectively.

27 The BS model is estimated as the constant implied volatility that minimizes the IVMSE in the in-sample option panel. The predictions of the BS model out-of-sample for any option are simply the estimated constant implied volatility.

28 More specifically, the level of the implied volatility surface is the average implied volatility for short-term ATM options, the term structure is the difference between the average implied volatility of long- and short-term ATM options, the skew is defined as the difference between the average implied volatility of short-term OTM put and OTM call options, and the skew term structure is the difference between the long- and short-term skew, where the long-term skew is defined analogously to the short-term skew.

29 More recently, Ait-Sahalia, Li, and Li (2021) propose stochastic volatility models designed to fit directly the shape characteristics of the implied volatility surface.

30 The same comment would apply to a standalone neural network, which is equivalent to correcting the BS model. This further highlights the usefulness of our model-guided approach.

31 See, for instance, the estimation of the Heston model in the option panel described in Section 2.3, where we estimate the structural parameters $\mathit{\xi }=(\overline{v},\kappa ,{\sigma }_v,\rho )$ which are fixed over time.

Additional information

Funding

Fan’s research was supported by NSFC grant No.71991470/71991471.