Machine learning for quantitative finance: fast derivative pricing,hedging and fitting

In this paper, we show how we can deploy machine learning techniques in the context of traditional quant problems. We illustrate that for many classical problems, we can arrive at speed-ups of several orders of magnitude by deploying machine learning techniques based on Gaussian process regression. The price we have to pay for this extra speed is some loss of accuracy. However, we show that this reduced accuracy is often well within reasonable limits and hence very acceptable from a practical point of view. The concrete examples concern fitting and estimation. In the fitting context, we fit sophisticated Greek profiles and summarize implied volatility surfaces. In the estimation context, we reduce computation times for the calculation of vanilla option values under advanced models, the pricing of American options and the pricing of exotic options under models beyond the Black–Scholes setting.     

The Effect of Exercise Date Uncertainty on Employee Stock Option Value

The IASC recently recommended that employee compensation in the form of stock options be measured at the 'fair value' based on an option pricing model and the value should be recognized in financial statements. This follows adoption of SFAS No. 123 in the United States, which requires firms to estimate the value of employee stock options using either a Black‐Scholes or binomial model. Most US firms used the B‐S model for their 1996 financial statements. This study assumes that option life follows a Gamma distribution, allowing the variance of option life to be separate from its expected life. The results indicate the adjusted Black‐Scholes model could overvalue employee stock options on the grant date by as much as 72 percent for nondividend paying firms and by as much as 84 percent for dividend paying firms. The results further demonstrate the sensitivity of ESO values to the volatility of the expected option life, a parameter that the B‐S model or a Poisson process cannot accommodate. The variability of option life has an especially big impact on ESO value for firms whose ESOs have a relatively short life (5 years, for example) and high employee turnover. For such firms, the results indicate a binomial option pricing model is more appropriate for estimating ESO value than the B‐S type model.     

Binomial Option Pricing Biases and Inconsistent Implied Volatilities

We evaluate the binomial option pricing methodology (OPM) by examining simulated portfolio strategies. A key aspect of our study involves sampling from the empirical distribution of observed equity returns. Using a Monte Carlo simulation, we generate equity prices under known volatility and return parameters. We price American–style put options on the equity and evaluate the risk–adjusted performance of various strategies that require writing put options with different maturities and moneyness characteristics. The performance of these strategies is compared to an alternative strategy of investing in the underlying equity. The relative performance of the strategies allows us to identify biases in the binomial OPM leading to the well–known volatility smile . By adjusting option prices so as to rule out dominated option strategies in a mean–variance context, we are able to reduce the pricing errors of the OPM with respect to option prices obtained from the LIFFE. Our results suggest that a simple recalibration of inputs may improve binomial OPM performance.     

Index Option Pricing Models with Stochastic Volatility and Stochastic Interest Rates

This paper specifies a multivariate stochasticvolatility (SV) model for the S & P500 index and spot interest rateprocesses. We first estimate the multivariate SV model via theefficient method of moments (EMM) technique based on observations ofunderlying state variables, and then investigate the respective effects of stochastic interest rates, stochastic volatility, and asymmetric S & P500 index returns on option prices. We compute option prices using both reprojected underlying historical volatilities and the implied risk premiumof stochastic volatility to gauge each model's performance through direct comparison with observed market option prices on the index. Our major empirical findings are summarized as follows. First, while allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or leverage effect does help to explain the skewness of the volatility smile, allowing for stochastic interest rates has minimal impact on option prices in our case. Second, similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premium for stochastic volatility of asset returns. Based on the implied volatility risk premium, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information from the options market in pricing options. Finally, both the model diagnostics and option pricing errors in our study suggest that the Gaussian SV model is not sufficientin modeling short-term kurtosis of asset returns, an SV model withfatter-tailed noise or jump component may have better explanatory power.     

Is the realized volatility good for option pricing during the recent financial crisis?

The contributions of this paper are threefold. The first contribution is the proposed logarithmic HAR (log-HAR) option-pricing model, which is more convenient compared with other option pricing models associated with realized volatility in terms of simpler estimation procedure. The second contribution is the test of the empirical implications of heterogeneous autoregressive model of the realized volatility (HAR)-type models in the S&P 500 index options market with comparison of the non-linear asymmetric GARCH option-pricing model, which is the best model in pricing options among generalized autoregressive conditional heteroskedastic-type models. The third contribution is the empirical analysis based on options traded from July 3, 2007 to December 31, 2008, a period covering a recent financial crisis. Overall, the HAR-type models successfully predict out-of-sample option prices because they are based on realized volatilities, which are closer to the expected volatility in financial markets. However, mixed results exist between the log-HAR and the heterogeneous auto-regressive gamma models in pricing options because the former is better than the latter in times of turmoil, whereas it is worse during the rather stable periods.     

Empirical Performance of Alternative Option Pricing Models

Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance.     

Option pricing with random volatilities in complete markets

This article presents the theory of option pricing with random volatilities in complete markets. As such, it makes two contributions. First, the newly developed martingale measure technique is used to synthesize results dating from Merton (1973) through Eisenberg, (1985, 1987). This synthesis illustrates how Merton's formula, the CEV formula, and the Black-Scholes formula are special cases of the random volatility model derived herein. The impossibility of obtaining a self-financing trading strategy to duplicate an option in incomplete markets is demonstrated. This omission is important because option pricing models are often used for risk management, which requires the construction of synthetic options.Second, we derive a new formula, which is easy to interpret and easy to program, for pricing options given a random volatility. This formula (for a European call option) is seen to be a weighted average of Black-Scholes values, and is consistent with recent empirical studies finding evidence of mean-reversion in volatilities.Helpful comments from an anonymous referee are greatly appreciated.     

Market Risk and Model Risk for a Financial Institution Writing Options

Derivatives valuation and risk management involve heavy use of quantitative models. To develop a quantitative assessment of model risk as it affects the basic option writing strategy that might be followed by a financial institution, we conduct an empirical simulation, with and without hedging, using data from 1976 to 1996. Results indicate that imperfect models and inaccurate volatility forecasts create sizable risk exposure for option writers. We consider to what extent the damage due to model risk can be limited by pricing options using a higher volatility than the best estimate from historical data.     

Assessing the performance of symmetric and asymmetric implied volatility functions

This study examines several alternative symmetric and asymmetric model specifications of regression-based deterministic volatility models to identify the one that best characterizes the implied volatility functions of S&P 500 Index options in the period 1996–2009. We find that estimating the models with nonlinear least squares, instead of ordinary least squares, always results in lower pricing errors in both in- and out-of-sample comparisons. In-sample, asymmetric models of the moneyness ratio estimated separately on calls and puts provide the overall best performance. However, separating calls from puts violates the put-call-parity and leads to severe model mis-specification problems. Out-of-sample, symmetric models that use the logarithmic transformation of the strike price are the overall best ones. The lowest out-of-sample pricing errors are observed when implied volatility models are estimated consistently to the put-call-parity using the joint data set of out-of-the-money options. The out-of-sample pricing performance of the overall best model is shown to be resilient to extreme market conditions and compares quite favorably with continuous-time option pricing models that admit stochastic volatility and random jump risk factors.     

Oil futures volatility predictability: New evidence based on machine learning models

This paper comprehensively examines the connection between oil futures volatility and the financial market based on a model-rich environment, which contains traditional predicting models, machine learning models, and combination models. The results highlight the efficiency of machine learning models for oil futures volatility forecasting, particularly the ensemble models and neural network models. Most interestingly, we consider the “forecast combination puzzle” in machine learning models, and find that combination models continue to have more satisfactory performances in all types of situations. We also discuss the model interpretability and each indicator's contribution to the prediction. Our paper provides new insights for machine learning methods' applications in futures market volatility prediction, which is helpful for academics, policy-makers, and investors.     

An empirical comparison of GARCH option pricing models

Recent empirical studies have shown that GARCH models can be successfully used to describe option prices. Pricing such contracts requires knowledge of the risk neutral cumulative return distribution. Since the analytical forms of these distributions are generally unknown, computationally intensive numerical schemes are required for pricing to proceed. Heston and Nandi (2000) consider a particular GARCH structure that permits analytical solutions for pricing European options and they provide empirical support for their model. The analytical tractability comes at a potential cost of realism in the underlying GARCH dynamics. In particular, their model falls in the affine family, whereas most GARCH models that have been examined fall in the non-affine family. This article takes a closer look at this model with the objective of establishing whether there is a cost to restricting focus to models in the affine family. We confirm Heston and Nandi's findings, namely that their model can explain a significant portion of the volatility smile. However, we show that a simple non affine NGARCH option model is superior in removing biases from pricing residuals for all moneyness and maturity categories especially for out-the-money contracts. The implications of this finding are examined. JEL Classification G13     

VIX option pricing and CBOE VIX Term Structure: A new methodology for volatility derivatives valuation

This study integrates CBOE VIX Term Structure and VIX futures to simplify VIX option pricing in multifactor models. Exponential and hump volatility functions with one- to three-factor models of the VIX evolution are used to examine their pricing for VIX options across strikes and maturities. The results show that using exponential volatility functions presents an effective choice as pricing models for VIX calls, whereas hump volatility functions provide efficient out-of-sample valuation for most VIX puts, in particular with deep in-the-money and deep out-of-the-money. Pricing errors for calls can be further reduced with a two-factor model.     

Disagreement and equilibrium option trading volume

Using a complete market equilibrium model, we present results concerning the effect disagreement has on equilibrium option trading volume and positioning. We find that if agents agree on volatility, total option volume is independent of wealth distribution and average optimism. We also find option volume increasing in drift disagreement and decreasing in risk aversion and volatility. Pessimists are shown to write most options. With volatility disagreement, the results are less clear; however, we show agents with high volatility beliefs write deep out of the money options and buy close to the money options. Numerical comparative statics are also performed. This revised version was published online in June 2006 with corrections to the Cover Date.     

PRICING AND HEDGING SHORT STERLING OPTIONS USING NEURAL NETWORKS

This paper compares the performance of artificial neural networks (ANNs) with that of the modified Black model in both pricing and hedging short sterling options. Using high‐frequency data, standard and hybrid ANNs are trained to generate option prices. The hybrid ANN is significantly superior to both the modified Black model and the standard ANN in pricing call and put options. Hedge ratios for hedging short sterling options positions using short sterling futures are produced using the standard and hybrid ANN pricing models, the modified Black model, and also standard and hybrid ANNs trained directly on the hedge ratios. The performance of hedge ratios from ANNs directly trained on actual hedge ratios is significantly superior to those based on a pricing model, and to the modified Black model. Copyright © 2012 John Wiley & Sons, Ltd.     

Equity value,implied cost of equity and shareholders’ real options

This study investigates the effects of shareholders’ real options on (i) firm financial performance and (ii) estimations of the implied cost of equity. After measuring the equity value of steady‐state operations using the residual income model, and the abandonment and expansion options using the Black‐Scholes option pricing model, I find that firms with a large expansion (abandonment) option value experience better (worse) financial performance than those with a small such value. I also find that ignoring these options results in a downward bias in implied cost of equity estimates by an average of 1.23 percentage points.     

VIX期权定价与校正

VIX期权作为波动率衍生品能为金融机构提供有效的市场风险对冲工具。文献中对VIX期权定价的实证分析误差都很大,原因在于模型的选取误差以及校正方法和样本选取不妥。通过在VIX模型中加入均值回复因素和跳因素,可以使VIX过程更加合理,也可以使VIX期权定价精度更高。通过对VIX期权市场中间报价进行校正,得到了4个文献模型的参数估计,并比较4个模型的定价精度和正向隐含波动率偏斜拟合效果。     

Investor ambiguity,systemic banking risk and economic activity: The case of too-big-to-fail

This paper examines the relationship between investors' ambiguity in the financial options market and systemic banks' risk. Eliciting ambiguity information from option pricing data on the twelve major U.S. banks between 2003 and 2010, we show that higher behavioral deviations from risk-neutral and Bayesian valuation (i.e., investor ambiguity) are associated with higher systemic banks' downside, market and credit risks. Consistent with behavioral explanations, we confirm the detrimental effect of ambiguity on financial market outcomes and find strong evidence of ambiguity among call and put option holders. Variance decomposition indicates that such a pattern of behavior explains a significant proportion of U.S. banking risk variance. This effect is more pronounced during periods of economic turbulence and bank stress (i.e., the 2007–2009 crisis), and holds after controlling for size, tail risk, implied volatility, and volatility of volatility dynamics. We also document that ambiguity from the financial market has a depressing impact on real economic activity, including capacity utilization, non-farm payrolls and overall economic performance. Our findings are robust to alternative specifications of ambiguity such as multiple priors and expected utilities with uncertain probabilities.     

Application of Heston’s Model to the Chinese Stock Market

This article applies Heston’s (1993) stochastic volatility model to the Chinese stock market indices and subsequently assesses its pricing performance. A two-step estimation procedure is adopted to calibrate Heston’s model. First, we find that the option price is affected by both the moneyness and the maturity. Second, Heston’s model is more likely to overprice options, whereas the BS model tends to underestimate options. Finally, Heston’s model, by employing volatility as a random process, significantly improves the pricing accuracy compared to the BS model. Therefore, Heston’s model is tractable to analyze the Chinese stock market indices, and there is volatility risk that must not be overlooked in the Chinese stock market.     

Evaluation of Black-Scholes and GARCH Models Using Currency Call Options Data

This paper empirically examines the performance of Black-Scholes and Garch-M call option pricing models using call options data for British Pounds, Swiss Francs and Japanese Yen. The daily exchange rates exhibit an overwhelming presence of volatility clustering, suggesting that a richer model with ARCH/GARCH effects might have a better fit with actual prices. We perform dominant tests and calculate average percent mean squared errors of model prices. Our findings indicate that the Black-Scholes model outperforms the GARCH models. An implication of this result is that participants in the currency call options market do not seem to price volatility clusters in the underlying process.