Multiplicative noise,fast convolution and pricing

In this work we detail the application of a fast convolution algorithm to compute high-dimensional integrals in the context of multiplicative noise stochastic processes. The algorithm provides a numerical solution to the problem of characterizing conditional probability density functions at arbitrary times, and we apply it successfully to quadratic and piecewise linear diffusion processes. The ability to reproduce statistical features of financial return time series, such as thickness of the tails and scaling properties, makes these processes appealing for option pricing. Since exact analytical results are lacking, we exploit the fast convolution as a numerical method alternative to Monte Carlo simulation both in the objective and risk-neutral settings. In numerical sections we document how fast convolution outperforms Monte Carlo both in speed and efficiency terms.     

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985 Leland, HE. 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [Crossref], [Web of Science ®] , [Google Scholar], 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.     

Risk measure pricing and hedging in incomplete markets

This article attempts to extend the complete market option pricing theory to incomplete markets. Instead of eliminating the risk by a perfect hedging portfolio, partial hedging will be adopted and some residual risk at expiration will be tolerated. The risk measure (or risk indifference) prices charged for buying or selling an option are associated to the capital required for dynamic hedging so that the risk exposure will not increase. The associated optimal hedging portfolio is decided by minimizing a convex measure of risk. I will give the definition of risk-efficient options and confirm that options evaluated by risk measure pricing rules are indeed risk-efficient. Relationships to utility indifference pricing and pricing by valuation and stress measures will be discussed. Examples using the shortfall risk measure and average VaR will be shown. The work of Mingxin Xu is supported by the National Science Foundation under grant SES-0518869. I would like to thank Steven Shreve for insightful comments, especially his suggestions to extend the pricing idea from using shortfall risk measure to coherent ones, and to study its relationship to utility based derivative pricing. The comments from the associate editor and the anonymous referee have reshaped the paper into its current version. The paper has benefited from discussions with Freddy Delbaen, Jan Večeř, David Heath, Dmitry Kramkov, Peter Carr, and Joel Avrin.     

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black–Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed form formula for integration. Moreover, these two methods solve the backward dynamic programing problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of the binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite that this model is only bidimensional, the whole history of the process impacts on the price, and how to handle all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.     

Pricing options on illiquid assets with liquid proxies using utility indifference and dynamic-static hedging

This work addresses the problem of optimal pricing and hedging of a European option on an illiquid asset Z using two proxies: a liquid asset S and a liquid European option on another liquid asset Y. We assume that the S-hedge is dynamic while the Y-hedge is static. Using the indifference pricing approach, we derive a Hamilton–Jacobi–Bellman equation for the value function. We solve this equation analytically (in quadrature) using an asymptotic expansion around the limit of perfect correlation between assets Y and Z. While in this paper we apply our framework to an incomplete market version of Merton’s credit-equity model, the same approach can be used for other asset classes (equity, commodity, FX, etc.), e.g. for pricing and hedging options with illiquid strikes or illiquid exotic options.     

Machine learning methods in finance: Recent applications and prospects

We study how researchers can apply machine learning (ML) methods in finance. We first establish that the two major categories of ML (supervised and unsupervised learning) address fundamentally different problems than traditional econometric approaches. Then, we review the current state of research on ML in finance and identify three archetypes of applications: (i) the construction of superior and novel measures, (ii) the reduction of prediction error, and (iii) the extension of the standard econometric toolset. With this taxonomy, we give an outlook on potential future directions for both researchers and practitioners. Our results suggest many benefits of ML methods compared to traditional approaches and indicate that ML holds great potential for future research in finance.     

Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models

We present a neural network-based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models—including second-generation stochastic volatility models and the rough volatility family—and a range of derivative contracts. Neural networks in this work are used in an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. The form in which information from available data is extracted and used influences network performance: The grid-based algorithm used for calibration is inspired by representing the implied volatility and option prices as a collection of pixels. We highlight how this perspective opens new horizons for quantitative modelling. The calibration bottleneck posed by a slow pricing of derivative contracts is lifted, and stochastic volatility models (classical and rough) can be handled in great generality as the framework also allows taking the forward variance curve as an input. We demonstrate the calibration performance both on simulated and historical data, on different derivative contracts and on a number of example models of increasing complexity, and also showcase some of the potentials of this approach towards model recognition. The algorithm and examples are provided in the Github repository GitHub: NN-StochVol-Calibrations.     

Fundamentalists,chartists and asset pricing anomalies

This paper investigates whether excess volatility of asset prices and serial correlations of stock monthly returns may be explained by the interactions between fundamentalists and chartists. Fundamentalists forecast future prices cum dividends through an adaptive learning rule. In contrast, chartists forecast future prices based on the observation of past price movements. Numerical simulations reveal that the interplay of fundamentalists and chartists robustly generates excess volatility of asset prices, volatility clustering, trends in prices (i.e. positive serial correlations of returns) over short horizons and oscillations in prices (i.e. negative serial correlations of returns) over long horizons, often observed in financial data. Moreover, we find that the memory of the learning rule plays a key role in explaining the above-mentioned stylized facts. In particular, we establish that excess volatility of asset prices; volatility clustering and autocorrelation of returns at different horizons emerge when fundamentalists have short memory. However, volatility clustering as well as short-run and long-run dependencies, observed in financial time series, are more pronounced when fundamentalists have longer memory.     

Identifying,valuing and hedging of embedded options in non-maturity deposits

Non-maturity deposits like savings accounts or demand deposits contain significant option risks caused by the bank’s discretionary pricing and the customers’ withdrawal right. Option risks follow from inherent non-linear factor exposures. I propose an ordinal response model for deposit rate jumps to identify non-linear factor exposures and a discrete-time term structure model to value the resulting option risks and to derive hedge measures “outside the model”. My delta profile resembles a constant maturity swap, but vega and gamma are more pronounced, which demonstrates that the widespread practice of static hedging with zero bonds is inadequate.     

Nonlinear asset pricing in Chinese stock market: A deep learning approach

The redesign of asset pricing models failed to integrate the frequent financial phenomenon that stock markets exhibit a non-linear long- and short-term memory structure. The difficulty lies in developing a nonlinear pricing structure capable of depicting the memory influence of the pricing variable. This paper presents a Long- and Short-Term Memory Neural Network Model (LSTM) to capture the non-linear pricing structure among five elements in the Chinese stock market, including market portfolio return, market capitalisation, book-to-market ratio, earnings factor, and investment factor. The long–short-term memory structure implies that the autocorrelation function of the stock return series decays slowly and has a long-term characteristic. The LSTM model surpasses the standard Fama–French five-factor model in terms of out-of-sample goodness-of-fit and long–short strategy performance. The empirical findings indicate that the LSTM nonlinear model properly represents the nonlinear relationships between the five components.     

Generative Bayesian neural network model for risk-neutral pricing of American index options

Financial models with stochastic volatility or jumps play a critical role as alternative option pricing models for the classical Black–Scholes model, which have the ability to fit different market volatility structures. Recently, machine learning models have elicited considerable attention from researchers because of their improved prediction accuracy in pricing financial derivatives. We propose a generative Bayesian learning model that incorporates a prior reflecting a risk-neutral pricing structure to provide fair prices for the deep ITM and the deep OTM options that are rarely traded. We conduct a comprehensive empirical study to compare classical financial option models with machine learning models in terms of model estimation and prediction using S&P 100 American put options from 2003 to 2012. Results indicate that machine learning models demonstrate better prediction performance than the classical financial option models. Especially, we observe that the generative Bayesian neural network model demonstrates the best overall prediction performance.     

Machine learning loss given default for corporate debt

We apply multiple machine learning (ML) methods to model loss given default (LGD) for corporate debt using a common dataset that is cross-sectional but collected over different time periods and shows much variation over time. We investigate the efficacy of three cross-validation (CV) schemes for hyper-parameter tuning and bootstrap aggregation (Bagging) in preventing out-of-time model performance deterioration. The three CV methods are shuffled K-fold, unshuffled K-fold and sequential blocked, which completely destroys, keeps some and completely retains the chronological order in the data, respectively. We find that it is important to keep the chronological order in the data when creating the training and testing samples, and the more the chronological order that can be retained, the more stable the out-of-time ML LGD model performance. By contrast, although bagging improves out-of-time fit in some cases, its effectiveness is rather marginal relative to that from the unshuffled K-fold and sequential blocked CV methods. Substantial uncertainty in relative out-of-time performance remains, however, thus ongoing model performance monitoring and benchmarking are still essential for sound model risk management for corporate LGD and other ML models.     

Growth uncertainty,generalized disappointment aversion and production-based asset pricing

We study a production economy with regime switching in the conditional mean and volatility of productivity growth. The representative agent has generalized disappointment aversion (GDA) preferences. We show that volatility risk in productivity growth carries a positive and sizable risk premium in levered equity. Our model can endogenously generate long-run risks in the volatility of consumption growth observed in the data. We show that introducing leverage with a procyclical dividend process consistent with the data is critical for the GDA preferences to have a large impact on equity returns.     

Information,hedging demand,and institutional investors: Evidence from the Taiwan Futures Exchange

This paper examines the effect of hedging demand by various types of institutional investor on subsequent returns and volatility. Using data from the Taiwan Futures Exchange, empirical results indicate that the hedging demand of foreign investors has a significant negative impact on subsequent returns and volatility. In addition, trading strategies based on the extreme hedging demand of foreigners are positively correlated with trading performance. Furthermore, there is evidence to show that returns (volatility) also affect the subsequent hedging demand of foreign investors, suggesting a feedback relation. Finally, the hedging demand of foreign investors has a greater impact on subsequent returns and volatility after global financial turmoil. Accordingly, this paper concludes that foreign investors are informed hedgers in the Taiwan futures market, especially after global financial turmoil.     

Bond and option pricing for interest rate model with clustering effects

This paper analyzes an interest rate model with self-exciting jumps, in which a jump in the interest rate model increases the intensity of jumps in the same model. This self-exciting property leads to clustering effects in the interest rate model. We obtain a closed-form expression for the conditional moment-generating function when the model coefficients have affine structures. Based on the Girsanov-type measure transformation for general jump-diffusion processes, we derive the evolution of the interest rate under the equivalent martingale measure and an explicit expression of the zero-coupon bond pricing formula. Furthermore, we give a pricing formula for the European call option written on zero-coupon bonds. Finally, we provide an interpretation for the clustering effects in the interest rate model within a simple framework of general equilibrium. Indeed, we construct an interest rate model, the equilibrium state of which coincides with the interest rate model with clustering effects proposed in this paper.     

The QLBS Q-Learner goes NuQLear: fitted Q iteration,inverse RL,and option portfolios

The QLBS model is a discrete-time option hedging and pricing model that is based on Dynamic Programming (DP) and Reinforcement Learning (RL). It combines the famous Q-Learning method for RL with the Black–Scholes (–Merton) (BSM) model's idea of reducing the problem of option pricing and hedging to the problem of optimal rebalancing of a dynamic replicating portfolio for the option, which is made of a stock and cash. Here we expand on several NuQLear (Numerical Q-Learning) topics with the QLBS model. First, we investigate the performance of Fitted Q Iteration for an RL (data-driven) solution to the model, and benchmark it versus a DP (model-based) solution, as well as versus the BSM model. Second, we develop an Inverse Reinforcement Learning (IRL) setting for the model, where we only observe prices and actions (re-hedges) taken by a trader, but not rewards. Third, we outline how the QLBS model can be used for pricing portfolios of options, rather than a single option in isolation, thus providing its own, data-driven and model-independent solution to the (in)famous volatility smile problem of the Black–Scholes model.     

Drivers of economic and financial integration: A machine learning approach

We propose a new approach to identifying drivers of economic and financial integration, separately, and across emerging and developed countries. Our advanced machine learning technique allows for nonlinear relationships, corrects for over-fitting, and is less prone to noise. It also can tackle a large number of highly correlated explanatory variables and controls for multicollinearity. Results suggest that general economic growth, increasing international trade, and contained population growth have helped emerging countries catch up to the level of the economic integration of developed countries. However, slow financial development and a high level of investment riskiness have hindered the speed of emerging countries’ financial integration. Furthermore, the results suggest that integration is a gradual process and is not driven by cyclical or transitory events.     

Matthew F. Dixon,Igor Halperin,and Paul Bilokon: Machine learning in finance from theory to practice

Financial Markets and Portfolio Management -     

Dynamic spillover effects among green bond,renewable energy stocks and carbon markets during COVID-19 pandemic: Implications for hedging and investments strategies

This study has been inspired by the emergence of socially responsible investment practices in mainstream investment activity as it examines the transmission of return patterns between green bonds, carbon prices, and renewable energy stocks, using daily data spanning from 4th January 2015 to 22nd September 2020. In this study, our dataset comprises the price indices of S&P Green Bond, Solactive Global Solar, Solactive Global Wind, S&P Global Clean Energy and Carbon. We employ the TVP-VAR approach to investigate the return spillovers and connectedness, and various portfolio techniques including minimum variance portfolio, minimum correlation portfolio and the recently developed minimum connectedness portfolio to test portfolio performance. Additionally, a LASSO dynamic connectedness model is used for robustness purposes. The empirical results from the TVP-VAR indicate that the dynamic total connectedness across the assets is heterogeneous over time and economic event dependent. Moreover, our findings suggest that clean energy dominates all other markets and is seen to be the main net transmitter of shocks in the entire network with Green Bonds and Solactive Global Wind, emerging to be the major recipients of shocks in the system. Based on the hedging effectiveness, we show that bivariate and multivariate portfolios significantly reduce the risk of investing in a single asset except for Green Bonds. Finally, the minimum connectedness portfolio reaches the highest Sharpe ratio implying that information concerning the return transmission process is helpful for portfolio creation. The same pattern has been observed during the COVID-19 pandemic period.