Discrete-time Option Research Articles

A New Closed-Form Discrete-Time Option Pricing Model with Stochastic Volatility

A New Closed-Form Discrete-Time Option Pricing Model with Stochastic Volatility

Volatility Bursts: A Discrete-Time Option Model with Multiple Volatility Components

Abstract I propose an affine discrete-time model, called Vector Autoregressive Gamma with volatility Bursts (VARG-B) in which volatility experiences, in addition to frequent and small changes, periods of sudden and extreme movements generated by a latent factor which evolves according to the Autoregressive Gamma Zero process. A key advantage of the discrete-time specification is the possibility of estimating the model using Kalman Filter techniques. Moreover, the VARG-B model leads to a fully analytic conditional Laplace transform which boils down to a closed-form option pricing formula. When estimated on S&P500 index options and returns, the new model provides more accurate option pricing and modeling of the IV surface compared with some alternative models.

Volatility Bursts: A Discrete-time Option Model with Multiple Volatility Components

I propose an affine discrete-time model, called Vector Autoregressive Gamma with volatility Bursts (VARG-B) in which volatility experiences, in addition to frequent and small changes, periods of sudden and extreme movements generated by a latent factor which evolves according to the Autoregressive Gamma Zero process. A key advantage of the discrete-time specification is that it makes it possible to estimate the model via the Extended Kalman Filter. Moreover, the VARG-B model leads to a fully analytic conditional Laplace transform, resulting in a closed-form option pricing formula. When estimated on S&P500 index options and returns the new model provides more accurate option pricing and modelling of the IV surface compared with some alternative models.

Refutation of a Rumor by the Mass Media: Mathematical Model and Numerical Experiments

The process is considered, in which an unreliable rumor spreads in society, which is opposed by the broadcasting of the mass media. In this case, the unreliability of hearing is understood so that the information of the media contains a refutation and thereby inoculates individuals, that is, makes them immune to hearing. At the same time, individuals who have managed to accept the rumor cease to trust the media and thereby become unavailable for persuasion. For this process, a mathematical model is proposed in two versions. The variant with continuous time reveals some of the mathematical properties of the model. The discrete time option is more convenient for analyzing real processes since it allows one to estimate the parameters of the model. To assess these parameters, data on the ratings of the main socio-political programs of Russian TV channels were used. Several scenario calculations of the model with these parameters are presented. The main conclusion is that if the information disseminated by the media is not viral, that is, it is not retold by viewers to their neighbors in society, then the media are unable to resist rumors.

Value-at-Risk and Expected Shortfall approaches for option premiums and the probability of default estimation based on ARMA models

The need to address the issue of risk management has given rise to a number of models for estimation the probability of default, as well as a special tool that allows to sell credit risk – a credit default swap (CDS). From the moment it appeared in 1994 until the crisis of 2008, that the CDS market was actively growing, and then sharply contracted. Currently, there is practically no CDS market in emerging economies (including Russia). This article is to improve the existing CDS valuation models by using discrete-time models that allow for more accurate assessment and forecasting of the selected asset dynamics, as well as new option pricing models that take into account the degree of risk acceptance by the option seller. This article is devoted to parametric discrete-time option pricing models that provide more accurate results than the traditional Black-Scholes continuous-time model. Improvement in the quality of assessment is achieved due to three factors: a more detailed consideration of the properties of the time series of the underlying asset (in particular, autocorrelation and heavy tails), the choice of the optimal number of parameters and the use of Value-at-Risk approach. As a result of the study, expressions were obtained for the premiums of European put and call options for a given level of risk under the assumption that the return on the underlying asset follows a stationary ARMA process with normal or Student's errors, as well as an expression for the credit spread under similar assumptions. The simplicity of the ARMA process underlying the model is a compromise between the complexity of model calibration and the quality of describing the dynamics of assets in the stock market. This approach allows to take into account both discreteness in asset pricing and take into account the current structure and the presence of interconnections for the time series of the asset under consideration (as opposed to the Black–Scholes model), which potentially allows better portfolio management in the stock market.

QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds

This article presents a discrete-time option pricing model that is rooted in reinforcement learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model can go model-free and learn to price and hedge an option directly from data, without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for the optimal pricing and hedging of options. Once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based reinforcement learning. Further, due to the simplicity and tractability of our model, which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relationship to the original BSM model, we suggest that our model could be used in the benchmarking of different RL algorithms for financial trading applications. TOPICS:Derivatives, options Key Findings • Reinforcement learning (RL) is the most natural way for pricing and hedging of options that relies directly on data and not on a specific model of asset pricing. • The discrete-time RL approach to option pricing generalizes classical continuous-time methods, enables tracking mis-hedging risk, which disappears in the formal continuous-time limit, and provides a consistent framework for using options for both hedging and speculation. • A simple quadratic reward function, which presents a minimal extension of the classical Black-Scholes framework when combined with the Q-learning method of RL, gives rise to a particularly simple computational scheme where option pricing and hedging are semi-analytical, as they amount to multiple uses of a conventional least-squares regression.

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.

Option-implied filtering: evidence from the GARCH option pricing model

One crucial task of option price modeling is to estimate latent state variables. This paper emphasizes the importance of incorporating option implied information to update latent state variables and sheds light on numerical developments to alleviate the cumbersome estimation process in option valuation. We propose a simple option-implied approximation to obtain the latent state variable and investigate its performance in option pricing. Specifically, we directly filter conditional variance from option implied volatilities (option-implied filtering) and compare its performance to that of a futures-based filtering technique and that of an option-based filtering technique with the Brownian Bridge process. Using a GARCH type discrete-time option pricing model and the crude oil option data, we demonstrate that the option-implied filtering technique significantly improves model fit and estimation efficiency, both in-sample and out-of-sample.

Mixed Fractional Merton Model to Evaluate European Options with Transaction Costs

This paper deals with the problem of discrete-time option pricing by the mixed fractional version of Merton model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. We also investigate the effect of the time-step δt and the Hurst parameter H on our pricing option model, which reveals that these parameters have high impact on option pricing. The properties of this model are also explained.

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) 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 a 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.

Volatility Bursts: A Discrete-Time Option Model with Multiple Volatility Components

I propose an affine discrete-time model, called Vector Autoregressive Gamma with volatility Bursts (VARG-B) in which volatility experiences, in addition to frequent and small changes, periods of sudden and extreme movements generated by a latent factor which evolves according to the Autoregressive Gamma Zero process. A key advantage of the discrete-time specification is that it makes it possible to estimate the model via the Extended Kalman Filter. Moreover, the VARG-B model leads to a fully analytic conditional Laplace transform, resulting in a closed-form option pricing formula. When estimated on S&P500 index options and returns the new model provides more accurate option pricing and modelling of the IV surface compared with some alternative models.

QLBS: Q-Learner in the Black-Scholes (-Merton) Worlds

This paper presents a discrete-time option pricing model that is rooted in Reinforcement Learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in the Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model is able to go model-free and learn to price and hedge an option directly from data generated from a dynamic replicating portfolio which is rebalanced at discrete times. If the world is according to BSM, our risk-averse Q-Learner converges, given enough training data, to the true BSM price and hedge ratio of the option in the continuous time limit, even if hedges applied at the stage of data generation are completely random (i.e. it can learn the BSM model itself, too!), because Q-Learning is an off-policy algorithm. If the world is different from a BSM world, the Q-Learner will find it out as well, because Q-Learning is a model-free algorithm. For finite time steps, the Q-Learner is able to efficiently calculate both the optimal hedge and optimal price for the option directly from trading data, and without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for optimal pricing and hedging of options, once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based Reinforcement Learning. Further, due to simplicity and tractability of our model which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relation to the original BSM model, we suggest that our model could be used for benchmarking of different RL algorithms for financial trading applications. A supplement to this paper can be found here: http://ssrn.com/abstract=3090608.

A generalized Brennan–Rubinstein approach for valuing options with stochastic interest rates

We construct a discrete-time option valuation model capable of taking into consideration multiple exercises and stochastic interests rates under a generalized Brennan–Rubinstein framework, and further apply the Geske and Johnson (1984) method for the valuation of American options. For implementation, we use only the once- and twice-exercisable option values to approximate American option values. Our numerical results show that the effects of stochastic interest rates are very large, particularly for out-of-the-money American put options with a high-risk underlying asset price. For options with a high-risk return on asset as the underlying assets, the effects of stochastic interest rates are negligible, even for options with long-term maturity. Finally, the sign of correlation between asset prices and bond prices plays an important role in determining whether the values of American put options under stochastic interest rates are larger or smaller than those under constant interest rates.

ANALYTICAL VALUATION OF VULNERABLE OPTIONS IN A DISCRETE-TIME FRAMEWORK

In this paper, we present a pricing model for vulnerable options in discrete time. A Generalized Autoregressive Conditional Heteroscedasticity process is used to describe the variance of the underlying asset, which is correlated with the returns of the asset. As for counterparty default risk, we study it in a reduced form model and the proposed model allows for the correlation between the intensity of default and the variance of the underlying asset. In this framework, we derive a closed-form solution for vulnerable options and investigate quantitative impacts of counterparty default risk on option prices.

Discrete-Time Option Pricing with Stochastic Liquidity

Classical option pricing theories are usually built on the law of one price, neglecting the impact of market liquidity that may contribute to significant bid-ask spreads. Within the framework of conic finance, we develop a stochastic liquidity model, extending the discrete-time constant liquidity model of Madan (2010). With this extension, we can replicate the term and skew structures of bid-ask spreads typically observed in option markets. We show how to implement such a stochastic liquidity model within our framework using multidimensional binomial trees and we calibrate it to call and put options on the S&P 500.

Long-Range Dependence in the Risk-Neutral Measure for the Market on Lehman Brothers Collapse

ABSTRACTThis paper discusses the long-range dependence in the risk-neutral stock return process of the S&P 500 index option market. To observe the long-range dependence together with fat-tails, I define the parametric model of fractional Lévy process. Since the continuous time fractional Lévy process allows arbitrage, I use discrete time option pricing model based on the fractional Lévy process. By model calibration, we can capture the long-range dependence in the S&P 500 index option market. The paper finds that the long range dependence becomes stronger for the volatile market caused by the Lehman Brothers Collapse, comparing with other less volatility markets.

Discrete-time option pricing with stochastic liquidity

Classical option pricing theories are usually built on the law of one price, neglecting the impact of market liquidity that may contribute to significant bid-ask spreads. Within the framework of conic finance, we develop a stochastic liquidity model, extending the discrete-time constant liquidity model of Madan (2010). With this extension, we can replicate the term and skew structures of bid-ask spreads typically observed in option markets. We show how to implement such a stochastic liquidity model within our framework using multidimensional binomial trees and we calibrate it to call and put options on the S&P 500.

An Elementary Approach to the Option Pricing Problem

Our goal here is to discuss the pricing problem of European and American options in discrete time using elementary calculus so as to be an easy reference for first year undergraduate students. Using the binomial model we compute the fair price of European and American options. We explain the notion of Arbitrage and the notion of the fair price of an option using common sense. We give a criterion that the holder can use to decide when it is appropriate to exercise the option. We prove the put-call parity formulas for both European and American options and we discuss the relation between American and European options. We give also the bounds for European and American options. We also discuss the portfolio's optimization problem and the fair value in the case where the holder can not produce the opposite portfolio.

Partial hedging of American options in discrete time and complete markets: convex duality and optimal Markov policies

In this paper we study the partial hedging of American options in discrete time and complete markets. This problem leads to a stochastic game maximizing over stopping times and minimizing over stochastic processes satisfying a budget constraint. In the first part of the paper we introduce a dual problem maximizing over two-dimensional vectors of randomized stopping times and parametrized by Lagrange multipliers. We show that the value function of the hedging problem is conjugate to the dual problem. The solution to the dual problem for a fixed Lagrange multiplier is given as the solution of a two-dimensional coupled singular control problem under which the American option is decomposed in terms of risk and cost. In the second part of the paper we show, under a Markov assumption, the existence of an optimal deterministic Markov policy with respect to an enlarged state process.

Large-Scale Empirical Tests of the Markov Tree Model

The Markov Tree model is a discrete-time option pricing model that accounts for short-term memory of the underlying asset. In this work, we compare the empirical performance of the Markov Tree model against that of the Black-Scholes model and Heston’s stochastic volatility model. Leveraging a total of five years of individual equity and index option data, and using three new methods for fitting the Markov Tree model, we find that the Markov Tree model makes smaller out-of-sample hedging errors than competing models. This comparison includes versions of Markov Tree and Black-Scholes models in which volatilities are strike- and maturity-dependent. Visualizing the errors over time, we find that the Markov Tree model yields more accurate and less risky single instrument hedges than Heston’s stochastic volatility model. A statistical resampling method indicates that the Markov Tree model’s superior hedging performance is due to its robustness with respect to noise in option data.