Journal of Risk and Financial Management

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19 pages, 549 KiB

Open AccessArticle

On the Asymptotic Behavior of the Optimal Exercise Price Near Expiry of an American Put Option under Stochastic Volatility

by Wenting Chen and Song-Ping Zhu

J. Risk Financial Manag. 2022, 15(5), 189; https://doi.org/10.3390/jrfm15050189 - 19 Apr 2022

Viewed by 1994

Abstract

The behavior of the optimal exercise price of American puts near expiry has been well studied under the Black–Scholes model as a result of a series of publications. However, the behavior of the optimal exercise price under a stochastic volatility model, such as [...] Read more.

The behavior of the optimal exercise price of American puts near expiry has been well studied under the Black–Scholes model as a result of a series of publications. However, the behavior of the optimal exercise price under a stochastic volatility model, such as the Heston model, has not been reported at all. Adopting the method of matched asymptotic expansions, this paper addresses the asymptotic behavior of American put options on a dividend-paying underlying with stochastic volatility near expiry. Through our analyses, we are able to show that the option price will be quite different from that evaluated under the Black–Scholes model, while the leading-order term of the optimal exercise price remains almost the same as the constant volatility case if the spot volatility is given the same value as the constant volatility in the Black–Scholes model. Results from numerical experiments also suggest that our analytical formulae derived from the asymptotic analysis are quite reasonable approximations for options with remaining times to expiry in the order of days or weeks. Full article

(This article belongs to the Special Issue Option Pricing)

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20 pages, 545 KiB

Open AccessArticle

Pricing Product Options and Using Them to Complete Markets for Functions of Two Underlying Asset Prices

by Dilip B. Madan and King Wang

J. Risk Financial Manag. 2021, 14(8), 355; https://doi.org/10.3390/jrfm14080355 - 4 Aug 2021

Cited by 4 | Viewed by 2078

Abstract

Options paying the product of put and/or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The [...] Read more.

Options paying the product of put and/or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The first perspective uses a geometric two-dimensional Brownian motion model. The second inverts two-dimensional characteristic functions. The third uses a bootstrapped physical measure to propose a risk charge minimizing hedge using options on the two underlying assets. The options are priced at the cost of the hedge plus the risk charge. Full article

(This article belongs to the Special Issue Option Pricing)

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21 pages, 1663 KiB

Open AccessArticle

American Option Pricing with Importance Sampling and Shifted Regressions

by Francois-Michel Boire, R. Mark Reesor and Lars Stentoft

J. Risk Financial Manag. 2021, 14(8), 340; https://doi.org/10.3390/jrfm14080340 - 22 Jul 2021

Cited by 4 | Viewed by 2967

Abstract

This paper proposes a new method for pricing American options that uses importance sampling to reduce estimator bias and variance in simulation-and-regression based methods. Our suggested method uses regressions under the importance measure directly, instead of under the nominal measure as is the [...] Read more.

This paper proposes a new method for pricing American options that uses importance sampling to reduce estimator bias and variance in simulation-and-regression based methods. Our suggested method uses regressions under the importance measure directly, instead of under the nominal measure as is the standard, to determine the optimal early exercise strategy. Our numerical results show that this method successfully reduces the bias plaguing the standard importance sampling method across a wide range of moneyness and maturities, with negligible change to estimator variance. When a low number of paths is used, our method always improves on the standard method and reduces average root mean squared error of estimated option prices by

22.5 %

. Full article

(This article belongs to the Special Issue Option Pricing)

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12 pages, 385 KiB

Open AccessArticle

Pricing and Hedging American-Style Options with Deep Learning

by Sebastian Becker, Patrick Cheridito and Arnulf Jentzen

J. Risk Financial Manag. 2020, 13(7), 158; https://doi.org/10.3390/jrfm13070158 - 19 Jul 2020

Cited by 29 | Viewed by 7422

Abstract

In this paper we introduce a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a lower bound for the price. Then it calculates an upper bound, a point estimate and [...] Read more.

In this paper we introduce a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a lower bound for the price. Then it calculates an upper bound, a point estimate and confidence intervals. Finally, it constructs an approximate dynamic hedging strategy. We test the approach on different specifications of a Bermudan max-call option. In all cases it produces highly accurate prices and dynamic hedging strategies with small replication errors. Full article

(This article belongs to the Special Issue Option Pricing)

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7 pages, 269 KiB

Open AccessCommunication

Pricing American Options with a Non-Constant Penalty Parameter

by Anna Clevenhaus, Matthias Ehrhardt, Michael Günther and Daniel Ševčovič

J. Risk Financial Manag. 2020, 13(6), 124; https://doi.org/10.3390/jrfm13060124 - 13 Jun 2020

Cited by 3 | Viewed by 2077

Abstract

As the American early exercise results in a free boundary problem, in this article we add a penalty term to obtain a partial differential equation, and we also focus on an improved definition of the penalty term for American options. We replace the [...] Read more.

As the American early exercise results in a free boundary problem, in this article we add a penalty term to obtain a partial differential equation, and we also focus on an improved definition of the penalty term for American options. We replace the constant penalty parameter with a time-dependent function. The novelty and advantage of our approach consists in introducing a bounded, time-dependent penalty function, enabling us to construct an efficient, stable, and adaptive numerical approximation scheme, while in contrast, the existing standard approach to the penalisation of the American put option-free boundary problem involves a constant penalty parameter. To gain insight into the accuracy of our proposed extension, we compare the solution of the extension to standard reference solutions from the literature. This illustrates the improvement of using a penalty function instead of a penalising constant. Full article

(This article belongs to the Special Issue Option Pricing)

11 pages, 257 KiB

Open AccessArticle

Testing the Information-Based Trading Hypothesis in the Option Market: Evidence from Share Repurchases

by Ihsan Badshah, Hardjo Koerniadi and James Kolari

J. Risk Financial Manag. 2019, 12(4), 179; https://doi.org/10.3390/jrfm12040179 - 29 Nov 2019

Cited by 1 | Viewed by 3038

Abstract

The informed options trading hypothesis posits that option prices lead stock prices. In this paper, we extended the research on this hypothesis to open-market share repurchases. Empirical tests showed that the implied volatility spread was not significantly related to buy-and-hold abnormal stock returns. [...] Read more.

The informed options trading hypothesis posits that option prices lead stock prices. In this paper, we extended the research on this hypothesis to open-market share repurchases. Empirical tests showed that the implied volatility spread was not significantly related to buy-and-hold abnormal stock returns. However, further evidence reveal a significant relationship between implied volatility spread and subsequent stock return volatility around open-market share repurchase events. We concluded that option traders have private information on the volatility of stock returns and superior information processing ability that accounts for prescient pricing behavior in options relative to stocks. Full article

(This article belongs to the Special Issue Option Pricing)

12 pages, 1156 KiB

Open AccessArticle

Correcting the Bias in the Practitioner Black-Scholes Method

by Yun Yin and Peter G. Moffatt

J. Risk Financial Manag. 2019, 12(4), 157; https://doi.org/10.3390/jrfm12040157 - 26 Sep 2019

Cited by 1 | Viewed by 3415

Abstract

We address a number of technical problems with the popular Practitioner Black-Scholes (PBS) method for valuing options. The method amounts to a two-stage procedure in which fitted values of implied volatilities (IV) from a linear regression are plugged into the Black-Scholes formula to [...] Read more.

We address a number of technical problems with the popular Practitioner Black-Scholes (PBS) method for valuing options. The method amounts to a two-stage procedure in which fitted values of implied volatilities (IV) from a linear regression are plugged into the Black-Scholes formula to obtain predicted option prices. Firstly we ensure that the prediction from stage one is positive by using log-linear regression. Secondly, we correct the bias that results from the transformation applied to the fitted values (i.e., the Black-Scholes formula) being a highly non-linear function of implied volatility. We apply the smearing technique in order to correct this bias. An alternative means of implementing the PBS approach is to use the market option price as the dependent variable and estimate the parameters of the IV equation by the method of non-linear least squares (NLLS). A problem we identify with this method is one of model incoherency: the IV equation that is estimated does not correspond to the set of option prices used to estimate it. We use the Monte Carlo method to verify that (1) standard PBS gives biased option values, both in-sample and out-of-sample; (2) using standard (log-linear) PBS with smearing almost completely eliminates the bias; (3) NLLS gives biased option values, but the bias is less severe than with standard PBS. We are led to conclude that, of the range of possible approaches to implementing PBS, log-linear PBS with smearing is preferred on the basis that it is the only approach that results in valuations with negligible bias. Full article

(This article belongs to the Special Issue Option Pricing)

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15 pages, 315 KiB

Open AccessArticle

Volatility Integration in Spot, Futures and Options Markets: A Regulatory Perspective

by Shailesh Rastogi and Chaitaly Athaley

J. Risk Financial Manag. 2019, 12(2), 98; https://doi.org/10.3390/jrfm12020098 - 9 Jun 2019

Cited by 8 | Viewed by 4417

Abstract

The aim of this paper is to study the integration of volatility in the three markets, viz. spot, futures and options, in order to provide input for hedging purposes and the formulation of policies for derivatives. The generalized method of moments (GMM) is [...] Read more.

The aim of this paper is to study the integration of volatility in the three markets, viz. spot, futures and options, in order to provide input for hedging purposes and the formulation of policies for derivatives. The generalized method of moments (GMM) is used to capture the simultaneous equation modelling of volatility in the three markets. The integration of the volatility in the three markets is also tested for structural breaks. The main finding of the paper is that the volatility in the options market is not associated with volatility in spot and futures market. However, volatility in spot and futures markets are associated with each other. As a consequence, investors can use options for hedging purposes and policy makers do not need to be concerned about the imminent impact of options markets on spot markets. To the best of the authors’ knowledge, there is no other study which discusses the integration of volatility in the three markets. Moreover, the finding of this paper that the options market behaves differently compared to the futures market has also not been discussed in earlier studies. Full article

(This article belongs to the Special Issue Option Pricing)

21 pages, 2542 KiB

Open AccessArticle

Arbitrage Free Approximations to Candidate Volatility Surface Quotations

by Dilip B. Madan and Wim Schoutens

J. Risk Financial Manag. 2019, 12(2), 69; https://doi.org/10.3390/jrfm12020069 - 21 Apr 2019

Cited by 5 | Viewed by 3292

Abstract

It is argued that the growth in the breadth of option strikes traded after the financial crisis of 2008 poses difficulties for the use of Fourier inversion methodologies in volatility surface calibration. Continuous time Markov chain approximations are proposed as an alternative. They [...] Read more.

It is argued that the growth in the breadth of option strikes traded after the financial crisis of 2008 poses difficulties for the use of Fourier inversion methodologies in volatility surface calibration. Continuous time Markov chain approximations are proposed as an alternative. They are shown to be adequate, competitive, and stable though slow for the moment. Further research can be devoted to speed enhancements. The Markov chain approximation is general and not constrained to processes with independent increments. Calibrations are illustrated for data on 2695 options across 28 maturities for

S P Y

as at 8 February 2018. Full article

(This article belongs to the Special Issue Option Pricing)

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