Limit theorems for prices of options written on semi-Markov processes

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

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Analytical Approximations of American Call Options with Discrete Dividends

Journal of Derivatives and Quantitative Studies ◽

10.1108/jdqs-03-2018-b0001 ◽

2018 ◽

Vol 26 (3) ◽

pp. 283-310

Author(s):

Kwangil Bae

Keyword(s):

Brownian Motion ◽

Stock Prices ◽

Stock Price ◽

Computing Time ◽

Option Price ◽

Geometric Brownian Motion ◽

Option Prices ◽

Call Options ◽

Analytical Approximations ◽

Discrete Dividends

In this study, we assume that stock prices follow piecewise geometric Brownian motion, a variant of geometric Brownian motion except the ex-dividend date, and find pricing formulas of American call options. While piecewise geometric Brownian motion can effectively incorporate discrete dividends into stock prices without losing consistency, the process results in the lack of closed-form solutions for option prices. We aim to resolve this by providing analytical approximation formulas for American call option prices under this process. Our work differs from other studies using the same assumption in at least three respects. First, we investigate the analytical approximations of American call options and examine European call options as a special case, while most analytical approximations in the literature cover only European options. Second, we provide both the upper and the lower bounds of option prices. Third, our solutions are equal to the exact price when the size of the dividend is proportional to the stock price, while binomial tree results never match the exact option price in any circumstance. The numerical analysis therefore demonstrates the efficiency of our method. Especially, the lower bound formula is accurate, and it can be further improved by considering second order approximations although it requires more computing time.

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HOLDER-EXTENDIBLE EUROPEAN OPTION: CORRECTIONS AND EXTENSIONS

The ANZIAM Journal ◽

10.1017/s1446181115000097 ◽

2015 ◽

Vol 56 (4) ◽

pp. 359-372 ◽

Cited By ~ 1

Author(s):

PAVEL V. SHEVCHENKO

Keyword(s):

Brownian Motion ◽

Interest Rate ◽

Real Life ◽

Option Price ◽

Geometric Brownian Motion ◽

Fixed Amount ◽

Underlying Asset ◽

Closed Form Solutions ◽

Pricing Options ◽

Interest Rate Volatility

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

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Option pricing under the fractional stochastic volatility model

ANZIAM Journal ◽

10.21914/anziamj.v63.15204 ◽

2021 ◽

Vol 63 ◽

pp. 123-142

Author(s):

Yuecai Han ◽

Zheng Li ◽

Chunyang Liu

Keyword(s):

Brownian Motion ◽

Option Pricing ◽

Fractional Brownian Motion ◽

Stochastic Volatility ◽

Option Price ◽

Stochastic Volatility Model ◽

Option Prices ◽

European Call Option ◽

Volatility Model ◽

The Impact

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

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OPTION PRICING UNDER THE FRACTIONAL STOCHASTIC VOLATILITY MODEL

The ANZIAM Journal ◽

10.1017/s1446181121000225 ◽

2021 ◽

pp. 1-20

Author(s):

Y. HAN ◽

Z. LI ◽

C. LIU

Keyword(s):

Brownian Motion ◽

Option Pricing ◽

Fractional Brownian Motion ◽

Stochastic Volatility ◽

Option Price ◽

Stochastic Volatility Model ◽

Option Prices ◽

European Call Option ◽

Volatility Model ◽

The Impact

Abstract We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented.

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A SIMPLE TIME-CONSISTENT MODEL FOR THE FORWARD DENSITY PROCESS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024913500489 ◽

2013 ◽

Vol 16 (08) ◽

pp. 1350048

Author(s):

HENRIK HULT ◽

FILIP LINDSKOG ◽

JOHAN NYKVIST

Keyword(s):

Brownian Motion ◽

Particle Filtering ◽

Option Price ◽

Time Varying ◽

Option Prices ◽

Price Data ◽

Consistent Model ◽

Forward Contract ◽

Future Value ◽

Density Process

In this paper, a simple model for the evolution of the forward density of the future value of an asset is proposed. The model allows for a straightforward initial calibration to option prices and has dynamics that are consistent with empirical findings from option price data. The model is constructed with the aim of being both simple and realistic, and avoid the need for frequent re-calibration. The model prices of n options and a forward contract are expressed as time-varying functions of an (n + 1)-dimensional Brownian motion and it is investigated how the Brownian trajectory can be determined from the trajectories of the price processes. An approach based on particle filtering is presented for determining the location of the driving Brownian motion from option prices observed in discrete time. A simulation study and an empirical study of call options on the S&P 500 index illustrate that the model provides a good fit to option price data.

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PRICING EXOTIC OPTIONS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800143037 ◽

2000 ◽

Vol 14 (3) ◽

pp. 317-326 ◽

Cited By ~ 3

Author(s):

Sheldon M. Ross ◽

J. George Shanthikumar

Keyword(s):

Brownian Motion ◽

Convex Function ◽

Option Price ◽

Geometric Brownian Motion ◽

Exotic Options ◽

European Option ◽

No Arbitrage ◽

Efficient Simulation ◽

Lookback Options ◽

Fixed Set

We show that if the payoff of a European option is a convex function of the prices of the security at a fixed set of times, then the geometric Brownian motion risk neutral option price is increasing in the volatility of the security. We also give efficient simulation procedures for determining the no-arbitrage prices of European barrier, Asian, and lookback options.

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A LOGISTIC NONLINEAR BLACK-SCHOLES-MERTON PARTIAL DIFFERENTIAL EQUATION: EUROPEAN OPTION

International Journal of Research -GRANTHAALAYAH ◽

10.29121/granthaalayah.v6.i6.2018.1393 ◽

2018 ◽

Vol 6 (6) ◽

pp. 480-487

Author(s):

Joseph Otula Nyakinda

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Brownian Motion ◽

Stock Price ◽

Option Price ◽

Logistic Models ◽

Geometric Brownian Motion ◽

Partial Differential ◽

Illiquid Markets ◽

Black Scholes

Nonlinear Black-Scholes equations provide more accurate values by taking into account more realistic assumptions, such as transaction costs, illiquid markets, risks from an unprotected portfolio or large investor's preferences, which may have an impact on the stock price, the volatility, the drift and the option price itself. Most modern models are represented by nonlinear variations of the well-known Black-Scholes Equation. On the other hand, asset security prices may naturally not shoot up indefinitely (exponentially) leading to the use of Verhulst’s Logistic equation. The objective of this study was to derive a Logistic Nonlinear Black Scholes Merton Partial Differential equation by incorporating the Logistic geometric Brownian motion. The methodology involves, analysis of the geometric Brownian motion, review of logistic models, process and lemma, stochastic volatility models and the derivation of the linear and nonlinear Black-Scholes-Merton partial differential equation. Illiquid markets have also been analyzed alongside stochastic differential equations. The result of this study may enhance reliable decision making based on a rational prediction of the future asset prices given that in reality the stock market may depict a nonlinear pattern.

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