scholarly journals APPROXIMATING EXPECTED VALUE OF AN OPTION WITH NON-LIPSCHITZ PAYOFF IN FRACTIONAL HESTON-TYPE MODEL

2020 ◽  
Vol 23 (05) ◽  
pp. 2050031
Author(s):  
YULIYA MISHURA ◽  
ANTON YURCHENKO-TYTARENKO
Keyword(s):  
Option Pricing ◽  
Rate Of Convergence ◽  
Explicit Formula ◽  
Malliavin Calculus ◽  
Asset Price ◽  
Payoff Function ◽  
Type Model ◽  
Alternative Formula ◽  
Discretization Schemes ◽  
Smooth Functional

In this paper, we consider option pricing in a framework of the fractional Heston-type model with [Formula: see text]. As it is impossible to obtain an explicit formula for the expectation [Formula: see text] in this case, where [Formula: see text] is the asset price at maturity time and [Formula: see text] is a payoff function, we provide a discretization schemes [Formula: see text] and [Formula: see text] for volatility and price processes correspondingly and study convergence [Formula: see text] as the mesh of the partition tends to zero. The rate of convergence is calculated. As we allow [Formula: see text] to be non-Lipschitz and/or to have discontinuities of the first kind which can cause errors if [Formula: see text] is replaced by [Formula: see text] under the expectation straightforwardly, we use Malliavin calculus techniques to provide an alternative formula for [Formula: see text] with smooth functional under the expectation.

scholarly journals Geometric Average Asian Option Pricing with Paying Dividend Yield under Non-Extensive Statistical Mechanics for Time-Varying Model

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 828 ◽  
Author(s):  
Jixia Wang ◽  
Yameng Zhang
Keyword(s):  
Statistical Mechanics ◽  
Option Pricing ◽  
Asset Price ◽  
Closed Form Solution ◽  
Real Data ◽  
Form Solution ◽  
Asian Option ◽  
Time Varying ◽  
Dividend Yield ◽  
Geometric Average

This paper is dedicated to the study of the geometric average Asian call option pricing under non-extensive statistical mechanics for a time-varying coefficient diffusion model. We employed the non-extensive Tsallis entropy distribution, which can describe the leptokurtosis and fat-tail characteristics of returns, to model the motion of the underlying asset price. Considering that economic variables change over time, we allowed the drift and diffusion terms in our model to be time-varying functions. We used the I t o ^ formula, Feynman–Kac formula, and P a d e ´ ansatz to obtain a closed-form solution of geometric average Asian option pricing with a paying dividend yield for a time-varying model. Moreover, the simulation study shows that the results obtained by our method fit the simulation data better than that of Zhao et al. From the analysis of real data, we identify the best value for q which can fit the real stock data, and the result shows that investors underestimate the risk using the Black–Scholes model compared to our model.

scholarly journals A LOW-BIAS SIMULATION SCHEME FOR THE SABR STOCHASTIC VOLATILITY MODEL

2012 ◽  
Vol 15 (02) ◽  
pp. 1250016 ◽  
Author(s):  
BIN CHEN ◽  
CORNELIS W. OOSTERLEE ◽  
HANS VAN DER WEIDE
Keyword(s):  
Stochastic Volatility ◽  
Asset Price ◽  
Realistic Model ◽  
Stochastic Volatility Model ◽  
Model Parameters ◽  
Fixed Income ◽  
Financial Industry ◽  
Volatility Model ◽  
Discretization Schemes ◽  
Simulation Scheme

The Stochastic Alpha Beta Rho Stochastic Volatility (SABR-SV) model is widely used in the financial industry for the pricing of fixed income instruments. In this paper we develop a low-bias simulation scheme for the SABR-SV model, which deals efficiently with (undesired) possible negative values in the asset price process, the martingale property of the discrete scheme and the discretization bias of commonly used Euler discretization schemes. The proposed algorithm is based the analytic properties of the governing distribution. Experiments with realistic model parameters show that this scheme is robust for interest rate valuation.

A RECOMBINING TREE METHOD FOR OPTION PRICING WITH STATE-DEPENDENT SWITCHING RATES

2016 ◽  
Vol 19 (02) ◽  
pp. 1650012 ◽  
Author(s):  
J. X. JIANG ◽  
R. H. LIU ◽  
D. NGUYEN
Keyword(s):  
Option Pricing ◽  
Regime Switching ◽  
Asset Price ◽  
Numerical Results ◽  
Price Process ◽  
State Dependent ◽  
Switching Models ◽  
Markovian Regime Switching ◽  
Tree Method ◽  
Markovian Regime

This paper develops simple and efficient tree approaches for option pricing in switching jump diffusion models where the rates of switching are assumed to depend on the underlying asset price process. The models generalize many existing models in the literature and in particular, the Markovian regime-switching models with jumps. The proposed trees grow linearly as the number of tree steps increases. Conditions on the choices of key parameters for the tree design are provided that guarantee the positivity of branch probabilities. Numerical results are provided and compared with results reported in the literature for the Markovian regime-switching cases. The reported numerical results for the state-dependent switching models are new and can be used for comparison in the future.

scholarly journals Good Volatility, Bad Volatility, and Option Pricing

2018 ◽  
Vol 54 (2) ◽  
pp. 695-727 ◽  
Author(s):  
Bruno Feunou ◽  
Cédric Okou
Keyword(s):  
Option Pricing ◽  
Stock Price ◽  
Asset Price ◽  
Jump Diffusion ◽  
Quadratic Variation ◽  
Pricing Model ◽  
Economic Gain ◽  
Underlying Asset ◽  
Model Free ◽  
Option Pricing Model

Advances in variance analysis permit the splitting of the total quadratic variation of a jump-diffusion process into upside and downside components. Recent studies establish that this decomposition enhances volatility predictions and highlight the upside/downside variance spread as a driver of the asymmetry in stock price distributions. To appraise the economic gain of this decomposition, we design a new and flexible option pricing model in which the underlying asset price exhibits distinct upside and downside semivariance dynamics driven by the model-free proxies of the variances. The new model outperforms common benchmarks, especially the alternative that splits the quadratic variation into diffusive and jump components.

TREE METHOD FOR OPTION PRICING UNDER STOCHASTIC VARIANCE

2000 ◽  
Vol 03 (03) ◽  
pp. 557-557
Author(s):  
DRAGAN ŠESTOVIĆ
Keyword(s):  
Option Pricing ◽  
Asset Price ◽  
Three Dimensional ◽  
Diffusion Limit ◽  
Lattice Method ◽  
Binomial Tree ◽  
Long Run ◽  
Hedging Portfolio ◽  
Risk Neutral Probabilities ◽  
Tree Method

We develop a recombining tree method for pricing of options by using a general two-factor stochastic-variance (SV) diffusion model for asset price dynamics. We show that it is possible to construct a riskless hedge by including additional short-term options in the hedging portfolio. This procedure gives us Partial Differential Equation (PDE) that can be solved by using standard numerical techniques giving us a unique option's price. We show that the option's price does not depend on the long run volatility forecast but only on the parameters of the model, which are related to the volatility of variance. We show one particular transformation of PDE to the Finite Difference Equation (FDE) that leads to the three-dimensional lattice method similar to the standard binomial-tree method. Our tree grows in the price-variance space and similarly to the binomial-tree, the coefficients of the FDE can be interpreted as risk neutral probabilities for jumping between the tree nodes. By investigating an error accumulation in the tree we found the stability criterion and the method that can be applied in order to achieve a stabile procedure. Our option pricing method can be used for European and American options and various payoffs. Since the procedure converges quickly and can be easily implemented, we believe that it could be useful for practitioners. The SV model we used here was shown earlier to be a diffusion limit of various GARCH-type models giving the possibility of using parameters obtained in the discrete-time GARCH framework as an input for our option pricing method.

scholarly journals Lookback Option Pricing Problems of Uncertain Mean-Reverting Stock Model

2021 ◽  
Vol 25 (5) ◽  
pp. 539-545
Author(s):  
Zhaopeng Liu ◽  
Keyword(s):  
Interest Rate ◽  
Option Pricing ◽  
Asset Price ◽  
Option Price ◽  
Cir Model ◽  
Lookback Option ◽  
Proposed Model ◽  
Stock Model ◽  
Path Dependent ◽  
The Interest Rate

A lookback option is a path-dependent option, offering a payoff that depends on the maximum or minimum value of the underlying asset price over the life of the option. This paper presents a new mean-reverting uncertain stock model with a floating interest rate to study the lookback option price, in which the processing of the interest rate is assumed to be the uncertain counterpart of the Cox–Ingersoll–Ross (CIR) model. The CIR model can reflect the fluctuations in the interest rate and ensure that such rate is positive. Subsequently, lookback option pricing formulas are derived through the α-path method and some mathematical properties of the uncertain option pricing formulas are discussed. In addition, several numerical examples are given to illustrate the effectiveness of the proposed model.

scholarly journals Option Pricing for Path-Dependent Options with Assets Exposed to Multiple Defaults Risk

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Taoshun He
Keyword(s):  
Option Pricing ◽  
Default Risk ◽  
Asset Price ◽  
Default Time ◽  
Original Technique ◽  
Endogenous Default ◽  
Lookback Options ◽  
Option Model ◽  
Analytical Formulas ◽  
Path Dependent

In the present paper, we derive analytical formulas for barrier and lookback options with underlying assets exposed to multiple defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price drop to zero and the exogenous counterparty default risk induces a drop in the asset price, but the asset can still be traded after this default time. An original technique is developed to valuate the barrier and lookback options by first conditioning on the predefault and the afterdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with the default-free option model and exogenous counterparty default risk option model.

scholarly journals Malliavin differentiability of the Heston volatility and applications to option pricing

2008 ◽  
Vol 40 (01) ◽  
pp. 144-162 ◽  
Author(s):  
Elisa Alòs ◽  
Christian-Oliver Ewald
Keyword(s):  
Option Pricing ◽  
Explicit Expression ◽  
Stochastic Volatility ◽  
Malliavin Calculus ◽  
Numerical Results ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Malliavin Derivative ◽  
Volatility Model

We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore, we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of Alòs (2006) in order to derive approximate option pricing formulae in the context of the Heston model. Numerical results are given.

Binomial tree method for option pricing: Discrete Carr and Madan formula approach

2021 ◽  
pp. 2150024
Author(s):  
Yoshifumi Muroi ◽  
Ryota Saeki ◽  
Shintaro Suda
Keyword(s):  
Option Pricing ◽  
Broad Class ◽  
Asset Price ◽  
Option Prices ◽  
European Option ◽  
Tree Model ◽  
Binomial Tree ◽  
Tree Models ◽  
Binomial Tree Model ◽  
Tree Method

This paper suggests a new Fourier analysis approach to evaluate the option prices and its sensitivities (Greeks) using the binomial tree model. In the last half of this paper, we show that option prices are efficiently and effectively evaluated using a semi-closed form formula for European option prices. We can compute option prices in a broad class of jump-diffusion models because we calculate the characteristic function for an underlying asset price numerically. Furthermore, we also compute the price of European options in the exp-Levy model. This numerical experiment gives new insights into option pricing in the nonparametric Levy model. The option prices and sensitivities can be computed very accurately and efficiently, even in binomial tree models with jumps.

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