REPLICATION SCHEME FOR THE PRICING OF EUROPEAN OPTIONS

2021 ◽  
pp. 2150014
Author(s):  
HIDEHARU FUNAHASHI
Keyword(s):  
Diffusion Process ◽  
Density Function ◽  
Numerical Experiments ◽  
Analytical Formula ◽  
Option Price ◽  
Option Prices ◽  
European Options ◽  
European Option ◽  
Volatility Models ◽  
Replication Scheme

This paper proposes an efficient method for calculating European option prices under local, stochastic, and fractional volatility models. Instead of directly calculating the density function of a target underlying asset, we replicate it from a simpler diffusion process with a known analytical solution for the European option. For this purpose, we derive six functions that characterize the density function of a diffusion process, for both the original and simpler processes and match these functions so that the latter mimics the former. Using the analytical formula, we then approximate the option price of the target asset. By comparison with previous works and numerical experiments, we show that the accuracy of our approximation is high, and the calculation is fast enough for practical purposes; hence, it is suitable for calibration purposes.

scholarly journals PRICING AND HEDGING OF VIX OPTIONS FOR BARNDORFF-NIELSEN AND SHEPHARD MODELS

2019 ◽  
Vol 22 (08) ◽  
pp. 1950043 ◽  
Author(s):  
TAKUJI ARAI
Keyword(s):  
Fourier Transform ◽  
Stochastic Volatility ◽  
Numerical Experiments ◽  
Option Price ◽  
Call Option ◽  
Stochastic Volatility Models ◽  
Risky Assets ◽  
Volatility Models ◽  
Vix Options ◽  
Non Gaussian

The VIX call options for the Barndorff-Nielsen and Shephard models will be discussed. Derivatives written on the VIX, which is the most popular volatility measurement, have been traded actively very much. In this paper, we give representations of the VIX call option price for the Barndorff-Nielsen and Shephard models: non-Gaussian Ornstein–Uhlenbeck type stochastic volatility models. Moreover, we provide representations of the locally risk-minimizing strategy constructed by a combination of the underlying riskless and risky assets. Remark that the representations obtained in this paper are efficient to develop a numerical method using the fast Fourier transform. Thus, numerical experiments will be implemented in the last section of this paper.

AN APPROXIMATION METHOD FOR PRICING CONTINUOUS BARRIER OPTIONS UNDER MULTI-ASSET LOCAL STOCHASTIC VOLATILITY MODELS

2020 ◽  
pp. 2050051
Author(s):  
KENICHIRO SHIRAYA
Keyword(s):  
Risk Management ◽  
Stochastic Volatility ◽  
Approximation Method ◽  
Numerical Experiments ◽  
Analytical Approximation ◽  
Barrier Options ◽  
European Options ◽  
Stochastic Volatility Models ◽  
Volatility Models

This paper presents a new approximation method for pricing multi-asset continuous single-barrier options. Barrier options are frequently traded, and it is necessary for practitioners to evaluate these precisely and quickly, both for competitiveness, and for risk management. However, it is a difficult task under local stochastic volatility models. To the best of our knowledge, this paper is the first to provide an analytical approximation for continuous barrier options prices in a multi-asset environment. In numerical experiments, we examine the validity of the formula by using parameters calibrated to EURUSD European options.

CONVERGENCE SPEED OF GARCH OPTION PRICE TO DIFFUSION OPTION PRICE

2009 ◽  
Vol 12 (03) ◽  
pp. 359-391 ◽  
Author(s):  
JIN-CHUAN DUAN ◽  
YAZHEN WANG ◽  
JIAN ZOU
Keyword(s):  
Discrete Time ◽  
Option Price ◽  
Garch Model ◽  
Convergence Speed ◽  
Time Interval ◽  
Square Root ◽  
Option Prices ◽  
European Option ◽  
The Garch Model

It is well known that as the time interval between two consecutive observations shrinks to zero, a properly constructed GARCH model will weakly converge to a bivariate diffusion. Naturally the European option price under the GARCH model will also converge to its bivariate diffusion counterpart. This paper investigates the convergence speed of the GARCH option price. We show that the European option prices under the two corresponding models are equal up to an order near the square root of the length of discrete time interval.

scholarly journals MOST-LIKELY-PATH IN ASIAN OPTION PRICING UNDER LOCAL VOLATILITY MODELS

2018 ◽  
Vol 21 (05) ◽  
pp. 1850029 ◽  
Author(s):  
LOUIS-PIERRE ARGUIN ◽  
NIEN-LIN LIU ◽  
TAI-HO WANG
Keyword(s):  
Large Deviation ◽  
Time Window ◽  
Brownian Bridge ◽  
Option Price ◽  
Transition Density ◽  
Small Time ◽  
Option Prices ◽  
Local Volatility ◽  
Volatility Models ◽  
Time Price

This paper addresses the problem of approximating the price of options on discrete and continuous arithmetic averages of the underlying, i.e. discretely and continuously monitored Asian options, in local volatility models. A “path-integral”-type expression for option prices is obtained using a Brownian bridge representation for the transition density between consecutive sampling times and a Laplace asymptotic formula. In the limit where the sampling time window approaches zero, the option price is found to be approximated by a constrained variational problem on paths in time-price space. We refer to the optimizing path as the most-likely path (MLP). An approximation for the implied normal volatility follows accordingly. The small-time asymptotics and the existence of the MLP are also rigorously recovered using large deviation theory.

AN ANALYTICAL APPROXIMATION FOR EUROPEAN OPTION PRICES UNDER STOCHASTIC INTEREST RATES

2015 ◽  
Vol 18 (04) ◽  
pp. 1550026 ◽  
Author(s):  
HIDEHARU FUNAHASHI
Keyword(s):  
Interest Rates ◽  
Analytical Approximation ◽  
Option Prices ◽  
European Option ◽  
Stochastic Interest Rates ◽  
Option Markets ◽  
Stochastic Nature ◽  
Volatility Models ◽  
Stochastic Interest ◽  
High Volatility

This paper extends the Wiener–Itô chaos expansion approach proposed by Funahashi & Kijima (2015) to an equity-interest-rate hybrid model for the pricing of European contingent claims with special emphasis on calibration to the option markets. Our model can capture the volatility skew and smile of option markets, as well as the stochastic nature of interest rates. Further, the proposed method is applicable to widely used option pricing models such as local volatility models (LVM), stochastic volatility models (SVM), and their combinations with the stochastic nature of interest rates; hence, it is suitable for practical purposes. Through numerical examples, we show that our approximation is quite accurate even for long-maturity and/or high-volatility cases.

Capturing implied correlation skew from options prices via multiscale stochastic volatility models

2020 ◽  
Vol 07 (04) ◽  
pp. 2050042
Author(s):  
T. Pellegrino
Keyword(s):  
Stochastic Volatility ◽  
Calibration Procedure ◽  
Second Order ◽  
Model Parameters ◽  
Option Prices ◽  
European Options ◽  
Stochastic Volatility Models ◽  
Volatility Models ◽  
Order Expansion ◽  
Implied Correlation

The aim of this paper is to derive a second-order asymptotic expansion for the price of European options written on two underlying assets, whose dynamics are described by multiscale stochastic volatility models. In particular, the second-order expansion of option prices can be translated into a corresponding expansion in implied correlation units. The resulting approximation for the implied correlation curve turns out to be quadratic in the log-moneyness, capturing the convexity of the implied correlation skew. Finally, we describe a calibration procedure where the model parameters can be estimated using option prices on individual underlying assets.

scholarly journals EUROPEAN OPTIONS SENSITIVITY WITH RESPECT TO THE CORRELATION FOR MULTIDIMENSIONAL HESTON MODELS

2014 ◽  
Vol 17 (03) ◽  
pp. 1450015 ◽  
Author(s):  
LOKMAN A. ABBAS-TURKI ◽  
DAMIEN LAMBERTON
Keyword(s):  
Stock Prices ◽  
Asymptotic Approximation ◽  
Heston Model ◽  
Option Prices ◽  
European Options ◽  
European Option ◽  
Price Function ◽  
Spread Option ◽  
Correlation Parameters ◽  
Theoretical Results

We study the sensitivity of European option prices with respect to correlation parameters in the multi-asset Heston model. The differentiability of the price function with respect to the correlation is proved by using the regularity of the flow of the Cox–Ingersoll–Ross model. In the bidimensional case and when the Feller condition is satisfied, we establish an asymptotic approximation for the derivative of the price with respect to the correlation for short maturities. This approximation is used to discuss monotony issues for exchange and spread option prices. Monotony properties are also obtained for some values of "the volatility of the volatility parameter" and of the correlation between stock prices and their volatilities. We conclude with a large number of simulations that confirm the theoretical results.

scholarly journals DUPIRE'S EQUATION FOR BUBBLES

2012 ◽  
Vol 15 (06) ◽  
pp. 1250041 ◽  
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Option Price ◽  
Local Martingale ◽  
Option Prices ◽  
Uniqueness Of Solutions ◽  
Underlying Asset ◽  
Put Options ◽  
Price Process ◽  
Usual Form ◽  
Volatility Models ◽  
Strict Local Martingale

We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.

A REGULARIZED FOURIER TRANSFORM APPROACH FOR VALUING OPTIONS UNDER STOCHASTIC DIVIDEND YIELDS

2010 ◽  
Vol 13 (02) ◽  
pp. 211-240 ◽  
Author(s):  
BAYE M. DIA
Keyword(s):  
Fourier Transform ◽  
Option Pricing ◽  
Inverse Fourier Transform ◽  
Option Price ◽  
Square Root ◽  
Option Prices ◽  
European Options ◽  
Dividend Yield ◽  
New Approach ◽  
Main Effects

This paper studies the option pricing problem in a class of models in which dividend yields follow a time-homogeneous diffusion. Within this framework, we develop a new approach for valuing options based on the use of a regularized Fourier transform. We derive a pricing formula for European options which gives the option price in the form of an inverse Fourier transform and propose two methods for numerically implementing this formula. As an application of this pricing approach, we introduce the Ornstein-Uhlenbeck and the square-root dividend yield models in which we explicitly solve the pricing problem for European options. Finally we highlight the main effects of a stochastic dividend yield on option prices.

MONOTONICITY IN THE VOLATILITY OF SINGLE-BARRIER OPTION PRICES

2006 ◽  
Vol 09 (06) ◽  
pp. 987-996 ◽  
Author(s):  
JONATAN ERIKSSON
Keyword(s):  
Option Price ◽  
Rate Of Return ◽  
Barrier Options ◽  
Option Prices ◽  
Barrier Option ◽  
Volatility Models ◽  
Risk Free Rate ◽  
Log Normal ◽  
Free Rate ◽  
Normal Stock

We generalize earlier results on barrier options for puts and calls and log-normal stock processes to general local volatility models and convex contracts. We show that Γ ≥ 0, that Δ has a unique sign and that the option price is increasing with the volatility for convex contracts in the following cases: • If the risk-free rate of return dominates the dividend rate, then it holds for up-and-out options if the contract function is zero at the barrier and for down-and-in options in general. • If the risk-free rate of return is dominated by the dividend rate, then it holds for down-and-out options if the contract function is zero at the barrier and for up-and-in options in general. We apply our results to show that a hedger who misspecifies the volatility using a time-and-level dependent volatility will super-replicate any claim satisfying the above conditions if the misspecified volatility dominates the true (possibly stochastic) volatility almost surely.

Sign in / Sign up

Export Citation Format

Share Document