Some Simulation Results of the Put-Call Symmetry Method Applied to Stochastic Volatility Models (original) (raw)

Estimating option greeks under the stochastic volatility using simulation

Physica A: Statistical Mechanics and its Applications, 2018

As the Black-Scholes (BS) equation being widely used to price options, which is based on a hypothesis that the underlying (bonds & stocks) volatility is constant. Many scholars proposed the extended version of this formula to predict the behavior of the volatility. So stochastic volatility model is the improved version in which fixed volatility is replaced. The purpose of this study is to adopt one of the famous stochastic volatility models, Heston Model (1993), to price European call options. Put option values can easily obtained by callput parity if it is needed. Simulation has proved to be a valuable tool for estimating options price derivatives i.e. "Greeks". This paper proposes the method for the simulation of stock prices and variance under the Heston stochastic volatility model. We consider three different models based on the Heston model. We present two direct methods a Path-wise method and Likelihood ratio method for estimating the derivatives of Options. Then we compare it with Black-Scholes equation, and make a sensitivity analysis for its parameters by using estimator's approaches.

Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model

Applied and Computational Mathematics

Stochastic volatility models were introduced because option prices have been mis-priced using Black-Scholes model. In this work, focus is made on pricing European put option in a Geometric Brownian Motion (GBM) stochastic volatility model with uncorrelated stock and volatility. The option is priced using two numerical methods (Crank-Nicolson and Alternating Direction Implicit (ADI) finite difference). Numerical schemes were considered because the closed form solution to the model could not be obtained. The change in option value due to changes in volatility, maturity time and market price of volatility risk are considered and comparison between the efficiency of the numerical methods by computing the CPU time was made.

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

SSRN Electronic Journal, 2014

We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for β = 1, namely ∂τ u = σ 2 1 2 (∂ 2 x u − ∂xu) + νρ∂x∂σu + 1 2 ν 2 ∂ 2 σ u + κ(θ − σ)∂σ, by choosing ν as small parameter. This yields u = u 0 + νu 1 + ν 2 u 2 +. . ., with u j independent of ν. The terms u j are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of u that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" τ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of ν, similar to Hagan's formula, but including also the mean reverting term. We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility. Contents 2 O. GRISHCHENKO, X. HAN, AND V. NISTOR 2. Applications: approximations of the derivatives and of the implied volatility 18 2.1. Approximation of derivatives 18 2.2. Implied Volatility 19 3. Model calibration and market tests 20 3.1. Description of the data 20 3.2. General description of the method 20 3.3. The first type of market data tests: implied volatility 21 3.4. The second type of market data tests: actual prices 23 3.5. The third type of market data tests: log-prices 24 4. Numerical tests 25 4.1. The residual of the approximations: substituting in the PDE 25 4.2. Comparison of our implied volatility formula with Hagan's formula for maket data 26 4.3. A Monte Carlo simulation 27 4.4. Comparison with the Finite Difference approximate solution 27 5. Extensions of the method 31 5.1. The log-normal model with mean-reverting volatility 32 5.2. The SABR model (with general β) 34 References 35 mean-reverting stochastic volatility model.

Stochastic Volatility Models with Application in Option Pricing

2010

Derivative pricing, model calibration, and sensitivity analysis are the three main problems in financial modeling. The purpose of this study is to present an algorithm to improve the pricing process, the calibration process, and the sensitivity analysis of the double Heston model, in the sense of accuracy and efficiency. Using the optimized caching technique, our study reduces the pricing computation time by about 15%. Another contribution of this thesis is: a novel application of the Automatic Differentiation (AD) algorithms in order to achieve a more stable, more accurate, and faster sensitivity analysis for the double Heston model (compared to the classical finite difference methods). This thesis also presents a novel hybrid model by combing the heuristic method Differentiation Evolution, and the gradient method Levenberg-Marquardt algorithm. Our new hybrid model significantly accelerates the calibration process. iii

Geometrical Approximation method and stochastic volatility market models

RePEc: Research Papers in Economics, 2010

We propose to discuss a new technique to derive an good approximated solution for the price of a European Vanilla options, in a market model with stochastic volatility. In particular, the models that we have considered are the Heston and SABR(for β = 1). These models allow arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation.

Numerical Solution of the Heston Stocastic Volatility Model

IOSR Journal of Mathematics

This paper has considered the numerical solution of the Heston stochastic volatility model (HSVM) using the Elzaki transform method (ETM). The proposed method seeks the approximate solution of the HSVM by implementing its properties on the HSVM. The ETM proposes the solution as a rapid convergent series that represents the precise interpretation of the HSVM in real life situations. Also, the reckless interest rate is choice as -2.01 in correspondence with [4]. Numerical evidences were obtained with the help of Maple 18 software, and are compared with the homotopy perturbation method (HPM) and variational iteration method (VIM) found in the literature [4]

A Comparative Analysis of the Black-Scholes- Merton Model and the Heston Stochastic Volatility Model

GANIT: Journal of Bangladesh Mathematical Society

This paper compares the performance of two different option pricing models, namely, the Black-Scholes-Merton (B-S-M) model and the Heston Stochastic Volatility (H-S-V) model. It is known that the most popular B-S-M Model makes the assumption that volatility of an asset is constant while the H-S-V model considers it to be random. We examine the behavior of both B-S-M and H-S-V formulae with the change of different affecting factors by graphical representations and hence assimilate them. We also compare the behavior of some of the Greeks computed by both of these models with changing stock prices and hence constitute 3D plots of these Greeks. All the numerical computations and graphical illustrations are generated by a powerful Computer Algebra System (CAS), MATLAB. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 127-140

Comparative Study of Two Extensions of Heston Stochastic Volatility Model

arXiv: Mathematical Finance, 2019

In the option valuation literature, the shortcomings of one factor stochastic volatility models have traditionally been addressed by adding jumps to the stock price process. An alternate approach in the context of option pricing and calibration of implied volatility is the addition of a few other factors to the volatility process. This paper contemplates two extensions of the Heston stochastic volatility model. Out of which, one considers the addition of jumps to the stock price process (a stochastic volatility jump diffusion model) and another considers an additional stochastic volatility factor varying at a different time scale (a multiscale stochastic volatility model). An empirical analysis is carried out on the market data of options with different strike prices and maturities, to compare the pricing performance of these models and to capture their implied volatility fit. The unknown parameters of these models are calibrated using the non-linear least square optimization. It ha...

Mathematical Analysis of Financial Model on Market Price with Stochastic Volatility

Journal of Mathematical Finance

The Heston model is one of the most popular stochastic volatility models for option pricing to measure the volatility of different parameters in the financial market. In this work, we study the statistical analysis of Heston Model by partial differential equations. The model proposed by Heston takes into account non-lognormal distribution of the assets returns, leverage effect and the important mean reverting property of volatility. We have assayed on the return distribution on the basis of different values of correlation parameter and volatility, then we measure the effects of parameters ρ (correlation coefficient) and σ (standard deviation) for different situation such as ρ > 0, σ > 0, ρ = 0, σ = 0, ρ < 0, σ < 0 etc. On return distribution of Heston Model which indicates market condition for buyers and sellers to buy and sell options. All solvers used in this analysis are implemented using MATLAB codes and the simulation results are presented graphically.