MODELING AND SIMULATION OF SEQUENTIAL AUCTIONS: PRICING AND CALIBRATION ALGORITHMS

We consider sequential auctions wherein seller and bidder agents need to price goods on sale at the 'right' market price. We propose algorithms based on a binomial model for both the seller and buyer. Then, we consider the problem of calibrating pricing models to market data. To this end, we studied a stochastic volatility model used for option pricing, derived, and analyzed Monte Carlo estimators for computing the gradient of a certain payoff function using Finite Differencing and Algorithmic Differentiation. We then assessed the accuracy and efficiency of both methods as well as their impacts into the optimization algorithm. Numerical results are presented and discussed. This work can benefit those engaged in electronic trading or investors in financial products with the need for fast and more precise predictions of future market data.

Download Full-text

SWITCHING TO NONAFFINE STOCHASTIC VOLATILITY: A CLOSED-FORM EXPANSION FOR THE INVERSE GAMMA MODEL

International Journal of Theoretical and Applied Finance ◽

10.1142/s021902491650031x ◽

2016 ◽

Vol 19 (05) ◽

pp. 1650031 ◽

Cited By ~ 6

Author(s):

NICOLAS LANGRENÉ ◽

GEOFFREY LEE ◽

ZILI ZHU

Keyword(s):

Closed Form ◽

Stochastic Volatility ◽

Stochastic Volatility Model ◽

Heston Model ◽

Time Dependent ◽

Market Data ◽

Affine Models ◽

Inverse Gamma ◽

Volatility Model ◽

Black Scholes

We examine the inverse gamma (IGa) stochastic volatility model with time-dependent parameters. This nonaffine model compares favorably in terms of volatility distribution and volatility paths to classical affine models such as the Heston model, while being as parsimonious (only four stochastic parameters). In practice, this means more robust calibration and better hedging, explaining its popularity among practitioners. Closed-form volatility-of-volatility expansions are obtained for the price of vanilla options, which allow for very fast pricing and calibration to market data. Specifically, the price of a European put option with IGa volatility is approximated by a Black–Scholes price plus a weighted combination of Black–Scholes Greeks, with weights depending only on the four time-dependent parameters of the model. The accuracy of the expansion is illustrated on several calibration tests on foreign exchange market data. This paper shows that the IGa model is as simple, more realistic, easier to implement and faster to calibrate than classical transform-based affine models. We therefore hope that the present work will foster further research on nonaffine models favored by practitioners such as the IGa model.

Download Full-text

European Option Pricing under Wishart Processes

Journal of Mathematics ◽

10.1155/2021/7411885 ◽

2021 ◽

Vol 2021 ◽

pp. 1-24

Author(s):

Raphael Naryongo ◽

Philip Ngare ◽

Anthony Waititu

Keyword(s):

Option Pricing ◽

Market Price ◽

Stochastic Volatility Model ◽

European Option ◽

Multifactor Model ◽

Asset Pricing Model ◽

European Option Pricing ◽

Volatility Model ◽

Wishart Processes ◽

The Fourier Transform

This study deals with a single risky asset pricing model whose volatility is described by Wishart affine processes. This multifactor model with two dependency matrices describing the correlation between the asset dynamic and Wishart processes makes it more flexible enough to fit the market data for short or long maturities. The aim of the study is to derive and solve the call option pricing problem under the double Wishart stochastic volatility model. The Fourier transform techniques combined with perturbation methods are employed in order to price the European call options. The numerical illustrations on pricing predictions show similar behavior of price movements under the double Wishart model with respect to the market price.

Download Full-text

Calibration and simulation of Heston model

Open Mathematics ◽

10.1515/math-2017-0058 ◽

2017 ◽

Vol 15 (1) ◽

pp. 679-704 ◽

Cited By ~ 4

Author(s):

Milan Mrázek ◽

Jan Pospíšil

Keyword(s):

Variance Reduction ◽

Calibration Procedure ◽

Optimization Techniques ◽

Stochastic Volatility Model ◽

Heston Model ◽

Approximation Formula ◽

Market Data ◽

Volatility Model ◽

Calibration Methods ◽

Real Market

Abstract We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods. We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.

Download Full-text

The Heston Model with Time-Dependent Correlation Driven by Isospectral Flows

Mathematics ◽

10.3390/math9090934 ◽

2021 ◽

Vol 9 (9) ◽

pp. 934

Author(s):

Long Teng

Keyword(s):

Financial Institutions ◽

Stochastic Volatility Model ◽

Heston Model ◽

Time Dependent ◽

Fast Computation ◽

Volatility Smile ◽

Financial Products ◽

Volatility Model ◽

Fixed Constant ◽

Dependent Correlation

In this work, we extend the Heston stochastic volatility model by including a time-dependent correlation that is driven by isospectral flows instead of a constant correlation, being motivated by the fact that the correlation between, e.g., financial products and financial institutions is hardly a fixed constant. We apply different numerical methods, including the method for backward stochastic differential equations (BSDEs) for a fast computation of the extended Heston model. An example of calibration to market data illustrates that our extended Heston model can provide a better volatility smile than the Heston model with other considered extensions.

Download Full-text

Viscosity Solutions and American Option Pricing in a Stochastic Volatility Model of the Ornstein-Uhlenbeck Type

Journal of Probability and Statistics ◽

10.1155/2010/863585 ◽

2010 ◽

Vol 2010 ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Alexandre F. Roch

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Option Pricing ◽

Stochastic Volatility ◽

Viscosity Solution ◽

Lipschitz Condition ◽

Payoff Function ◽

Stochastic Volatility Model ◽

American Option Pricing ◽

Volatility Model

We study the valuation of American-type derivatives in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We characterize the value of such derivatives as the unique viscosity solution of an integral-partial differential equation when the payoff function satisfies a Lipschitz condition.

Download Full-text

A SIMPLE CLOSED-FORM FORMULA FOR PRICING DISCRETELY-SAMPLED VARIANCE SWAPS UNDER THE HESTON MODEL

The ANZIAM Journal ◽

10.1017/s1446181114000236 ◽

2014 ◽

Vol 56 (1) ◽

pp. 1-27 ◽

Cited By ~ 1

Author(s):

SANAE RUJIVAN ◽

SONG-PING ZHU

Keyword(s):

Closed Form ◽

Stochastic Volatility ◽

Analytical Approach ◽

Stochastic Volatility Model ◽

Heston Model ◽

Model Parameters ◽

Variance Swaps ◽

Volatility Model ◽

The Right ◽

Closed Form Formula

AbstractWe develop a simplified analytical approach for pricing discretely-sampled variance swaps with the realized variance, defined in terms of the squared log return of the underlying price. The closed-form formula obtained for Heston’s two-factor stochastic volatility model is in a much simpler form than those proposed in literature. Most interestingly, we discuss the validity of our solution as well as some other previous solutions in different forms in the parameter space. We demonstrate that market practitioners need to be cautious, making sure that their model parameters extracted from market data are in the right parameter subspace, when any of these analytical pricing formulae is adopted to calculate the fair delivery price of a discretely-sampled variance swap.

Download Full-text

Pricing the Volatility Risk Premium with a Discrete Stochastic Volatility Model

Mathematics ◽

10.3390/math9172038 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2038

Author(s):

Petra Posedel Šimović ◽

Azra Tafro

Keyword(s):

Risk Premium ◽

Transition Probabilities ◽

Market Price ◽

Stochastic Volatility Model ◽

Volatility Risk ◽

Underlying Asset ◽

Asset Return ◽

Volatility Model ◽

Local Risk ◽

Market Shocks

Investors’ decisions on capital markets depend on their anticipation and preferences about risk, and volatility is one of the most common measures of risk. This paper proposes a method of estimating the market price of volatility risk by incorporating both conditional heteroscedasticity and nonlinear effects in market returns, while accounting for asymmetric shocks. We develop a model that allows dynamic risk premiums for the underlying asset and for the volatility of the asset under the physical measure. Specifically, a nonlinear in mean time series model combining the asymmetric autoregressive conditional heteroscedastic model with leverage (NGARCH) is adapted for modeling return dynamics. The local risk-neutral valuation relationship is used to model investors’ preferences of volatility risk. The transition probabilities governing the evolution of the price of the underlying asset are adjusted for investors’ attitude towards risk, presenting the asset returns as a function of the risk premium. Numerical studies on asset return data show the significance of market shocks and levels of asymmetry in pricing the volatility risk. Estimated premiums could be used in option pricing models, turning options markets into volatility trading markets, and in measuring reactions to market shocks.

Download Full-text