ASYMPTOTIC APPROXIMATIONS FOR PRICING DERIVATIVES UNDER MEAN-REVERTING PROCESSES

2016 ◽  
Vol 19 (05) ◽  
pp. 1650030 ◽  
Author(s):  
RICHARD JORDAN ◽  
CHARLES TIER
Keyword(s):  
Interest Rates ◽  
Implied Volatility ◽  
Asymptotic Approximations ◽  
Ray Method ◽  
Interest Rate Derivatives ◽  
Put Options ◽  
Merton Model ◽  
Volatility Model ◽  
Black Scholes ◽  
Efficient Alternative

The problem of fast pricing, hedging, and calibrating of derivatives is considered when the underlying does not follow the standard Black–Scholes–Merton model but rather a mean-reverting and deterministic volatility model. Mean-reverting models are often used for volatility, commodities, and interest-rate derivatives, while the deterministic volatility accounts for the nonconstant implied volatility. Trading desks often use numerical methods for real-time pricing, hedging, and calibration when implementing such models. A more efficient alternative is to use an analytic formula, even if only approximate. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations to the density function that can be used to derive simple formulas for pricing derivatives. Such approximations are usually only valid away from any boundaries, yet for some derivatives the values of the underlying near the boundaries are needed such as when interest rates are very low or for pricing put options. Hence, the ray approximation may not yield acceptable results. A new asymptotic approximation near boundaries is derived, which is shown to be of value for pricing certain derivatives. The results are illustrated by deriving new analytic approximations for European derivatives and their high accuracy is demonstrated numerically.

Modeling of implied volatility surfaces of nifty index options

2019 ◽  
Vol 06 (03) ◽  
pp. 1950028 ◽  
Author(s):  
Mihir Dash
Keyword(s):  
Implied Volatility ◽  
Strike Price ◽  
Index Options ◽  
Put Options ◽  
Call Options ◽  
Merton Model ◽  
Implied Volatility Surface ◽  
Black Scholes ◽  
Volatility Surface ◽  
Implied Volatility Surfaces

The implied volatility of an option contract is the value of the volatility of the underlying instrument which equates the theoretical option value from an option pricing model (typically, the Black–Scholes[Formula: see text]Merton model) to the current market price of the option. The concept of implied volatility has gained in importance over historical volatility as a forward-looking measure, reflecting expectations of volatility (Dumas et al., 1998). Several studies have shown that the volatilities implied by observed market prices exhibit a pattern very different from that assumed by the Black–Scholes[Formula: see text]Merton model, varying with strike price and time to expiration. This variation of implied volatilities across strike price and time to expiration is referred to as the volatility surface. Empirically, volatility surfaces for global indices have been characterized by the volatility skew. For a given expiration date, options far out-of-the-money are found to have higher implied volatility than those with an exercise price at-the-money. For short-dated expirations, the cross-section of implied volatilities as a function of strike is roughly V-shaped, but has a rounded vertex and is slightly tilted. Generally, this V-shape softens and becomes flatter for longer dated expirations, but the vertex itself may rise or fall depending on whether the term structure of at-the-money volatility is upward or downward sloping. The objective of this study is to model the implied volatility surfaces of index options on the National Stock Exchange (NSE), India. The study employs the parametric models presented in Dumas et al. (1998); Peña et al. (1999), and several subsequent studies to model the volatility surfaces across moneyness and time to expiration. The present study contributes to the literature by studying the nature of the stationary point of the implied volatility surface and by separating the in-the-money and out-of-the-money components of the implied volatility surface. The results of the study suggest that an important difference between the implied volatility surface of index call and put options: the implied volatility surface of index call options was found to have a minimum point, while that of index put options was found to have a saddlepoint. The results of the study also indicate the presence of a “volatility smile” across strike prices, with a minimum point in the range of 2.3–9.0% in-the-money for index call options and of 10.7–29.3% in-the-money for index put options; further, there was a jump in implied volatility in the transition from out-of-the-moneyness to in-the-moneyness, by 10.0% for index call options and about 1.9% for index put options.

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

2019 ◽  
Vol 39 ◽  
pp. 127-140
Author(s):  
Tahmid Tamrin Suki ◽  
ABM Shahadat Hossain
Keyword(s):  
Comparative Analysis ◽  
Stochastic Volatility ◽  
Stock Prices ◽  
Computer Algebra System ◽  
Stochastic Volatility Model ◽  
Affecting Factors ◽  
Pricing Models ◽  
Merton Model ◽  
Volatility Model ◽  
Black Scholes

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

scholarly journals Efficient Simulation for Pricing Barrier Options with Two-Factor Stochastic Volatility and Stochastic Interest Rate

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Zhang Sumei ◽  
Zhao Jieqiong
Keyword(s):  
Interest Rates ◽  
Stochastic Volatility ◽  
Implied Volatility ◽  
Differential Evolution Algorithm ◽  
Extended Model ◽  
Barrier Options ◽  
Simulation Algorithm ◽  
Volatility Model ◽  
Stochastic Interest ◽  
Simulation Scheme

This paper presents an extension of the double Heston stochastic volatility model by combining Hull-White stochastic interest rates. By the change of numeraire and quadratic exponential scheme, this paper develops a new simulation scheme for the extended model. By combining control variates and antithetic variates, this paper provides an efficient Monte Carlo simulation algorithm for pricing barrier options. Based on the differential evolution algorithm the extended model is calibrated to S&P 500 index options to obtain the model parameter values. Numerical results show that the proposed simulation scheme outperforms the Euler scheme, the proposed simulation algorithm is efficient for pricing barrier options, and the extended model is flexible to fit the implied volatility surface.

Who predicts dollar-rupee volatility better? A tale of two options markets

2019 ◽  
Vol 45 (9) ◽  
pp. 1292-1308
Author(s):  
Aparna Prasad Bhat
Keyword(s):  
Implied Volatility ◽  
Realized Volatility ◽  
Informational Efficiency ◽  
Significant Information ◽  
Content Type ◽  
Merton Model ◽  
Model Free ◽  
Ex Post ◽  
Black Scholes ◽  
Practical Implications

Purpose The purpose of this paper is to examine whether volatility implied from dollar-rupee options is an unbiased and efficient predictor of ex post volatility, and to determine which options market is a better predictor of future realized volatility and to ascertain whether the model-free measure of implied volatility outperforms the traditional measure derived from the Black–Scholes–Merton model. Design/methodology/approach The information content of exchange-traded implied volatility and that of quoted implied volatility for OTC options is compared with that of historical volatility and a GARCH(1, 1)-based volatility. Ordinary least squares regression is used to examine the unbiasedness and informational efficiency of implied volatility. Robustness of the results is tested by using two specifications of implied volatility and realized volatility and comparison across two markets. Findings Implied volatility from both OTC and exchange-traded options is found to contain significant information for predicting ex post volatility, but is neither unbiased nor informationally efficient. The implied volatility of at-the-money options derived using the Black–Scholes–Merton model is found to outperform the model-free implied volatility (MFIV) across both markets. MFIV from OTC options is found to be a better predictor of realized volatility than MFIV from exchange-traded options. Practical implications This study throws light on the predictive power of currency options in India and has strong practical implications for market practitioners. Efficient currency option markets can serve as effective vehicles both for hedging and speculation and can convey useful information to the regulators regarding the market participants’ expectations of future volatility. Originality/value This study is a comprehensive study of the informational efficiency of options on an emerging currency such as the Indian rupee. To the author’s knowledge, this is one of the first studies to compare the predictive ability of the exchange-traded and OTC markets and also to compare traditional model-dependent volatility with MFIV.

scholarly journals An investigation into the reasons for the pricing differences between a warrant and an option on the same stock in the South African derivatives market

2010 ◽  
Vol 8 (1) ◽  
pp. 379-389
Author(s):  
F.Y. Jordaan ◽  
J.H. van Rooyen
Keyword(s):  
South African ◽  
Implied Volatility ◽  
Calendar Period ◽  
The South ◽  
Merton Model ◽  
Proposed Model ◽  
Black Scholes ◽  
European Markets ◽  
Market Changes

This study set out to draw a pricing comparison between two similar contracts in the South African derivatives market. These contracts, a normal option and a warrant on the same underlying stock are considered. The research shows that although the two derivatives are the same in all respects, the premiums differ substantially when priced with the Black-Scholes-Merton model. It is clear that pricing has to take place over the same calendar period due to market changes when comparing the instruments. The Black-Scholes-Merton model was the proposed model to be used. However, due to certain limitations the Modified Black model was used as the best suited model. It was shown that warrant contracts always have a higher implied volatility and a higher premium than a comparable normal option per share of the same stock. These results werecompared with similar studies conducted in the European markets

FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES

2001 ◽  
Vol 04 (04) ◽  
pp. 651-675 ◽  
Author(s):  
JEAN-PIERRE FOUQUE ◽  
GEORGE PAPANICOLAOU ◽  
K. RONNIE SIRCAR
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Stochastic Volatility ◽  
Free Boundary Problem ◽  
Implied Volatility ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
One Dimensional ◽  
Volatility Model ◽  
Black Scholes

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a fixed boundary-value problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the effect of the volatility skew.

Which Option Pricing Model Is the Best? HF Data for Nikkei 225 Index Options

2019 ◽  
Vol 4 (51) ◽  
pp. 18-39
Author(s):  
Kokoszczyński Ryszard ◽  
Sakowski Paweł ◽  
Ślepaczuk Robert
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Model Performance ◽  
Stochastic Volatility Model ◽  
Percentage Error ◽  
Pricing Models ◽  
Index Options ◽  
Put Options ◽  
Pricing Errors ◽  
Volatility Model

Abstract In this study, we analyse the performance of option pricing models using 5-minutes transactional data for the Japanese Nikkei 225 index options. We compare 6 different option pricing models: the Black (1976) model with different assumptions about the volatility process (realized volatility with and without smoothing, historical volatility and implied volatility), the stochastic volatility model of Heston (1993) and the GARCH(1,1) model. To assess the model performance, we use median absolute percentage error based on differences between theoretical and transactional options prices. We present our results with respect to 5 classes of option moneyness, 5 classes of option time to maturity and 2 option types (calls and puts). The Black model with implied volatility (BIV) comes as the best and the GARCH(1,1) as the worst one. For both call and put options, we observe the clear relation between average pricing errors and option moneyness: high error values for deep OTM options and the best fit for deep ITM options. Pricing errors also depend on time to maturity, although this relationship depend on option moneyness. For low value options (deep OTM and OTM), we obtained lower errors for longer maturities. On the other hand, for high value options (ITM and deep ITM) pricing errors are lower for short times to maturity. We obtained similar average pricing errors for call and put options. Moreover, we do not see any advantage of much complex and time-consuming models. Additionally, we describe liquidity of the Nikkei225 option pricing market and try to compare the results we obtain here with a detailed study for Polish emerging option market (Kokoszczyński et al. 2010b).

EXPANSION FORMULAS FOR BIVARIATE PAYOFFS WITH APPLICATION TO BEST-OF OPTIONS ON EQUITY AND INFLATION

2014 ◽  
Vol 17 (02) ◽  
pp. 1450010 ◽  
Author(s):  
EMMANUEL GOBET ◽  
JULIEN HOK
Keyword(s):  
Interest Rates ◽  
Stock Price ◽  
Local Volatility ◽  
Local Volatility Model ◽  
Volatility Model ◽  
Asset Value ◽  
Black Scholes ◽  
Approximation Formulas ◽  
Volatility Function ◽  
Log Normal

A wide class of hybrid products are evaluated with a model where one of the underlying price follows a local volatility diffusion and the other asset value a log-normal process. Because of the generality for the local volatility function, the numerical pricing is usually much time consuming. Using proxy approximations related to log-normal modeling, we derive approximation formulas of Black–Scholes type for the price, that have the advantage of giving very rapid numerical procedures. This derivation is illustrated with the best-of option between equity and inflation where the stock price follows a local volatility model and the inflation rate a Hull–White process. The approximations possibly account for Gaussian HJM (Heath-Jarrow-Morton) models for interest rates. The experiments show an excellent accuracy.

scholarly journals Formulation Of A Rational Option Pricing Model using Artificial Neural Networks

2020 ◽  
Author(s):  
Kaustubh yadav ◽  
Anubhuti yadav
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Option Pricing ◽  
Numerical Solutions ◽  
Options Pricing ◽  
Volatility Model ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Efficient Alternative ◽  
Artificial Neural

This paper inquires on the options pricing modeling using Artificial Neural Networks to price Apple(AAPL) European Call Options. Our model is based on the premise that Artificial Neural Networks can be used as functional approximators and can be used as an alternative to the numerical methods to some extent, for a faster and an efficient solution. This paper provides a neural network solution for two financial models, the BlackScholes-Merton model, and the calibrated-Heston Stochastic Volatility Model, we evaluate our predictions using the existing numerical solutions for the same, the analytic solution for the Black-Scholes equation, COS-Model for Heston’s Stochastic Volatility Model and Standard Heston-Quasi analytic formula. The aim of this study is to find a viable time-efficient alternative to existing quantitative models for option pricing.

scholarly journals PENGUKURAN PROBABILITAS KEBANGKRUTAN OBLIGASI KORPORASI DENGAN SUKU BUNGA COX INGERSOLL ROSS MODEL MERTON (Studi Kasus Obligasi PT Indosat, Tbk)

2018 ◽  
Vol 7 (2) ◽  
pp. 175-186
Author(s):  
Muhammad Akhir Siregar ◽  
Mustafid Mustafid ◽  
Rukun Santoso
Keyword(s):  
Interest Rates ◽  
Default Risk ◽  
Structural Method ◽  
Probability Of Default ◽  
Due Date ◽  
Merton Model ◽  
Black Scholes ◽  
The Many ◽  
Investment Grade

Nowadays bonds become one of the many securities products that are being prefered by investors. Observing the level of the company's rating which good enough or in the criteria of investment grade can’t be a handle of investors. Investing in long-term period investors should understand the risks to be faced, one of investment credit risk on bonds is default risk, this risk is related to the possibility that the issuer fails to fulfill its obligations to the investor in due date. The measurement of the probability of default failure by the structural method approach introduced first by Black-Scholes (1973) than developed by Merton (1974).  In Bankruptcy model, merton’s model assumed the company get default (bankrupt) when the company can’t pay the coupon or face value in the due date. Interest rates on the Merton model assumed to be constant values replaced by Cox Ingersoll Ross (CIR) rates. The CIR rate is the fluctuating interest rate in each period and the change is a stochastic process. The empirical study was conducted on PT Indosat, Tbk's bonds issued in 2017 with a face value of 511 Billion in payment of obligations by the issuer for 10 years. Based on simulation results done with R software obtained probability of default value equal to 7,416132E-215 Indicates that PT Indosat Tbk is deemed to be able to fulfill its obligation payment at the end of the bond maturity in 2027. Keywords: Bond, CIR Rate, Merton Model, Ekuity, Probability of default

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