scholarly journals Option Pricing under Randomised GBM Models

2021 ◽  
Vol 9 (3) ◽  
pp. 7-26
Author(s):  
G. Campolieti ◽  
H. Kato ◽  
R. Makarov
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Scale Parameter ◽  
Transition Density ◽  
Elementary Functions ◽  
Positive Real ◽  
No Arbitrage ◽  
Inverse Gamma ◽  
Black Scholes ◽  
Transition Densities

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

Arbitrage Pricing

2019 ◽  
pp. 95-118
Author(s):  
Tomas Björk
Keyword(s):  
Continuous Time ◽  
Implied Volatility ◽  
Futures Contracts ◽  
Martingale Measures ◽  
Bank Account ◽  
No Arbitrage ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Financial Derivative ◽  
Delta Hedging

The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.

scholarly journals Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing

2018 ◽  
Vol 10 (6) ◽  
pp. 108
Author(s):  
Yao Elikem Ayekple ◽  
Charles Kofi Tetteh ◽  
Prince Kwaku Fefemwole
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Call Option ◽  
Option Prices ◽  
Options Market ◽  
European Call Option ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Real Market ◽  
Independent Path

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

scholarly journals Implied volatility estimation of bitcoin options and the stylized facts of option pricing

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Noshaba Zulfiqar ◽  
Saqib Gulzar
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Volatility Smile ◽  
Stylized Facts ◽  
Approximation Techniques ◽  
Risk Hedging ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Newton Raphson ◽  
Futures And Options

AbstractThe recently developed Bitcoin futures and options contracts in cryptocurrency derivatives exchanges mark the beginning of a new era in Bitcoin price risk hedging. The need for these tools dates back to the market crash of 1987, when investors needed better ways to protect their portfolios through option insurance. These tools provide greater flexibility to trade and hedge volatile swings in Bitcoin prices effectively. The violation of constant volatility and the log-normality assumption of the Black–Scholes option pricing model led to the discovery of the volatility smile, smirk, or skew in options markets. These stylized facts; that is, the volatility smile and implied volatilities implied by the option prices, are well documented in the option literature for almost all financial markets. These are expected to be true for Bitcoin options as well. The data sets for the study are based on short-dated Bitcoin options (14-day maturity) of two time periods traded on Deribit Bitcoin Futures and Options Exchange, a Netherlands-based cryptocurrency derivative exchange. The estimated results are compared with benchmark Black–Scholes implied volatility values for accuracy and efficiency analysis. This study has two aims: (1) to provide insights into the volatility smile in Bitcoin options and (2) to estimate the implied volatility of Bitcoin options through numerical approximation techniques, specifically the Newton Raphson and Bisection methods. The experimental results show that Bitcoin options belong to the commodity class of assets based on the presence of a volatility forward skew in Bitcoin option data. Moreover, the Newton Raphson and Bisection methods are effective in estimating the implied volatility of Bitcoin options. However, the Newton Raphson forecasting technique converges faster than does the Bisection method.

BLACK–SCHOLES–MERTON IN RANDOM TIME: A NEW STOCHASTIC VOLATILITY MODEL WITH PATH DEPENDENCE

2007 ◽  
Vol 10 (05) ◽  
pp. 847-872 ◽  
Author(s):  
DMITRY OSTROVSKY
Keyword(s):  
Brownian Motion ◽  
Implied Volatility ◽  
Time Derivative ◽  
Sufficient Conditions ◽  
Random Time ◽  
Average Price ◽  
European Options ◽  
No Arbitrage ◽  
Volatility Model ◽  
Black Scholes

A generalized Black–Scholes–Merton economy is introduced. The economy is driven by Brownian motion in random time that is taken to be continuous and independent of Brownian motion. European options are priced by the no-arbitrage principle as conditional averages of their classical values over the random time to maturity. The prices are path dependent in general unless the time derivative of the random time is Markovian. An explicit self-financing hedging strategy is shown to replicate all European options by dynamically trading in stock, money market, and digital calls on realized variance. The notion of the average price is introduced, and the average price of the call option is shown to be greater than the corresponding Black–Scholes price for all deep in- and out-of-the-money options under appropriate sufficient conditions. The model is implemented in limit lognormal random time. The significance of its multiscaling law is explained theoretically and verified numerically to be a determining factor of the term structure of implied volatility.

FINANCIAL MARKETS WITH NO RISKLESS (SAFE) ASSET

2017 ◽  
Vol 20 (08) ◽  
pp. 1750054
Author(s):  
SVETLOZAR T. RACHEV ◽  
STOYAN V. STOYANOV ◽  
FRANK J. FABOZZI
Keyword(s):  
Option Pricing ◽  
Financial Markets ◽  
Price Dynamics ◽  
Jump Diffusions ◽  
Risky Assets ◽  
No Arbitrage ◽  
Market Completeness ◽  
Black Scholes ◽  
Stochastic Volatilities

We study markets with no riskless (safe) asset. We derive the corresponding Black–Scholes–Merton option pricing equations for markets where there are only risky assets which have the following price dynamics: (i) continuous diffusions; (ii) jump-diffusions; (iii) diffusions with stochastic volatilities, and; (iv) geometric fractional Brownian and Rosenblatt motions. No-arbitrage and market-completeness conditions are derived in all four cases.

An Empirical Test of the Black-Scholes Option Pricing Model and the Implied Variance: A Confidence Interval Approach

1987 ◽  
Vol 2 (4) ◽  
pp. 355-369 ◽  
Author(s):  
Haim Levy ◽  
Young Hoon Byun
Keyword(s):  
Confidence Interval ◽  
Option Pricing ◽  
Implied Volatility ◽  
Empirical Test ◽  
Market Price ◽  
Pricing Model ◽  
Black Scholes ◽  
Option Pricing Model ◽  
S Model ◽  
Over Time

The empirical studies on the Black-Scholes (B-S) option pricing model have reported that the model tends to exhibit systematic biases with respect to the exercise price, time to expiration, and the stock's volatility. This paper attempts to test the B-S model with a new approach: derive the confidence interval of the model call option value based on the confidence interval of the. estimated variance. The test reports that even when the variance's confidence interval is considered, a systematic deviation between the theoretical “range” of the option price values and the observed market price still exist. If the stock variance is constant over time, the interpretation of the results is that the B-S model is wrong. However, if stock variance changes over time, the interpretation of the results is that the implied volatility in options market prices had a tendency to be significantly higher than the estimate that could have been obtained from historical data.

scholarly journals A modification term for Black-Scholes model based on discrepancy calibrated with real market data

2021 ◽  
Vol 1 (4) ◽  
pp. 313-326
Author(s):  
Xiaozheng Lin ◽  
◽  
Meiqing Wang ◽  
Choi-Hong Lai ◽  
Keyword(s):  
Option Pricing ◽  
Implied Volatility ◽  
Option Prices ◽  
Market Prices ◽  
Market Data ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Real Market ◽  
S Model

<abstract><p>The Black-Scholes option pricing model (B-S model) generally requires the assumption that the volatility of the underlying asset be a piecewise constant. However, empirical analysis shows that there are discrepancies between the option prices obtained from the B-S model and the market prices. Most current modifications to the B-S model rely on modelling the implied volatility or interest rate. In contrast to the existing modifications to the Black-Scholes model, this paper proposes the concept of including a modification term to the B-S model itself. Using the actual discrepancies of the results of the Black-Scholes model and the market prices, the modification term related to the implied volatility is derived. Experimental results show that the modified model produces a better option pricing results when compare to market data.</p></abstract>

A Risk-Neutral Stochastic Volatility Model

1998 ◽  
Vol 01 (02) ◽  
pp. 289-310 ◽  
Author(s):  
Yingzi Zhu ◽  
Marco Avellaneda
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Monte Carlo Study ◽  
Stochastic Volatility Model ◽  
No Arbitrage ◽  
Volatility Model ◽  
Black Scholes ◽  
Risk Neutral ◽  
The Difference ◽  
Log Normal

We construct a risk-neutral stochastic volatility model using no-arbitrage pricing principles. We then study the behavior of the implied volatility of options that are deep in and out of the money according to this model. The motivation of this study is to show the difference in the asymptotic behavior of the distribution tails between the usual Black–Scholes log-normal distribution and the risk-neutral stochastic volatility distribution. In the second part of the paper, we further explore this risk-neutral stochastic volatility model by a Monte-Carlo study on the implied volatility curve (implied volatility as a function of the option strikes) for near-the-money options. We study the behavior of this "smile" curve under different choices of parameter for the model, and determine how the shape and skewness of the "smile" curve is affected by the volatility of volatility ("V-vol") and the correlation between the underlying asset and its volatility.

ON THE RELATIONSHIP BETWEEN THE CALL PRICE SURFACE AND THE IMPLIED VOLATILITY SURFACE CLOSE TO EXPIRY

2009 ◽  
Vol 12 (04) ◽  
pp. 427-441 ◽  
Author(s):  
MICHAEL ROPER ◽  
MAREK RUTKOWSKI
Keyword(s):  
Implied Volatility ◽  
Final Section ◽  
Small Time ◽  
Market Model ◽  
Free Markets ◽  
No Arbitrage ◽  
Implied Volatility Surface ◽  
Black Scholes ◽  
Volatility Surface ◽  
The Relationship

We examine the asymptotic behaviour of the call price surface and the associated Black-Scholes implied volatility surface in the small time to expiry limit under the condition of no arbitrage. In the final section, we examine a related question of existence of a market model with non-convergent implied volatility. We show that there exist arbitrage free markets in which implied volatility may fail to converge to any value, finite or infinite.

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