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.

scholarly journals PRICING DERIVATIVES IN HERMITE MARKETS

2019 ◽  
Vol 22 (06) ◽  
pp. 1950031
Author(s):  
STOYAN V. STOYANOV ◽  
SVETLOZAR T. RACHEV ◽  
STEFAN MITTNIK ◽  
FRANK J. FABOZZI
Keyword(s):  
Brownian Motion ◽  
Financial Markets ◽  
Fractional Brownian Motion ◽  
Trading Strategies ◽  
Risky Assets ◽  
No Arbitrage ◽  
Arbitrage Opportunities ◽  
Black Scholes ◽  
Hedging Portfolio ◽  
New Framework

We present a new framework for Hermite fractional financial markets, generalizing the fractional Brownian motion (FBM) and fractional Rosenblatt markets. Considering pure and mixed Hermite markets, we introduce a strategy-specific arbitrage tax on the rate of transaction volume acceleration of the hedging portfolio as the prices of risky assets change, allowing us to transform Hermite markets with arbitrage opportunities to markets with no arbitrage opportunities within the class of Markov trading strategies. We derive PDEs for the price of such strategies in the presence of an arbitrage tax in pure Hermite, mixed Hermite, and Black–Scholes–Merton diffusion markets.

scholarly journals An Interval of No-Arbitrage Prices in Financial Markets with Volatility Uncertainty

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Hanlei Hu ◽  
Zheng Yin ◽  
Weipeng Yuan
Keyword(s):  
Brownian Motion ◽  
Financial Markets ◽  
Stock Price ◽  
Contingent Claims ◽  
Price Dynamics ◽  
Complex Situation ◽  
No Arbitrage ◽  
Price Process ◽  
Arbitrage Opportunities

In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometric G-Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.

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.

STATISTICAL ARBITRAGE IN THE MULTI-ASSET BLACK–SCHOLES ECONOMY

2017 ◽  
Vol 12 (01) ◽  
pp. 1750004
Author(s):  
AHMET GÖNCÜ ◽  
ERDINC AKYILDIRIM
Keyword(s):  
Model Parameters ◽  
Expected Return ◽  
Risky Assets ◽  
Quantitative Finance ◽  
No Arbitrage ◽  
Statistical Arbitrage ◽  
Arbitrage Opportunities ◽  
Complete Market ◽  
Black Scholes ◽  
Analytical Formulas

In this study, we consider the statistical arbitrage definition given in Hogan, S, R Jarrow, M Teo and M Warachka (2004). Testing market efficiency using statistical arbitrage with applications to momentum and value strategies, Journal of Financial Economics, 73, 525–565 and derive the statistical arbitrage condition in the multi-asset Black–Scholes economy building upon the single asset case studied in Göncü, A (2015). Statistical arbitrage in the Black Scholes framework. Quantitative Finance, 15(9), 1489–1499. Statistical arbitrage profits can be generated if there exists at least one asset in the economy that satisfies the statistical arbitrage condition. Therefore, adding a no-statistical arbitrage condition to no-arbitrage pricing models is not realistic if not feasible. However, with an example we show that what excludes statistical arbitrage opportunities in the Black–Scholes economy, and possibly in other complete market models, is the presence of uncertainty or stochasticity in the model parameters. Furthermore, we derive analytical formulas for the expected value and probability of loss of the statistical arbitrage portfolios and compute optimal boundaries to sell the risky assets in the portfolio by maximizing the expected return with a constraint on the probability of loss.

Completeness and Hedging

2019 ◽  
pp. 119-127
Author(s):  
Tomas Björk
Keyword(s):  
Risky Assets ◽  
Market Completeness ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Absence Of Arbitrage

The concept of market completeness is discussed in some detail and we prove that the Black–Scholes model is complete. We also discuss how completeness and absence of arbitrage is related to the number of risky assets and the number of random sources in the model.

Real-Estate Derivatives

2017 ◽  
Author(s):  
Radu S. Tunaru
Keyword(s):  
Real Estate ◽  
Financial Markets ◽  
Investment Banks ◽  
Financial Instruments ◽  
Stress Tests ◽  
Real World Data ◽  
No Arbitrage ◽  
New Frontier ◽  
Property Price ◽  
Comparative Description

This book brings together the latest concepts and models in real-estate derivatives, the new frontier in financial markets. The importance of real-estate derivatives in managing property price risk that has destabilized economies frequently in the last hundred years has been brought into the limelight by Robert Shiller over the last three decades. In spite of his masterful campaign for the introduction of real-estate derivatives, these financial instruments are still in a state of infancy. This book aims to provide a state-of-the-art overview of real-estate derivatives at this moment in time, covering the description of these financial products, their applications, and the most important models proposed in the literature in this area. In order to facilitate a better understanding of the situations when these products can be successfully used, ancillary topics such as real-estate indices, mortgages, securitization, and equity release mortgages are also discussed. The book is designed to pay attention to the econometric aspects of realestate index prices, time series, and also to financial engineering no-arbitrage principles governing pricing of derivatives. The emphasis is on understanding the financial instruments through their mechanics and comparative description. The examples are based on real-world data from exchanges or frommajor investment banks or financial houses in London. The numerical analysis is easily replicable with Excel and Matlab. This is the most advanced published book in this area, combining practical relevance with intellectual rigour. Real-estate derivatives will become important for managing macro risks in order to pass stress tests imposed by regulators.

scholarly journals Quanto Pricing beyond Black–Scholes

2021 ◽  
Vol 14 (3) ◽  
pp. 136
Author(s):  
Holger Fink ◽  
Stefan Mittnik
Keyword(s):  
Option Pricing ◽  
Goodness Of Fit ◽  
Calibration Procedure ◽  
Barrier Options ◽  
Pricing Models ◽  
Modeling Framework ◽  
Double Barrier ◽  
Parameter Stability ◽  
Comprehensive Data ◽  
Black Scholes

Since their introduction, quanto options have steadily gained popularity. Matching Black–Scholes-type pricing models and, more recently, a fat-tailed, normal tempered stable variant have been established. The objective here is to empirically assess the adequacy of quanto-option pricing models. The validation of quanto-pricing models has been a challenge so far, due to the lack of comprehensive data records of exchange-traded quanto transactions. To overcome this, we make use of exchange-traded structured products. After deriving prices for composite options in the existing modeling framework, we propose a new calibration procedure, carry out extensive analyses of parameter stability and assess the goodness of fit for plain vanilla and exotic double-barrier options.

Sign in / Sign up

Export Citation Format

Share Document