728 research outputs found
Perturbation Expansion for Option Pricing with Stochastic Volatility
We fit the volatility fluctuations of the S&P 500 index well by a Chi
distribution, and the distribution of log-returns by a corresponding
superposition of Gaussian distributions. The Fourier transform of this is,
remarkably, of the Tsallis type. An option pricing formula is derived from the
same superposition of Black-Scholes expressions. An explicit analytic formula
is deduced from a perturbation expansion around a Black-Scholes formula with
the mean volatility. The expansion has two parts. The first takes into account
the non-Gaussian character of the stock-fluctuations and is organized by powers
of the excess kurtosis, the second is contract based, and is organized by the
moments of moneyness of the option. With this expansion we show that for the
Dow Jones Euro Stoxx 50 option data, a Delta-hedging strategy is close to being
optimal.Comment: 33 pages, 13 figures, LaTeX
Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation
We apply an exponential time integration scheme
combined with a central difference scheme on a piecewise uniform mesh with
respect to the spatial variable to evaluate a generalized Black-Scholes equation. We show that the scheme is second-order convergent for both time
and spatial variables. It is proved that the scheme is unconditionally stable.
Numerical results support the theoretical results
FX Smile in the Heston Model
The Heston model stands out from the class of stochastic volatility (SV)
models mainly for two reasons. Firstly, the process for the volatility is
non-negative and mean-reverting, which is what we observe in the markets.
Secondly, there exists a fast and easily implemented semi-analytical solution
for European options. In this article we adapt the original work of Heston
(1993) to a foreign exchange (FX) setting. We discuss the computational aspects
of using the semi-analytical formulas, performing Monte Carlo simulations,
checking the Feller condition, and option pricing with FFT. In an empirical
study we show that the smile of vanilla options can be reproduced by suitably
calibrating three out of five model parameters.Comment: Chapter prepared for the 2nd edition of Statistical Tools for Finance
and Insurance, P.Cizek, W.Haerdle, R.Weron (eds.), Springer-Verlag,
forthcoming in 201
Some extensions of the Black-Scholes and Cox-Ingersoll-Ross models
In this thesis we will study some financial problems concerning the option pricing in complete and incomplete markets and the bond pricing in the short-term interest rates framework. We start from well known models in pricing options or zero-coupon bonds, as the Black-Scholes model and the Cox-Ingersoll-Ross model and study some their generalizations.
In particular, in the first part of the thesis, we study a generalized Black-Scholes equation to derive explicit or approximate solutions of an option pricing problem in incomplete market where the incompleteness is generated by the presence of a non-traded asset. Our aim is to give a closed form representation of the indifference price by using the analytic tool of (C0) semigroup theory.
The second part of the thesis deals with the problem of forecasting future interest rates from observed financial market data. We propose a new numerical methodology for the CIR framework, which we call the CIR# model, that well fits the term structure of short interest rates as observed in a real market
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