495 research outputs found
Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes
In this paper, we study the Kelly criterion in the continuous time framework
building on the work of E.O. Thorp and others. The existence of an optimal
strategy is proven in a general setting and the corresponding optimal wealth
process is found. A simple formula is provided for calculating the optimal
portfolio for a set of price processes satisfying some simple conditions.
Properties of the optimal investment strategy for assets governed by multiple
Ornstein-Uhlenbeck processes are studied. The paper ends with a short
discussion of the implications of these ideas for financial markets.Comment: presented at Complex'2009 (Shanghai, Feb. 23-25
Order of convergence of regression parameter estimates in models with infinite variance
AbstractA semimartingale driven continuous time linear regression model is studied. Assumptions concerning errors allow us to consider also models with infinite variance. The order of the almost sure convergence of a class of estimates which includes least squares estimates is given. In the presence of errors with heavy tails a modification of least squares estimates is suggested and shown to be better than the latter
Continuous-Time Term Structure Models
pagehe problem of term structure of interest rates modelling is considered in a continuous-time framework. The emphasis is on the bond prices, forward bond prices or LIBOR rates, rather than on the instantaneous rates as in the traditional models. Forward and spot probability measures are introduced in this general setup. Two conditions of no-arbitrage are examined. A unique process of savings account implied by an arbitrage-free family of bond prices is identified.term structure of interest rates, forward measure, martingale
Derivative pricing methodology in continuous-time models
AbstractWe show that the fundamental methodology (and practice) of evaluation of derivative securities in continuous-time models is consistent with discrete-time theory, in which a derivative price is based on the principle that adding this security to the market does not create a violation of the basic economic principle: no riskless profit with zero investment
Lognormality of Rates and Term Structure Models
A term structure model with lognormal type volatility structure is proposed. The Heath, Jarrow and Morton (HJM) framework, coupled with the theory of stochastic evolution equations in infinite dimensions, is used to show that the resulting rates are well defined (they do not explode) and remain positive. They are bounded from below and above by lognormal processes. The model can be used to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond. This extends results obtained by Sandmann and Sondermann (1993), (1994) for Markovian lognormal short rates to (non-Markovian) lognormal forward rates.Term structure of interest rates, lognormal volatility structure, Heath, Jarrow and Morton models.
From Smile Asymptotics to Market Risk Measures
The left tail of the implied volatility skew, coming from quotes on
out-of-the-money put options, can be thought to reflect the market's assessment
of the risk of a huge drop in stock prices. We analyze how this market
information can be integrated into the theoretical framework of convex monetary
measures of risk. In particular, we make use of indifference pricing by dynamic
convex risk measures, which are given as solutions of backward stochastic
differential equations (BSDEs), to establish a link between these two
approaches to risk measurement. We derive a characterization of the implied
volatility in terms of the solution of a nonlinear PDE and provide a small
time-to-maturity expansion and numerical solutions. This procedure allows to
choose convex risk measures in a conveniently parametrized class, distorted
entropic dynamic risk measures, which we introduce here, such that the
asymptotic volatility skew under indifference pricing can be matched with the
market skew. We demonstrate this in a calibration exercise to market implied
volatility data.Comment: 24 pages, 4 figure
Option Pricing Formulas based on a non-Gaussian Stock Price Model
Options are financial instruments that depend on the underlying stock. We
explain their non-Gaussian fluctuations using the nonextensive thermodynamics
parameter . A generalized form of the Black-Scholes (B-S) partial
differential equation, and some closed-form solutions are obtained. The
standard B-S equation () which is used by economists to calculate option
prices requires multiple values of the stock volatility (known as the
volatility smile). Using which well models the empirical distribution
of returns, we get a good description of option prices using a single
volatility.Comment: final version (published
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