52 research outputs found
Long Term Risk: A Martingale Approach
This paper extends the long-term factorization of the stochastic discount
factor introduced and studied by Alvarez and Jermann (2005) in discretetime
ergodic environments and by Hansen and Scheinkman (2009) and Hansen (2012) in
Markovian environments to general semimartingale environments. The transitory
component discounts at the stochastic rate of return on the long bond and is
factorized into discounting at the long-term yield and a positive
semimartingale that extends the principal eigenfunction of Hansen and
Scheinkman (2009) to the semimartingale setting. The permanent component is a
martingale that accomplishes a change of probabilities to the long forward
measure, the limit of T-forward measures. The change of probabilities from the
data generating to the long forward measure absorbs the long-term risk-return
trade-off and interprets the latter as the long-term risk-neutral measure
Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach
We propose an efficient method to evaluate callable and putable bonds under a
wide class of interest rate models, including the popular short rate diffusion
models, as well as their time changed versions with jumps. The method is based
on the eigenfunction expansion of the pricing operator. Given the set of call
and put dates, the callable and putable bond pricing function is the value
function of a stochastic game with stopping times. Under some technical
conditions, it is shown to have an eigenfunction expansion in eigenfunctions of
the pricing operator with the expansion coefficients determined through a
backward recursion. For popular short rate diffusion models, such as CIR,
Vasicek, 3/2, the method is orders of magnitude faster than the alternative
approaches in the literature. In contrast to the alternative approaches in the
literature that have so far been limited to diffusions, the method is equally
applicable to short rate jump-diffusion and pure jump models constructed from
diffusion models by Bochner's subordination with a L\'{e}vy subordinator
Time-Changed Ornstein-Uhlenbeck Processes And Their Applications In Commodity Derivative Models
This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU
diffusions time changed by L\'{e}vy subordinators. We construct their sample
path decomposition, show that they possess mean-reverting jumps, study their
equivalent measure transformations, and the spectral representation of their
transition semigroups in terms of Hermite expansions. As an application, we
propose a new class of commodity models with mean-reverting jumps based on
subordinate OU process. Further time changing by the integral of a CIR process
plus a deterministic function of time, we induce stochastic volatility and time
inhomogeneity, such as seasonality, in the models. We obtain analytical
solutions for commodity futures options in terms of Hermite expansions. The
models are consistent with the initial futures curve, exhibit Samuelson's
maturity effect, and are flexible enough to capture a variety of implied
volatility smile patterns observed in commodities futures options
Lookback Options and Diffusion Hitting Times: A Spectral Expansion Approach
Abstract. Lookback options have payoffs dependent on the maximum and/or minimum of the underlying price attained during the option's lifetime. Based on the relationship between diffusion maximum and minimum and hitting times and the spectral decomposition of diffusion hitting times, this paper gives an analytical characterization of lookback option prices in terms of spectral expansions. In particular, analytical solutions for lookback options under the constant elasticity of variance (CEV) diffusion are obtained
The Path Integral Approach to Financial Modeling and Options Pricing
In this paper we review some applications of the path integral methodology of quantum mechanics to financial modeling and options pricing. A path integral is defined as a limit of the sequence of finite-dimensional integrals, in a much the same way as the Riemannian integral is defined as a limit of the sequence of finite sums. The risk-neutral valuation formula for path-dependent options contingent upon multiple underlying assets admits an elegant representation in terms of path integrals (Feynman–Kac formula). The path integral representation of transition probability density (Green's function) explicitly satisfies the diffusion PDE. Gaussian path integrals admit a closed-form solution given by the Van Vleck formula. Analytical approximations are obtained by means of the semiclassical (moments) expansion. Difficult path integrals are computed by numerical procedures, such as Monte Carlo simulation or deterministic discretization schemes. Several examples of path-dependent options are treated to illustrate the theory (weighted Asian options, floating barrier options, and barrier options with ladder-like barriers).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44345/1/10614_2004_Article_137534.pd
Step options and forward contracts.
http://deepblue.lib.umich.edu/bitstream/2027.42/6320/5/ban5830.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/6320/4/ban5830.0001.001.tx
- …