5,001 research outputs found
Functional central limit theorems for rough volatility
We extend Donsker's approximation of Brownian motion to fractional Brownian
motion with Hurst exponent and to Volterra-like processes. Some
of the most relevant consequences of our `rough Donsker (rDonsker) Theorem' are
convergence results for discrete approximations of a large class of rough
models. This justifies the validity of simple and easy-to-implement Monte-Carlo
methods, for which we provide detailed numerical recipes. We test these against
the current benchmark Hybrid scheme \cite{BLP15} and find remarkable agreement
(for a large range of values of~). This rDonsker Theorem further provides a
weak convergence proof for the Hybrid scheme itself, and allows to construct
binomial trees for rough volatility models, the first available scheme (in the
rough volatility context) for early exercise options such as American or
Bermudan.Comment: 30 pages, 11 figure
Markov cubature rules for polynomial processes
We study discretizations of polynomial processes using finite state Markov
processes satisfying suitable moment matching conditions. The states of these
Markov processes together with their transition probabilities can be
interpreted as Markov cubature rules. The polynomial property allows us to
study such rules using algebraic techniques. Markov cubature rules aid the
tractability of path-dependent tasks such as American option pricing in models
where the underlying factors are polynomial processes.Comment: 29 pages, 6 Figures, 2 Tables; forthcoming in Stochastic Processes
and their Application
On the robustness of least-squares Monte Carlo (LSM) for pricing American derivatives
This paper analyses the robustness of Least-Squares Monte Carlo, a technique recently proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least-squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options provide evidence that a) this approach is very robust to the choice of different alternative polynomials and b) few basis functions are required. However, these conclusions are not reached when analyzing more complex derivatives.Least-Squares Monte Carlo, option pricing, American options
Executive stock option exercise with full and partial information on a drift change point
We analyse the optimal exercise of an executive stock option (ESO) written on
a stock whose drift parameter falls to a lower value at a change point, an
exponentially distributed random time independent of the Brownian motion
driving the stock. Two agents, who do not trade the stock, have differing
information on the change point, and seek to optimally exercise the option by
maximising its discounted payoff under the physical measure. The first agent
has full information, and observes the change point. The second agent has
partial information and filters the change point from price observations. This
scenario is designed to mimic the positions of two employees of varying
seniority, a fully informed executive and a partially informed less senior
employee, each of whom receives an ESO. The partial information scenario yields
a model under the observation filtration in which the
stock drift becomes a diffusion driven by the innovations process, an
-Brownian motion also driving the stock under
, and the partial information optimal stopping value
function has two spatial dimensions. We rigorously characterise the free
boundary PDEs for both agents, establish shape and regularity properties of the
associated optimal exercise boundaries, and prove the smooth pasting property
in both information scenarios, exploiting some stochastic flow ideas to do so
in the partial information case. We develop finite difference algorithms to
numerically solve both agents' exercise and valuation problems and illustrate
that the additional information of the fully informed agent can result in
exercise patterns which exploit the information on the change point, lending
credence to empirical studies which suggest that privileged information of bad
news is a factor leading to early exercise of ESOs prior to poor stock price
performance.Comment: 48 pages, final version, accepted for publication in SIAM Journal on
Financial Mathematic
Tree methods
International audienceTree methods are among the most popular numerical methods to price financial derivatives. Mathematically speaking, they are easy to understand and do not require severe implementation skills to obtain algorithms to price financial derivatives. Tree methods basically consist in approximating the diffusion process modeling the underlying asset price by a discrete random walk. In this contribution, we provide a survey of tree methods for equity options, which focus on multiplicative binomial Cox-Ross-Rubinstein model
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