6,650 research outputs found
Intrinsic expansions for averaged diffusion processes
We show that the rate of convergence of asymptotic expansions for solutions
of SDEs is generally higher in the case of degenerate (or partial) diffusion
compared to the elliptic case, i.e. it is higher when the Brownian motion
directly acts only on some components of the diffusion. In the scalar case,
this phenomenon was already observed in (Gobet and Miri 2014) using Malliavin
calculus techniques. In this paper, we provide a general and detailed analysis
by employing the recent study of intrinsic functional spaces related to
hypoelliptic Kolmogorov operators in (Pagliarani et al. 2016). Relevant
applications to finance are discussed, in particular in the study of
path-dependent derivatives (e.g. Asian options) and in models incorporating
dependence on past information
A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion
In this paper we discuss a closed-form approximation of the likelihood
functions of an arbitrary diffusion process. The approximation is based on an
exponential ansatz of the transition probability for a finite time step , and a series expansion of the deviation of its logarithm from that of a
Gaussian distribution. Through this procedure, dubbed {\em exponent expansion},
the transition probability is obtained as a power series in . This
becomes asymptotically exact if an increasing number of terms is included, and
provides remarkably accurate results even when truncated to the first few (say
3) terms. The coefficients of such expansion can be determined
straightforwardly through a recursion, and involve simple one-dimensional
integrals.
We present several examples of financial interest, and we compare our results
with the state-of-the-art approximation of discretely sampled diffusions
[A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that
the exponent expansion provides a similar accuracy in most of the cases, but a
better behavior in the low-volatility regime. Furthermore the implementation of
the present approach turns out to be simpler.
Within the functional integration framework the exponent expansion allows one
to obtain remarkably good approximations of the pricing kernels of financial
derivatives. This is illustrated with the application to simple path-dependent
interest rate derivatives. Finally we discuss how these results can also be
used to increase the efficiency of numerical (both deterministic and
stochastic) approaches to derivative pricing.Comment: 28 pages, 7 figure
Basket Options Valuation for a Local Volatility Jump-Diffusion Model with the Asymptotic Expansion Method
In this paper we discuss the basket options valuation for a jump-diffusion
model. The underlying asset prices follow some correlated local volatility
diffusion processes with systematic jumps. We derive a forward partial integral
differential equation (PIDE) for general stochastic processes and use the
asymptotic expansion method to approximate the conditional expectation of the
stochastic variance associated with the basket value process. The numerical
tests show that the suggested method is fast and accurate in comparison with
the Monte Carlo and other methods in most cases.Comment: 16 pages, 4 table
Multi-asset Spread Option Pricing and Hedging
We provide two new closed-form approximation methods for pricing spread options on a basket of risky assets: the extended Kirk approximation and the second-order boundary approximation. Numerical analysis shows that while the latter method is more accurate than the former, both methods are extremely fast and accurate. Approximations for important Greeks are also derived in closed form. Our approximation methods enable the accurate pricing of a bulk volume of spread options on a large number of assets in real time, which offers traders a potential edge in a dynamic market environment.multi-asset spread option, closed-form approximation
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