1,111 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
On the probability density function of baskets
The state price density of a basket, even under uncorrelated Black-Scholes
dynamics, does not allow for a closed from density. (This may be rephrased as
statement on the sum of lognormals and is especially annoying for such are used
most frequently in Financial and Actuarial Mathematics.) In this note we
discuss short time and small volatility expansions, respectively. The method
works for general multi-factor models with correlations and leads to the
analysis of a system of ordinary (Hamiltonian) differential equations.
Surprisingly perhaps, even in two asset Black-Scholes situation (with its flat
geometry), the expansion can degenerate at a critical (basket) strike level; a
phenomena which seems to have gone unnoticed in the literature to date.
Explicit computations relate this to a phase transition from a unique to more
than one "most-likely" paths (along which the diffusion, if suitably
conditioned, concentrates in the afore-mentioned regimes). This also provides a
(quantifiable) understanding of how precisely a presently out-of-money basket
option may still end up in-the-money.Comment: Appeared in: Large Deviations and Asymptotic Methods in Finance,
Springer proceedings in Mathematics & Statistics, Editors: Friz, P.K.,
Gatheral, J., Gulisashvili, A., Jacquier, A., Teichmann, J., 2015, with minor
typos remove
Approximate solution to a hybrid model with stochastic volatility: a singular-perturbation strategy
We study a hybrid model of Schobel-Zhu-Hull-White-type from a singular-perturbation-analysis perspective. The merit of the paper is twofold: On one hand, we find boundary conditions for the deterministic non-linear degenerate parabolic partial differential equation for the evolution of the stock price. On the other hand, we combine two-scales regular- and singular-perturbation techniques to find an approximate solution to the pricing PDE. The aim is to produce an expression that can be evaluated numerically very fast
On a free boundary problem for an American put option under the CEV process
We consider an American put option under the CEV process. This corresponds to
a free boundary problem for a PDE. We show that this free bondary satisfies a
nonlinear integral equation, and analyze it in the limit of small = , where is the interest rate and is the volatility. We
use perturbation methods to find that the free boundary behaves differently for
five ranges of time to expiry.Comment: 14 pages, 0 figure
"Effects of Stochastic Interest Rates and Volatility on Contingent Claims (Revised Version)"
We investigate the effects of the stochastic interest rates and the volatility f the underlying asset price on the contingent claim prices including futures and options prices. The futures price can be decomposed into the forward price and the additional terms and the options price can be decomposed into the Black-Scholes formula and several additional terms via the asymptotic expansion approach in the small disturbance asymptotics developed by Kunitomo and Takahashi(1995,1998,2001), which is based on Malliavin-Watanabe Calculus in stochastic analysis. We illustrate our new formulae and their numerical accuracy by using some modi ed CIR type processes for the short term interest rates and stochastic volatility.
The Exact Value for European Options on a Stock Paying a Discrete Dividend
In the context of a Black-Scholes economy and with a no-arbitrage argument,
we derive arbitrarily accurate lower and upper bounds for the value of European
options on a stock paying a discrete dividend. Setting the option price error
below the smallest monetary unity, both bounds coincide, and we obtain the
exact value of the option.Comment: 14 pages,3 figure
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