European Call Option Research Articles

Nonconvergence in the Variation of the Hedging Strategy of a European Call Option

In this paper we consider the variation of the hedging strategy of a European call option when the underlying asset follows a binomial tree. In a binomial tree model the hedging strategy of a European call option converges to a continuous process when the number of time points increases so that the price process of the underlying asset converges to a Brownian motion, the Bachelier model. However, the variation of the hedging strategy need not converge to the variation of the limit process. In fact, it is shown that the asymptotic variation of the hedging strategy may be of any order.

Option pricing under model and parameter uncertainty using predictive densities

The theoretical price of a financial option is given by the expectation of its discounted expiry time payoff. The computation of this expectation depends on the density of the value of the underlying instrument at expiry time. This density depends on both the parametric model assumed for the behaviour of the underlying, and the values of parameters within the model, such as volatility. However neither the model, nor the parameter values are known. Common practice when pricing options is to assume a specific model, such as geometric Brownian Motion, and to use point estimates of the model parameters, thereby precisely defining a density function. We explicitly acknowledge the uncertainty of model and parameters by constructing the predictive density of the underlying as an average of model predictive densities, weighted by each model's posterior probability. A model's predictive density is constructed by integrating its transition density function by the posterior distribution of its parameters. This is an extension to Bayesian model averaging. Sampling importance-resampling and Monte Carlo algorithms implement the computation. The advantage of this method is that rather than falsely assuming the model and parameter values are known, inherent ignorance is acknowledged and dealt with in a mathematically logical manner, which utilises all information from past and current observations to generate and update option prices. Moreover point estimates for parameters are unnecessary. We use this method to price a European Call option on a share index.

PARALLEL PSEUDOSPECTRAL SOLUTION OF FINANCIAL PARTIAL DIFFERENTIAL EQUATIONS

We apply the Pseudospectral method to two fundamental financial equations: the Black-Scholes equation and the Cox Ingersoil Ross model of the term structure of interest rates. The former is used to price a European Call Option and the latter to price a zero coupon bond. Chebyshev polynomials are used as the basis functions and Chebyshev collocation points for the space discretisation. The Crank-Nicolson scheme is used for the time differencing. We have developed a C++ program to solve general second order linear parabolic equations, A parallel quasi-minimal residual version of the Bi-Conjugate Gradient stabilised algorithm is applied to solve the linear system on the AP3000, a parallel computer. The regular space domain and the smooth solutions often encountered in finance suggest the suitability of using this higher order technique.

A black-scholes formula for option pricing with dividends

We obtain a Black-Scholes formula for the arbitrage-free pricing of European Call options with constant coefficients when the underlying stock generates dividends. To hedge the Call option, we will always borrow money from bank. We see the influence of the dividend term on the option pricing via the comparison theorem of BSDE(backward stochastic differential equation [5], [7]). We also consider the option pricing problem in terms of the borrowing rate R which is not equal to the interest rater. The corresponding Black-Scholes formula is given. We notice that it is in fact the borrowing rate that plays the role in the pricing formula.