In this paper, the Legendre wavelet neural network with extreme learning machine is proposed for numerical solution of time fractional Black–Scholes model. way, operational matrix derivative based on two-dimensional derived and employed to solve European options pricing problem. This scheme converts problem into calculation a set algebraic equations. The constructed; meanwhile, algorithm adopte...

Abstract We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according dynamics continuous-time Markov chain two states. The proof is based on reduction original equivalent free-boundary and solution latter by means smooth-fit normal-reflection conditions. show that boundaries are determined ...

In the finance market, it is well known that price change of underlying fractal transmission system can be modeled with Black-Scholes equation. This article deals finding approximate analytic solutions for time-fractional equation fractional integral boundary condition a European option pricing problem in Katugampola derivative sense. It generalizes both Riemann–Liouville and Hadamard derivativ...

In common finance literature, Black-Scholes partial differential equation of option pricing is usually derived with no-arbitrage principle. Considering an asset market, Merton applied the Hamilton-Jacobi-Bellman techniques of his continuous-time consumption-portfolio problem, deriving general equilibrium relationships among the securities in the asset market. In special case where the interest ...

We reconsider the replication problem for contingent claims in a complete market under a general framework. Since there are various limitations in the Black–Scholes pricing formula, we propose a new method to obtain an explicit self–financing trading strategy expression for replications of claims in a general model. The main advantage of our method is that we propose using an orthogonal expansi...

Since Black and Scholes published their seminal paper [2] in 1973, the pricing of options by means of deterministic partial differential equations or inequalities has become standard practise in computational finance. An option gives the right (but not the obligation) to buy (call option) or sell (put option) a share for a certain value (the exercise price K) at a certain time T (exercise date)...

My research interests are in stochastic processes and mathematical finance. In particular, my dissertation provides a weak existence result which has application to the financial engineer’s calibration problem and essentially generalizes an earlier result of Gyöngy [9]. In this research statement I will briefly review the calibration problem, show how my work fits into the literature, and discu...

After the celebrated Black-Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid to stochastic volatility models where the volatility is randomly fluctuating driven by an additio...

Classical explicit finite difference schemes are unsuitable for the solution of the famous Black-Scholes partial differential equation, since they impose severe restrictions on the time step. Furthermore, they may produce spurious oscillations in the solution. We propose a new scheme that is free of spurious oscillations and guarantees the positivity of the solution for arbitrary stepsizes. The...