The aim of this study is to show that the Reduced Differential Transform Algorithm (RDTA) can be applied to highly nonlinear evolution equations appearing in quantitative finance. In particular, we compute exact solutions of nonlinear PDEs arising by relaxing the transaction-cost assumption in the illiquid Black-Scholes market. Moreover, we also aim to study the impact of the absence and presence of price slippage impact in the illiquid Black-Scholes model with transaction-cost.

In this work, we aim to study the well-posedness of a deterministic problem consisting of the non-linear heat equation of gradient type. The evolution equation emerges by projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree $2n-1$, onto the tangent space of the sphere $\mathcal{M}$ in a Hilbert space $\mathcal{H}$. We are going to deal with the question of existence and uniqueness based on the Faedo-Galerkin compactness method.

The objective of this paper is twofold. Firstly, to derive time-fractional evolution equation modeling the No-Arbitrage premium of Asian option (with arithmetic and geometric averages ) contingent upon an underlying asset that satisfies the fractional stochastic differential equation, in a setting when the strike price is fixed and floating. Secondly, we have computed the four versions of the put-call parities for Asian options, by solving the time-fractional Black-Scholes evolution modeling the difference of the premiums of put and call Asian options, through Fractional Reduced Differential Transform (FRDT) algorithm. We have also established the convergence and the error estimates for the FRDT Algorithm for the two independent variables.