Romancing with Black-Scholes Model: An Overview

Abstract

This paper is an overview of our previous and recent results associated with the modifications of Black-Scholes model and formula. There were and are attempts to extend the classical Black-Scholes model by introducing the randomness into the coefficients r, �, s (interest rate, drift and volatility, respectfully) or to describe them by SDEs, and so on, and then to find explicit or close-form formulas for the hedging strategies and option prices. If coefficients r, �, s depend on some parameter x ?X, where X is some state space, i.e., r(x), �(x), s(x) then we call it (B, S, X)-security markets or (B, S)-security markets in random environment X. We consider different (B, S, X)-securities markets, including Markov, semi-Markov cases, jumps, and analogues/generalizations of Black-Scholes formulas for them. Black-Scholes formula with delay is also presented. Then we consider two types of telegraph process, classical/symmetric and asymmetric, and give two applications of those telegraph processes in finance by presenting European call and put option prices. Moreover, We introduce a new type of Hawkes process, namely, the exponential one-dimensional general compound Hawkes process (E1DGCHP), consider its limit theorems (LLN and FCLT) and present analogue of Black-Scholes formula in this case. Numerical examples are presented for both telegraph processes and E1DGCHP.