On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model

Notes : Black - Scholes - Merton Model ( IEOR 4707 ,

denote an increment of the BM (with ds > 0). We also use N(μ, σ2) to denote a normal distribution with mean μ and variance σ2. Recall some of the key properties of BM: (i) B0 = 0; (ii) independent increments, i.e., dBs and dBt are independent, for any s + ds ≤ t; (iii) stationary increments, i.e., dBs follows a normal distribution N(0, ds). Note this last distribution depends only on the length...

QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds

This paper presents a discrete-time option pricing model that is rooted in Reinforcement Learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a riskadjusted Markov Decision Process for a discrete-time version of the classical Black-ScholesMerton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this ...

Calibration of the Local Volatility in a Generalized Black-Scholes Model Using Tikhonov Regularization

Following an approach introduced by Lagnado and Osher (1997), we study Tikhonov regularization applied to an inverse problem important in mathematical finance, that of calibrating, in a generalized Black–Scholes model, a local volatility function from observed vanilla option prices. We first establish W 1,2 p estimates for the Black–Scholes and Dupire equations with measurable ingredients. Appl...

The Black-Scholes-Merton Approach to Pricing Options

Correction to Black-Scholes Formula Due to Fractional Stochastic Volatility

Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is shown how the associated implied volatility has a term structure that is a f...