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Fractional Brownian markets with time-varying volatility and high-frequency data

Ananya Lahiri, Rituparna Sen

Abstract

Diffusion processes driven by Fractional Brownian motion (FBM) have often been considered in modeling stock price dynamics in order to capture the long range dependence of stock price observed in reality. Option prices for such models had been obtained by Necula (2002) under constant drift and volatility. We obtain option prices under time varying volatility model. The expression depends on volatility and the Hurst parameter in a complicated manner. We derive a central limit theorem for the quadratic variation as an estimator for volatility for both the cases, constant as well as time varying volatility. That will help us to find estimators of the option prices and to find their asymptotic distributions.

Fractional Brownian markets with time-varying volatility and high-frequency data

Abstract

Diffusion processes driven by Fractional Brownian motion (FBM) have often been considered in modeling stock price dynamics in order to capture the long range dependence of stock price observed in reality. Option prices for such models had been obtained by Necula (2002) under constant drift and volatility. We obtain option prices under time varying volatility model. The expression depends on volatility and the Hurst parameter in a complicated manner. We derive a central limit theorem for the quadratic variation as an estimator for volatility for both the cases, constant as well as time varying volatility. That will help us to find estimators of the option prices and to find their asymptotic distributions.

Paper Structure

This paper contains 22 sections, 12 theorems, 93 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Regarding the solution of the SDE, Wick Ito Skorohod Integral, $H\in(0,1)$
  4. The integral operator $M$
  5. Wiener Integral with respect to FBM, $H\in(0,1)$
  6. On the way to calculate European call option price
  7. Risk-neutral measure
  8. Complete Market
  9. Price of European Call Option
  10. Estimation of volatility from discrete observations
  11. Constant $\sigma$
  12. Simulation Studies for fixed $\sigma$
  13. Time varying $\sigma_t$
  14. Simulation studies (time varying volatility)
  15. Conclusions
  16. ...and 7 more sections

Key Result

Lemma 4.1

a) If $g_t=\exp(\int_0^t\sigma_s d\hat{B}^H_s- \frac{1}{2} \int_\mathbb{R}(M(\sigma_s\chi_{[0,t]}))^2ds)$ then for $T>t$, $\tilde{E}_{\hat{P}}[g_T| \mathcal{F}_t]=g_t$. b) If $F\in L^{1,2}(\hat{P})$ (similar to Definition A4 of Elliott and Van der Hoek (2003)), and $G_t= \int_0^t F_td\hat{B}^H_t$, t

Theorems & Definitions (22)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Lemma 5.1
  • proof
  • ...and 12 more