Exact Barrier Option Valuation with Arbitrary Functions for the Volatility

Estevão Rosalino Jr,Jack Baczynski,Allan Jonathan Silva,Dorival Leão

doi:10.5540/tema.2015.016.01.0061

Abstract

Focus, in the past four decades, has been obtaining closed-form expressions for the no-arbitrage prices and hedges of modified versions of the Europeanoptions, allowing the dynamic of the underlying assets to have non-constant pa-rameters.In this paper, we obtain a closed-form expression for the price and hedge of an up-and-out European barrier option, assuming that the volatility in the dynamicof the risky asset is an arbitrary deterministic function of time. Setting a con-stant volatility, the formulas recover the Black and Scholes results, which suggestsminimum computational effort.We introduce a novel concept of relative standard deviation for measuring the ex-posure of the practitioner to risk (enforced by a strategy). The notion that is found in the literature is different and looses the correct physical interpreta-tion. The measure serves aiding the practitioner to adjust the number of rebalancesduring the option’s lifetime.

Highlights

Under arbitrage-free assumptions, Black & Scholes [1] and Merton [17] pioneered the achievements on pricing and hedging derivatives in financial markets
We introduce a novel concept of relative standard deviation for measuring correctly the exposure of the practitioner to risk, or else, the efficiency of risk absorbtion assigned to the strategy
Σrel expresses the proportion of risk that must be assumed by the practitioner to that absorbed by the delta hedging strategy

Summary

INTRODUCTION

Under arbitrage-free assumptions, Black & Scholes [1] and Merton [17] pioneered the achievements on pricing and hedging derivatives in financial markets They considered an European call option and a market with one bond and one stock where the parameters in the dynamics that model the market have constant values. Harrison & Pliska [4] – among others – showed that, essentially, there is an equivalence between absence of arbitrage opportunities and the existence of an equivalent measure that renders the discounted underlying stock a martingale: under this measure, pricing a derivative is allowed to be naively obtained, in that average is applied to the discounted payoff, conditional to the present information Underpinned by these seminal results, significative advances followed in obtaining closed-form expressions for the exact prices and hedges of options, as can be verified, for instance, in the. The measure serves aiding the practitioner to adjust the number of rebalances during the option’s lifetime

PRICING AND HEDGING RESULTS

SIMULATIONS

Findings

CONCLUSION