Virtual Barrels by Ilia Bouchouev

Author:Ilia Bouchouev
Language: eng
Format: epub
ISBN: 9783031361517
Publisher: Springer Nature Switzerland

8.3 Delta Hedging and Option Replication

The breakthrough idea developed by Black and Scholes and independently by Merton was based on an important insight that follows from Itô’s lemma. Since a financial derivative, such as an option, is a function of a random variable, Itô’s lemma provides the rule relating a small change in the option price to a small change in the stock price. Importantly, both the stock and the option, which is a function of the stock, are driven by the same single source of uncertainty dz. Therefore, one should be able to combine the option and the stock in some smart way that eliminates this uncertainty, at least for a short period of time. If the entire risk can indeed be eliminated, then in the absence of an arbitrage, which rules out the existence of riskless profits, the value of a combined portfolio that includes an option and some quantity of the stock can only grow at the risk-free interest rate accrued on the initial investment.

The BSM framework was initially designed for the equity market and developed under the lognormal GBM assumption. Subsequently, it was tailored by one of the authors to the futures market.4 The BSM argument, however, remains intact for all diffusions of the form (8.5), and we now replicate their framework in a more general setting.

Let C(F, t) denote the price of a call option struck at K that expires at time T. The call depends on the stochastic futures price F and time t. To simplify the notation, in this chapter we suppress the price dependency on K and T, which are set contractually. To construct a mini portfolio of an option and futures, we need to know how C(F, t) changes over a small increment dt in response to the change in the futures price dF. This change is described by Itô’s formula (8.6). Applying it to the diffusion specification (8.5) results in the following stochastic equation for the option price:

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