What a delta hedge really does – a theoretical and pedagogical note

Abstract

The Black–Scholes description of delta hedging makes the instantaneous value of the short sale negative, but the value should be zero by the principle of no arbitrage. This violation of no-arbitrage makes it impossible to illustrate the Black–Scholes delta hedging of an endowment of one call by an example containing only legally realizable transactions, and this causes confusion as to what delta hedging really does. Like Cox and Ross (1976), we model the short sale as having cash proceeds, and after including these, its instantaneous net value is zero. From this fact, delta hedging yields the risk-free rate of return on the option's opening value, for as long as required, in both discrete and continuous time. The terms of the Black–Scholes equation can be interpreted as inflows or outflows of cash, whose values are fixed at the time of hedging, and which risk-averse investors correctly price as risk-free even under the objective probability measure. The Cox and Ross model of no-arbitrage can be re-interpreted as a model of delta hedging, giving the same result, and we also use it to directly derive the Black–Scholes equation for risk neutral investors.