Option delta ($\Delta$)

The option delta is the derivative of an option value with respect to the underlying asset price. In a smooth complete diffusion model, it is also the number of underlying units in the replicating delta hedge.

For contingent claim payoff $(S_T-K{)}_{+}^2$ in the non-dividend Black-Scholes model, differentiating the discounted expected value gives $\Delta =2e^{-r\tau }{S}^{-1}\mathbb{E}[(S_T^2-K{S}_T)1_{\{S_T\ge K\}}]$. Completing the square in the Gaussian exponent computes each truncated lognormal moment and yields the displayed formula, where $d_1=(\log (S/K)+(r+{\sigma }^2/2)\tau )/(\sigma \sqrt{\tau })$. Its terminal limit is $2(S-K{)}_{+}$.

A delta hedge holds ${\Delta }_t={\partial }_{s}V(t,S_t)$ units of the risky asset. In the Black-Scholes model, this choice cancels the claim's Brownian exposure.

The option delta is one of the Greeks (finance); hedging is the trading procedure that uses this sensitivity.

In the cash-numéraire market $d{S}_t=a(S_t)d{W}_t$, let a smooth call value solve $V_t+\frac{1}{2}{a}^2{V}_{SS}=0$. Its delta hedge is $U=V_S$. Differentiating the pricing equation giveswith this terminal condition interpreted away from $S=K$. If the terminal law has no atom at $K$, its assigned value at that point does not affect expectation pricing; a degenerate or absorbed diffusion can instead require a separate strike-boundary analysis. By Itô formula, $dU(t,S_t)=-a{a}^{\prime }{U}_{S}dt+a{U}_{S}d{W}_t$, while $dV(t,S_t)=U(t,S_t)d{S}_t$. Thus holding $U(t,S_t)$ shares and cash $V-SU$ replicates the terminal call. The drift in the option delta equation comes from differentiating the spatially varying diffusion coefficient, not from a drift in the original stock.

Suppose $a^{\prime }$ is bounded and let $d{Z}_t=Z_t{a}^{\prime }(S_t)d{W}_t$, $Z_0=1$. The Novikov condition makes $Z$ a strictly positive true martingale. For ${\pi }_t=U(t,S_t)$ solving the call delta equation for a driftless local volatility diffusion, the product formula givesThe two drift terms cancel through quadratic covariation. Thus $Z\pi$ is a local martingale. If it is a true martingale on the closed horizon, its terminal value is $Z_T{1}_{\{S_T\ge K\}}$, soThe upper bound uses $\mathbb{E}[Z_T\mid {\mathcal{F}}_t]=Z_t$. Equivalently, the option delta is the terminal-exceedance probability under the measure with density $Z_T$.