Discrete delta hedging and hedging error

Definition Fix trading dates $0 = t_0 < t_1 < \dots < t_n = T$ with steps $h_k = t_{k+1} - t_k$. Let $V(t, S)$ be the Black-Scholes price of a European option computed with a model volatility $\sigma_m > 0$ and constant $r$ and $q$, with Greeks $\Delta = V_S$, $\Gamma = V_{SS}$ and $\Theta = V_t$. A discrete delta hedge of a long option holds the option, is short $\Delta_k = V_S(t_k, S_{t_k})$ shares on $[t_k, t_{k+1})$, and keeps the balance in the money-market account, so that the option, the shares and this cash are worth zero together at $t_k$, with earlier gains set aside in the same account. The holding $\Delta_k$ is chosen from the information at $t_k$ and is not changed inside the step. The one-step residual $\Delta\Pi_k$ is the change in value of this position over $[t_k, t_{k+1}]$. With $r = q = 0$, writing $S = S_{t_k}$, $h = h_k$ and $\Delta S = \Delta S_k = S_{t_{k+1}} - S_{t_k}$, it is

With $r$ and $q$ nonzero it also contains the interest on the cash balance $\Delta_k S - V$ and the dividends owed on the short shares. For $t_k < T$ the price is smooth, and a second-order Taylor expansion with the state frozen at $(t_k, S)$ gives $\Delta V \approx \Theta h + \Delta_k\,\Delta S + \tfrac12\Gamma(\Delta S)^2$. The short shares cancel the delta term, and the carry adds about $\bigl((r - q)S\Delta_k - rV\bigr)h$. The model pricing PDE $\Theta + \tfrac12\sigma_m^2 S^2\Gamma + (r - q)S\Delta - rV = 0$ then leaves

The omitted terms, such as $\tfrac16 V_{SSS}(\Delta S)^3$ and the mixed term $\Theta_S\,h\,\Delta S$, are of higher order for small steps, but they matter for large moves and near expiry close to the strike, where the derivatives change quickly. A short option with the opposite share position has every sign reversed.

Definition The gamma exposure of a position is its second derivative in the stock price, $\Gamma = V_{SS}$ for one long option and $-\Gamma$ for one short option, in units of shares per currency unit. The delta hedge has zero gamma, so the gamma of the hedged position is that of the option. In the leading residual it multiplies the excess of the squared move over the model charge, and in terms of the return $\Delta S/S$

where the cash gamma $\tfrac12\Gamma S^2$ is in currency per unit of squared return. Gamma also sets the size of the next trade, since the new delta is $\Delta_{k+1} \approx \Delta_k + \Gamma\,\Delta S$ to first order. For a long call or put, $\Gamma > 0$, so after a rise the hedger shorts more shares at the higher price and after a fall buys shares back at the lower price.

Definition On the grid, the realized variance of the stock over $[0, T]$ is the sum of squared returns

or the same sum with log returns $\ln(S_{t_{k+1}}/S_{t_k})$, and $\operatorname{RV}/T$ is its annualized version. Suppose that under $\mathbb{P}$ the stock follows $dS_t = \mu S_t\,dt + \sigma_a S_t\,dW_t$ with an actual volatility $\sigma_a > 0$ and constant $\mu$. Then, given $\mathcal{F}_{t_k}$ and with $S = S_{t_k}$, the squared move over a step of length $h$ satisfies

where the exact mean is $S^2\bigl(e^{(2\mu + \sigma_a^2)h} - 2e^{\mu h} + 1\bigr)$ and the drift enters only at order $h^2$. The variance comes from $\Delta S \approx \sigma_a S\sqrt{h}\,Z$ with $\operatorname{Var}(Z^2) = 2$. As the mesh tends to zero, the realized variance tends in probability to the quadratic variation $\sigma_a^2 T$ of $\ln S$, and for an adapted volatility process $\sigma_t$ to $\int_0^T \sigma_t^2\,dt$, as in Quadratic variation and stochastic limits.

Definition The hedging error $E_T$ of the strategy is the value at $T$ of the position that buys the option at its model price $V(0, S_0)$ at time $0$, holds the delta hedge and the cash account until $T$, and receives the payoff. It equals the residuals accrued to $T$,

with $E_T = 0$ for exact replication, and an option writer who sells at the model price and holds the opposite hedge has the error $-E_T$. With $r = q = 0$ it is $E_T = (S_T - K)^{+} - V(0, S_0) - \sum_k \Delta_k\,\Delta S_k$ for a call. Under the diffusion of the previous card, the frozen-state approximations give for one step

A price jump $J$ inside a step leaves the exact residual

which is nonnegative for a long call or put because $V$ is convex in $S$, and which can be far from $\tfrac12\Gamma J^2$ for a large jump. A single stock position cannot offset jumps of every size, and a random volatility leaves a vega exposure. These are separate sources of error from the squared-move residual of a diffusion.

Definition The trading frequency is the number of rebalancing dates per unit of time, $1/h$ on an equally spaced grid. Over a fixed horizon $T$ with $n = T/h$ steps and the state frozen, the one-step noise has standard deviation about $(1/\sqrt{2})\lvert\Gamma\rvert\sigma_a^2 S^2 h$, and the conditionally uncorrelated steps add up to

so more frequent trading reduces the noise like $\sqrt{h}$ and leaves the variance mismatch unchanged. For calls and puts on an equally spaced grid with $\sigma_a = \sigma_m$, the root mean square of the total error shrinks in proportion to $\sqrt{h}$, a result stated here without proof. Trading has costs. With a proportional cost $\kappa$ per unit of traded value, a rebalance trades about $\lvert\Gamma\,\Delta S\rvert$ shares and costs about $\kappa\lvert\Gamma\rvert S^2\sigma_a\sqrt{2h/\pi}$ on average, so the total cost over the horizon grows like $1/\sqrt{h}$. Letting $h \to 0$ with a fixed $\kappa$ is therefore a different economic model from frictionless continuous trading, and choosing the trading frequency against costs is a separate optimization problem.

Definition Summing the leading residuals with $r = 0$ gives

with $\Gamma_k = \Gamma(t_k, S_{t_k})$, a comparison of each realized squared return with the model variance, weighted by the cash gamma at that date. For continuous hedging the comparison is exact under these assumptions. Trading is frictionless and continuous, under $\mathbb{P}$ the stock follows $dS_t = \mu_t S_t\,dt + \sigma_t S_t\,dW_t$ with adapted $\mu_t$ and $\sigma_t$, the long option is hedged by a short position of $V_S(t, S_t)$ shares computed from the model with constant $\sigma_m$, $r$ and $q$, cash earns $r$, and the dividends owed on the short shares are paid from cash. Itô’s formula for $V(t, S_t)$ uses $\sigma_t$, the model PDE uses $\sigma_m$, and the difference of the two drifts gives

for calls and puts, applying Itô on $[0, T - \varepsilon]$ and letting $\varepsilon \to 0$. With $\Gamma > 0$ and $\sigma_t > \sigma_m$ at all times the continuous hedge gains on every path, and with $\sigma_t < \sigma_m$ it loses. The amount depends on the path through $\Gamma(t, S_t)S_t^2$, which is largest near the strike close to expiry. On a finite grid the random squared-shock noise of the previous cards comes on top.