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Itô's lemma: why gamma exists

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Start with the idea

Ordinary calculus drops squared increments when they are too small. Brownian increments are different: their squares accumulate at order dt. The extra curvature term is the mathematical source of gamma exposure.

Symbols, units & horizon

V(S,t): derivative value in currency
S: underlying price
∂V/∂t or Θ: value change per year at fixed spot
∂V/∂S or Δ: delta, value units per price unit
∂²V/∂S² or Γ: gamma, value units per squared price unit
σ: return volatility in GBM
σ_imp: model implied volatility
σ_realised: volatility used in the local realised-move approximation
dt: years
dS: stochastic price increment
dW: Brownian increment with (dW)²=dt in Itô calculus
P&L: value change after subtracting the delta exposure, before financing and costs
V_t,V_S,V_SS: partial derivatives with respect to time, spot, and spot twice
σ_imp and σ_real: implied and realised return volatilities in the same time units

When and why to use this

Use the expansion to explain option P&L and why delta hedging removes only local first-order spot exposure. It separates theta, realised variation and hedge error.

In ordinary calculus, if $V$ depends on $S$ and $S$ moves by $d S$ , then $d V = V_{S} d S$ . Brownian paths are too rough for that: $(d W)^{2} = d t$ is not negligible, so a second-order term survives.

d V = \frac{\partial V}{\partial t} d t + \frac{\partial V}{\partial S} d S + \frac{1}{2} \frac{\partial^{2} V}{\partial S^{2}} σ^{2} S^{2} d t

Taylor expansion + quadratic variation

Retain the Brownian second-order term

Taylor-expand $d V \approx V_{t} d t + V_{S} d S + \frac{1}{2} V_{S S} (d S)^{2}$ . Terms involving dt² and dt·dW vanish in the Itô limit.
Substitute GBM and use $(d W)^{2} = d t$ in the quadratic-variation sense: $(d S)^{2} = σ^{2} S^{2} d t$ . This yields the displayed formula.
The statement about squared Brownian increments is a stochastic-calculus result, not an ordinary pointwise equality of numbers.

Work it by hand

For V=S², derivatives are Vₛ=2S, Vₛₛ=2 and Vₜ=0. Thus $d V = (2 μ + σ^{2}) S^{2} d t + 2 σ S^{2} d W$ .

The three partial derivatives have names on every options desk: $\frac{\partial V}{\partial t}$ is theta, $\frac{\partial V}{\partial S}$ is delta, $\frac{\partial^{2} V}{\partial S^{2}}$ is gamma. The last term says that a convex position earns from volatility, at rate $\frac{1}{2} Γ σ^{2} S^{2}$ , even with no drift. That is why an option has time value: it is the price of that convexity.

Hold a delta-hedged option and the $d S$ term cancels. What remains, per unit time, is

P&L \approx Θ d t + \frac{1}{2} Γ S^{2} σ_{realised}^{2} d t

Local approximation; carrying assumptions stated

Subtract delta and isolate the volatility contribution

An option’s local price change is $Θ d t + Δ d S + \frac{1}{2} Γ (d S)^{2}$ . Shorting Δ units of stock removes the linear term over that instant.
Approximating realised quadratic variation as $S^{2} σ_{r e a l}^{2} d t$ leaves the displayed theta-plus-gamma expression before carrying adjustments. In the zero-rate, no-dividend model, $Θ = - \frac{1}{2} Γ S^{2} σ_{i m p}^{2}$ .
Substitution gives local long-option hedged P&L $\frac{1}{2} Γ S^{2} (σ_{r e a l}^{2} - σ_{i m p}^{2}) d t$ . With rates/dividends, include financing and carry consistently.

Work it by hand

Γ=.02, S=100, realised vol=.25, implied=.20, dt=1/252 gives approximate long-option P&L $\frac{.5 (.02) 100^{2} (.0625 - .04)}{252} = $ 0.00893$ per option unit.

Under a zero-rate, no-dividend diffusion model with continuous delta hedging and a fixed pricing volatility, the local variance contribution is $\frac{1}{2} Γ S^{2} (σ_{real}^{2} - σ_{imp}^{2}) d t$ . With nonzero rates, account explicitly for financing the option and stock hedge before interpreting excess P&L. The expression weights variance differences by the changing gamma and spot along the path; it is not a guarantee based on an unweighted volatility comparison. Changing implied volatility, jumps, discrete hedging and transaction costs add exposures that this local approximation omits.

In your own words: why does a delta-hedged long option position lose money on a quiet day?

Delta hedging removes exposure to the direction of the move, leaving theta and gamma. Gamma earns in proportion to the square of the realised move; theta is paid regardless. The option was priced assuming a certain daily move (implied vol). On a quiet day the realised move is smaller than that, so gamma earnings fall short of theta, and the position bleeds. The premium you paid was for movement that didn't happen.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def ito_local_change(theta, delta, gamma, spot, volatility, dt, delta_spot):
    """Differential approximation, not a finite-move exact price."""
    return theta*dt+delta*delta_spot+.5*gamma*volatility**2*spot**2*dt

def delta_hedged_local_pnl(theta, gamma, spot, realized_volatility, dt):
    """Before cash-account financing, hedge costs and discrete-hedge error."""
    return theta*dt+.5*gamma*spot**2*realized_volatility**2*dt

print(delta_hedged_local_pnl(-5, .03, 100, .2, 1/252))
def zero_carry_variance_pnl(gamma, spot, realized_vol, implied_vol, dt):
    return .5*gamma*spot**2*(realized_vol**2-implied_vol**2)*dt

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