Itô's lemma

A chain rule with a catch

If S were a smooth function of time, the change of f(S) would be f'(S) * dS, the ordinary chain rule. But dS contains dW, and (dW)^2 = dt is not negligible: it is of the same order as dt. Expanding f(S + dS) to second order, the term 0.5 * f''(S) * (dS)^2 survives, and (dS)^2 = sigma^2 * S^2 * dt. That extra term is Itô's lemma, the main tool for deriving option prices.

def ito_drift(s, mu, sigma, f_t, f_s, f_ss):
    return f_t + mu * s * f_s + 0.5 * sigma ** 2 * s ** 2 * f_ss

# f(S) = S: the drift is just mu * S (no curvature, no extra term)
print(ito_drift(100.0, 0.08, 0.2, 0.0, 1.0, 0.0))             # 8.0
# f(S) = S squared: f_S = 2 S and f_SS = 2, so the curvature term appears
print(ito_drift(100.0, 0.08, 0.2, 0.0, 2 * 100.0, 2.0))        # 1600 + 400 = 2000.0

# The drift for several functions of the price, S = 100, mu = 8%, sigma = 20%
def ito_drift(s, mu, sigma, f_t, f_s, f_ss):
    return f_t + mu * s * f_s + 0.5 * sigma ** 2 * s ** 2 * f_ss

s, mu, sigma = 100.0, 0.08, 0.2
print("S      ", ito_drift(s, mu, sigma, 0.0, 1.0, 0.0))                  # 8.0
print("S^2    ", ito_drift(s, mu, sigma, 0.0, 2 * s, 2.0))                # 2000.0
print("S^3    ", ito_drift(s, mu, sigma, 0.0, 3 * s ** 2, 6 * s))         # 240000 + 120000 = 360000.0
print("sqrt(S)", round(ito_drift(s, mu, sigma, 0.0, 0.5 / s ** 0.5, -0.25 / s ** 1.5), 4))    # 0.35 : curvature pulls it down

The log price and the sigma squared over two

Apply the lemma to f = ln S. Here f_t = 0, f_S = 1/S and f_SS = -1/S^2, so the drift is mu - 0.5 * sigma^2 and the diffusion sigma: d ln S = (mu - sigma^2/2) dt + sigma dW. The log price is an arithmetic Brownian motion, which is exactly why the solution of GBM has mu - sigma^2/2 in the exponent. The convexity of exp and the concavity of ln are two sides of the same fact.

def ito_drift(s, mu, sigma, f_t, f_s, f_ss):
    return f_t + mu * s * f_s + 0.5 * sigma ** 2 * s ** 2 * f_ss

def log_drift(mu, sigma):
    s = 1.0                                        # the answer does not depend on s
    return ito_drift(s, mu, sigma, 0.0, 1.0 / s, -1.0 / s ** 2)

print(log_drift(0.08, 0.20))                       # 0.06   = 0.08 - 0.5 * 0.2**2
print(round(log_drift(0.08, 0.60), 4))             # -0.1   a high volatility makes the typical growth negative

Checking the lemma by simulation

You do not have to take the lemma on trust. Take many random one-step moves s_new = s + mu * s * dt + sigma * s * sqrt(dt) * z, compute f(s_new) - f(s) for a function f, average and divide by dt. The result converges to the drift from the lemma. With f = S^2 and the curvature term left out, the simulation visibly disagrees, which is the best way to see that the extra term is real.

import numpy as np

def mean_change_estimate(f, s, mu, sigma, dt, z):
    s_new = s + mu * s * dt + sigma * s * np.sqrt(dt) * z
    return float(np.mean(f(s_new) - f(s)) / dt)

half = np.random.default_rng(1).standard_normal(200_000)
z = np.concatenate([half, -half])                 # pairs of opposite draws: the noise at first order cancels
estimate = mean_change_estimate(lambda x: x ** 2, 100.0, 0.08, 0.2, 0.001, z)
print(round(estimate))                                       # about 2000
print(round(0.08 * 100.0 * 2 * 100.0))                       # 1600  the ordinary chain rule alone: the curvature term (400) is missing

• Ito's lemma: the drift of f(S, t) is f_t + mu * S * f_S + 0.5 * sigma^2 * S^2 * f_SS and the noise coefficient sigma * S * f_S.
• The curvature term exists because (dW)^2 = dt; for ln S it gives the drift mu - sigma^2 / 2 of the GBM solution.
• Verify by simulation: the average of f(s_new) - f(s) over dt matches the lemma; applied to an option value it yields Black–Scholes.

Next up: Asian options, whose payoff depends on the average price over the life of the option.

Lab · what's graded

• Find the drift of a function with ito_drift25 pts
• Find the noise coefficient with ito_diffusion25 pts
• Apply it to the log price with log_drift25 pts
• Check it by simulation with mean_change_estimate25 pts