Stochastic models for security prices: formula sheet

Key formulas

Random walk

S(n) = X₁ + X₂ + ... + X_n

The X_i are independent and identically distributed. E[S(n)] = nμ and Var[S(n)] = nσ².

Simple symmetric random walk

P(X = +1) = P(X = −1) = ½; E[S(n)] = 0; Var[S(n)] = n

Variance grows linearly with the number of steps.

Standard Brownian motion increments

B(t) − B(s) ~ N(0, t − s) for 0 ≤ s < t

The variance is the time gap, not the standard deviation. Increments over disjoint intervals are independent.

Distribution of B(t)

B(t) ~ N(0, t), since B(0) = 0

Standard deviation is √t.

Covariance

Cov(B(s), B(t)) = min(s, t)

Follows from independent increments. Useful for sums and averages of values at different times.

Brownian motion with drift

X(t) = μt + σB(t), so X(t) ~ N(μt, σ²t)

Increments X(t) − X(s) ~ N(μ(t − s), σ²(t − s)).

Quadratic variation

(dB)² = dt

The reason Ito's lemma has an extra term. Paths have non-zero quadratic variation but are not differentiable.

Scaling limit

Steps ±√h every h time units → B(t) as h → 0

Var at time t is (t ÷ h) × h = t.

Ito process (SDE)

dX(t) = μ(t, X) dt + σ(t, X) dW(t)

μ is the drift, σ is the volatility. W(t) is standard Brownian motion.

Multiplication rules

(dW)² = dt; dt · dW = 0; (dt)² = 0

These let you simplify (dX)² to σ² dt.

Ito's lemma (function of t and X)

df = (∂f/∂t + μ ∂f/∂x + ½ σ² ∂²f/∂x²) dt + σ ∂f/∂x dW

Partial derivatives are evaluated at (t, X(t)). Here dX = μ dt + σ dW.

Ito's lemma (function of X only)

df = (μ f′(X) + ½ σ² f″(X)) dt + σ f′(X) dW

Special case when f has no explicit t.

Geometric Brownian motion

dS = μS dt + σS dW

Solution: S(t) = S(0) exp{(μ − ½σ²)t + σW(t)}.

Log of GBM

d(ln S) = (μ − ½σ²) dt + σ dW

So ln S(t) is normal with mean ln S(0) + (μ − ½σ²)t and variance σ²t.

Brownian motion with drift

X(t) = X(0) + μt + σW(t)

From dX = μ dt + σ dW with constant μ and σ.

GBM stochastic differential equation

dS(t) = μS(t) dt + σS(t) dZ(t)

μ is the drift and σ the volatility, both constant. Z(t) is standard Brownian motion.

Solution of the SDE

S(t) = S(0) exp{(μ − σ²/2)t + σZ(t)}

Obtained by applying Ito's lemma to ln S(t).

Distribution of log price

ln S(t) ~ N(ln S(0) + (μ − σ²/2)t, σ²t)

Normal, so S(t) is lognormal.

Distribution of log return over (t, t+s)

ln[S(t+s)/S(t)] ~ N((μ − σ²/2)s, σ²s)

Independent of the past and independent across non-overlapping intervals.

Mean of the share price

E[S(t)] = S(0) e^(μt)

Uses E[e^Y] = exp(m + v/2) for Y ~ N(m, v).

Variance of the share price

Var[S(t)] = S(0)² e^(2μt) (e^(σ²t) − 1)

Grows with σ² and t.

Lognormal moments (general)

If ln X ~ N(m, v): E[X] = e^(m + v/2), Var[X] = e^(2m + v)(e^v − 1)

Use for any lognormal variable.

Lognormal median

Median of S(t) = S(0) exp{(μ − σ²/2)t}

Lower than the mean.

Probability statement

P(S(t) > k) = 1 − Φ[(ln(k/S(0)) − (μ − σ²/2)t) ÷ (σ√t)]

Standardise ln S(t) and use normal tables.

Share price model (GBM)

dS = μS dt + σS dW

Under the risk-neutral measure, replace μ with r (or r − q if the share pays a continuous dividend yield q).

Black-Scholes PDE (no dividends)

∂V/∂t + ½σ²S² ∂²V/∂S² + rS ∂V/∂S − rV = 0

With continuous dividend yield q, the third term becomes (r − q)S ∂V/∂S. The drift μ does not appear.

d1 and d2

d1 = [ln(S/K) + (r + σ²/2)T] ÷ (σ√T); d2 = d1 − σ√T

T is time to expiry in years. r is the continuously compounded risk-free rate. With dividend yield q, use r − q + σ²/2 in d1.

European call price

C = S·N(d1) − K·e^(−rT)·N(d2)

N is the standard normal cumulative distribution function. Share paying no dividends.

European put price

P = K·e^(−rT)·N(−d2) − S·N(−d1)

Or find C first and use put-call parity: P = C − S + K·e^(−rT).

Risk-neutral valuation

V = e^(−rT) × E_Q[payoff]

Under Q, S grows at r. The same price results as from the hedging argument.

Delta

Call: Δ = N(d1); Put: Δ = N(d1) − 1

Call delta lies between 0 and 1. Put delta lies between −1 and 0 (no dividends).

Gamma

Γ = φ(d1) ÷ (S σ √T)

φ(x) = e^(−x²/2) ÷ √(2π). Same for calls and puts.

Vega

ν = S φ(d1) √T

Same for calls and puts. This is the change per unit change in σ (for example 1.00 = 100 percentage points).

Call theta and rho

Θ_call = −S φ(d1) σ ÷ (2√T) − r K e^(−rT) N(d2); ρ_call = K T e^(−rT) N(d2)

Theta is per year. Divide by 365 for a daily figure if asked.

Greeks link

Θ + rSΔ + ½σ²S²Γ = rV

This is the PDE rewritten in Greeks. It is a useful check (no dividends).

Binomial step

S(t+Δt) = S(t) × u with probability p, or S(t) × d with probability 1 − p

Take u > d > 0. Prices stay positive.

Binomial moments for the one-step return

E[S(t+Δt) | S(t)] = S(t) × [pu + (1 − p)d]

Variance of the next price = S(t)² × p(1 − p)(u − d)².

Risk-neutral up probability

q = (e^(rΔt) − d) ÷ (u − d)

Valid for no arbitrage when d < e^(rΔt) < u. Use for pricing, not real-world forecasting.

Lognormal model (GBM)

dS = μS dt + σS dB, so ln S(t) ~ N(ln S(0) + (μ − σ²/2)t, σ²t)

Continuous paths, constant σ, no jumps.

Jump diffusion

dS/S = μ dt + σ dB + (J − 1) dN, where N is Poisson with rate λ

J is the random jump multiplier. The expected number of jumps in time t is λt.

Ornstein-Uhlenbeck process

dX = α(μ − X) dt + σ dB

α > 0 is the speed of reversion, μ the long-run mean.

OU mean and variance

E[X(t)] = μ + (X(0) − μ)e^(−αt); Var[X(t)] = σ²(1 − e^(−2αt)) ÷ (2α)

Long-run variance is σ² ÷ (2α). X(t) is normal.

Quick revision

• Brownian motion has independent, normally distributed increments: B(t) − B(s) ~ N(0, t − s) for t > s.
• Brownian motion paths are continuous but nowhere differentiable.
• Ito's lemma has an extra second-order term compared with ordinary calculus.
• Multiplication rules: (dB)² = dt, dt·dB = 0, (dt)² = 0.
• GBM: dS = μS dt + σS dB.
• Under GBM, ln S(t) is normal, so S(t) is lognormal.
• Under GBM, E[S(t)] = S(0) e^(μt).
• Black-Scholes assumes constant volatility and risk-free rate, no dividends in the basic form, no transaction costs and continuous trading.
• Call price: C = S N(d1) − K e^(−rT) N(d2).
• Put-call parity for European options on a non-dividend-paying share: C − P = S − K e^(−rT).
• Option prices use the risk-neutral drift r, not the real-world drift μ.
• Jump models add sudden price changes, and mean-reverting models pull the value back towards a long-run level.

Common mistakes

• Using the time gap as the standard deviation instead of the variance. Fix: The second parameter is variance. The standard deviation is √(t − s). Write the square root before standardising.
• Saying Brownian motion paths are differentiable because they are continuous. Fix: State that paths are continuous everywhere but differentiable nowhere. This is why stochastic calculus is needed.
• Leaving out the ½σ² f″ term. Fix: Always write the full second-order Taylor expansion first. Remind yourself that (dW)² = dt, so it is not negligible.
• Using σ instead of σ² in the correction term. Fix: Square the volatility. For GBM, the correction for ln S is −½σ², not −½σ.
• Using μt as the mean of the log return. Fix: Always write (μ − σ²/2)t for the log return mean. Use μt only for E[S(t)]/S(0) in the exponent.
• Writing the variance of the log return as σt instead of σ²t. Fix: The N(m, v) notation uses variance. The standard deviation is σ√t.
• Using μ, the real-world drift, in the pricing formula or PDE. Fix: Remember that μ cancels in the hedging argument. In pricing, use r (or r − q).
• Using a simple annual interest rate instead of the continuously compounded rate, or using T in months. Fix: Convert to a continuous rate (δ = ln(1 + i)) if needed and express T in years before computing.
• Using the real-world probability p to price an option on a binomial tree. Fix: Price with the risk-neutral probability q = (e^(rΔt) − d) ÷ (u − d). Use p only for real-world projections.
• Saying a mean-reverting model makes share prices predictable and so always profitable. Fix: Say the process reverts in expectation only. Random shocks still occur, and reversion speed must be estimated.

Exam tips

• Write the distribution in the form N(mean, variance) every time. Examiners give marks for stating it clearly.
• For written questions on properties, list all four defining properties and explain the link to random walks through the central limit theorem.
• Be ready to explain why non-differentiable paths mean ordinary calculus fails and Ito's lemma is needed.
• In computer-based paper work, simulate a path by summing independent N(0, h) increments and show the time step h and the formula used.
• Show the standardising step in full, and state any normal table value you use.
• Write Ito's lemma in full at the start of every answer. Examiners award marks for the correct formula, even if arithmetic slips later.
• Show each partial derivative on its own line. This protects method marks.
• For GBM questions, state clearly that ln S is normal and S is lognormal. This is a frequent written-paper requirement.