You now have the entire theoretical framework — Itô integrals, the Itô-Doeblin formula, Black-Scholes, correlation, and the Brownian Bridge. The exercises in this chapter are the proving ground: the problems that force you to use these tools under pressure, in the same way a new instrument is tested before it goes on stage. We break them into five clusters, each targeting a different skill.
Five clusters — five distinct skills. Master all five and you are ready for Risk-Neutral Pricing.
The problem: Show that if you trade an asset M(t) that is already a Martingale, your capital gains process I(t) = ∫₀ᵗ Δ(u) dM(u) is also a Martingale — no matter what strategy Δ you use.
The problem: The general Itô product rule says d(ΔS) = Δ dS + S dΔ + dΔ dS. But in the BSM self-financing portfolio, we write dX = Δ dS + r(X − ΔS) dt with no dΔ dS term. Why?
The answer: A self-financing portfolio satisfies dX = Δ dS + r(X − ΔS) dt by construction — it means you never inject or withdraw cash. Any rebalancing (changes in Δ) is funded by simultaneously adjusting the cash account. The dΔ dS term does not disappear mathematically — it is absorbed into the accounting: the gain from changing positions is exactly offset by the cost of funding that change. The self-financing condition enforces this balance as a constraint.
The self-financing condition is the bridge between discrete-time portfolio theory (where you explicitly track cash flows) and continuous-time stochastic calculus (where everything happens simultaneously). It is what allows us to write the clean BSM master equation — and it is the constraint that a real delta-hedger must maintain at every rebalancing step.
The problem: Given dS = αS dt + σS dW, find the explicit solution for S(t).
The problem: Solve dR = (α − βR) dt + σ dW explicitly.
This is the most important exercise for a practitioner. You take the closed-form BSM formula c = x·N(d₁) − Ke⁻ʳᵀN(d₂) and verify — by computing partial derivatives — that it actually satisfies the PDE: cₜ + rx·cₓ + ½σ²x²·cₓₓ = rc.
The BSM PDE is a balance sheet equation. Three forces — Theta (−), Delta (+), Gamma (+) — must always sum to the risk-free return r·c.
The verification tells you something profound: the Gamma term ½σ²x²·cₓₓ is the "cost of hedging" — the amount you must earn from the curvature of the option to cover the cost of continuous rebalancing. If Gamma is high (option is near the money), hedging is expensive and must be precisely offset by Theta. This Gamma-Theta balance is why options markets price implied volatility so carefully.
The problem: Given two correlated stocks dS₁ = σ₁S₁ dW₃ and dS₂ = σ₂S₂ dW₄ where W₃ and W₄ have correlation ρ, find the independent Brownian Motions W₁ and W₂ that drive them.
The method: Use the Cholesky decomposition of the correlation matrix. Set W₃ = W₁ and W₄ = ρW₁ + √(1−ρ²)W₂. Then:
Why it matters: Every multi-asset simulation starts here. To generate correlated random scenarios on a computer, you generate independent standard normals first, then apply the correlation mix. This decomposition is the engine inside every Monte Carlo simulator that runs multiple correlated assets.
The problem: If volatility σ(t) and correlation ρ(t) are changing continuously, show that the cross-variation formula dW₁·dW₂ = ρ(t) dt still holds locally.
The insight: Over an infinitesimal window dt, all smooth functions are effectively constant. Even if σ and ρ change from moment to moment, within each dt they behave as fixed parameters. The Multiplication Table applies with the current instantaneous values. This is why stochastic volatility models (where σ itself is a random process) are still tractable — you use the current σ at each step, not a global constant.
Exercise 4.15 asks you to construct a two-dimensional Brownian Motion from two independent one-dimensional motions. This is the foundation of every multi-factor risk model: the Fama-French three-factor model, the APT, and all modern factor-based strategies decompose asset returns into independent shocks mixed with a correlation structure. The stochastic calculus of Chapter 15 is the mathematical rigour behind what practitioners implement daily.
This exercise is a warning for anyone who comes to finance from physics or engineering. Two different conventions for defining stochastic integrals produce answers that differ by exactly ½T — and in finance, using the wrong convention leads to systematic mispricing.
Evaluates the integrand at the left endpoint of each interval — the value before the price move. This is the only physically realisable convention for trading: you must set your position before you see the price move.
∫₀ᵀ W dW = ½W²(T) − ½T
The −½T is the Itô correction — the "cost of jaggedness."
Evaluates the integrand at the midpoint of each interval — an average of beginning and end. Produces ordinary calculus rules, with no Itô correction.
∫₀ᵀ W ∘ dW = ½W²(T)
Cleaner formula — but requires knowledge of the future price within each interval.
Itô uses what you know (left endpoint). Stratonovich uses what you cannot yet know (midpoint). Only Itô is physically realisable in trading.
The difference between Itô and Stratonovich is exactly ½T for the integral ∫W dW. This is not a rounding error — it is a systematic gap that grows with time. A model that accidentally uses Stratonovich calculus in a finance context will consistently underprice options by an amount proportional to ½σ²T. In a 1-year option with σ = 20%, that is a mispricing of 2% of the notional — substantial by any standard.
Simulate many Brownian paths and compute both the Itô integral ∫W dW and the Stratonovich integral ∫W ∘ dW. Confirm that their average difference is exactly ½T — the Itô correction measured empirically.
Red histogram = Itô integral values | Blue histogram = Stratonovich values | Both centred but offset by exactly ½T
Before moving to Risk-Neutral Pricing, confirm you can answer each of these questions instinctively. They represent the minimum fluency needed for Chapter 5.
Chapter 17 — five clusters, five skills:
You are ready. The final readiness test is the −½σ² correction: if you can explain why d(log S) has this term, you have the full fluency needed for Risk-Neutral Pricing and Girsanov's Theorem — the magic that transforms the real world into a Martingale world, and makes fair option pricing possible.