A standard call expires worthless if the stock finishes below the strike. A lookback call has no such failure mode. Because its effective "strike" is the highest price ever reached during the contract's life, the terminal payoff is always non-negative. You are always buying at the worst possible historical price — and the option pays your regret.
The stock peaks at Y(T) then falls to S(T). The lookback payoff is the gap — always non-negative. It pays exactly the "regret" of not having sold at the peak.
The lookback is the option that every retail investor wishes they had. Imagine buying NIFTY in January and receiving a payoff equal to the January-to-December high minus the December closing price. Because markets are volatile, this payoff is usually substantial. The cost reflects this: lookbacks are typically 2–4× more expensive than vanilla calls on the same underlying.
The running maximum Y(t) = max{S(u): 0 ≤ u ≤ t} is unlike any process we have encountered in Chapters 1–27. It is singular: it stays perfectly flat for long periods, then suddenly increases when the stock hits a new all-time high. This unusual behaviour requires a special differential rule and produces a unique boundary condition in the PDE.
Y(t) (gold staircase) stays perfectly flat when S(t) is below the peak, then jumps to match S(t) at each new all-time high. The flat segments have zero measure — this is the "singular" growth.
The singular nature of Y(t) is what makes the lookback PDE different from the barrier PDE. Y(t) generates no diffusion term — it has zero quadratic variation. The PDE has no fᵧᵧ or fᵧ terms from Y itself. The only influence of Y enters through the boundary condition vᵧ(t,y,y) = 0 — the singular process is "felt" at the boundary, not in the interior.
The lookback option price v(t, x, y) depends on three variables: time t, current stock price x, and the running maximum y. The PDE in the interior (x < y) is the standard BSM equation. The magic — and mathematical subtlety — is at the boundary x = y, where the singular growth of Y imposes a precise condition.
The singular boundary condition vᵧ = 0 at x = y has a beautiful economic interpretation: the option's value should not jump discontinuously just because the stock hits a new record high. If it did, you could make an instantaneous profit by being in the option exactly at new-high moments. The condition vᵧ = 0 is the no-free-lunch requirement applied to the running maximum process.
Solving a PDE in three variables (t, x, y) requires a 3D grid — computationally expensive and mathematically complex. The lookback option has a special structure that enables a powerful simplification: linear scaling. If you double both the stock price and the maximum, the option value doubles exactly.
Dimension reduction is one of the most important computational techniques in quantitative finance. It exploits the symmetry of the pricing problem to collapse high-dimensional PDEs into tractable low-dimensional ones. The same technique (identifying the homogeneous scaling) applies to Asian options, exchange options, and any option whose payoff is a homogeneous function of the underlying assets.
After solving the 2D PDE for u(t, z) via Feynman-Kac and integrating against the joint density of (S(T), Y(T)) from Chapter 27, Shreve derives the closed-form lookback price. It resembles Black-Scholes but with additional terms that capture the path-dependency through the running maximum y.
The (σ²/2r) prefactor in the correction term reveals the critical insight: the lookback premium is directly proportional to σ². Double the volatility → four times the path-dependency premium. This is why lookback options are so expensive in volatile markets — a high-volatility stock is likely to reach a very high peak and then fall significantly, making the Y(T) − S(T) payoff large. In calm markets, the lookback approaches a vanilla call in price.
| Feature | Vanilla Call | Floating Lookback |
|---|---|---|
| Strike | Fixed K at inception | Floating Y(T) = historical max |
| Path dependent? | No — only S(T) matters | Yes — full path history matters |
| Can expire worthless? | Yes — if S(T) ≤ K | No — Y(T) − S(T) ≥ 0 always |
| State variables | t, S(t) — 2D | t, S(t), Y(t) — 3D (reducible to 2D) |
| Key boundary condition | v(T,x) = (x−K)⁺ | vᵧ(t,y,y) = 0 (singular) + v(T,x,y) = y−x |
| Sensitivity to σ | Linear in σ (Vega ≈ S·√T·N'(d₁)) | Quadratic in σ — σ² appears in the correction term |
| Relative cost | Baseline | 2–4× more expensive — the "perfect hindsight" premium |
| Hedge | Δ = N(d₁) shares | Δ = ∂v/∂x (more complex — depends on x and y) |
Simulate GBM paths and compare the three option types side by side. Watch how the lookback option always produces a non-negative payoff regardless of market direction, while the vanilla can expire worthless and the barrier can be killed. Increase volatility and observe the lookback premium grow quadratically.
Blue histogram = vanilla payoff distribution (mass at 0 = worthless expiry) | Gold histogram = lookback payoff distribution (zero mass at 0 — always pays) | Lookback distribution has no spike at zero
Chapter 28 — lookback options in five ideas:
What comes next: The Change of Numeraire technique — a powerful measure-change that simplifies complex payoffs by choosing a different "unit of account" (numeraire). It is the key tool for pricing exchange options, spread options, and Quanto derivatives.