Lookback Options: Buying at the Low, Selling at the High

Every trader has the same fantasy: buy at the absolute lowest price and sell at the absolute highest. You can buy a contract that pays you as if you did. That is the lookback option, Chapter 26.

The catch? Very expensive. Hindsight does not come cheap.

What Is a Lookback Option?

A payoff that depends on the realized maximum or minimum of the asset price. The extreme version pays $S_{\max} - S_{\min}$, literally “buy at the bottom, sell at the top.” In practice, more modest flavors exist.

Fixed Strike vs. Floating Strike

Just like Asian options, lookbacks come in two varieties.

Floating strike replaces the strike with the max or min. A call pays $\max(S(T) - S_{\min}, 0)$. You buy at the lowest price reached. A put pays $\max(S_{\max} - S(T), 0)$. You sell at the highest price reached.

Fixed strike replaces the stock price with the max or min. A call pays $\max(S_{\max} - E, 0)$. A put pays $\max(E - S_{\min}, 0)$.

Continuous Measurement

When the maximum is measured continuously, every price counts: $M(t) = \max_{0 \le \tau \le t} S(\tau)$. The maximum can only increase. When the stock is below the maximum, M stays constant. It only updates on new highs.

The PDE

The option value V(S, M, t) satisfies the Black-Scholes equation:

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$

This looks like regular Black-Scholes. Where is the extra term? For Asians, the running average always changes (dI = S dt). For lookbacks, the maximum changes only when S = M. For S < M, dM = 0. No extra term in the PDE.

The interesting part is the boundary condition at S = M:

$$\frac{\partial V}{\partial M} = 0 \quad \text{at } S = M$$

Why? At that instant, the probability of the current maximum still being the maximum at expiry is zero. Any tiny upward move creates a new maximum. First derived by Goldman, Sosin, and Gatto (1979).

Discrete Measurement

In practice, contracts specify sampling dates (monthly, weekly). Discrete sampling is cheaper (fewer chances to hit new highs) and easier to administer. The updating rule:

$$M_i = \max(M_{i-1}, S(t_i))$$

Between samples, the stock can go above the recorded maximum without triggering an update. The pricing follows Chapter 24’s algorithm with jump condition:

$$V(S, M, t_i^-) = V(S, \max(M, S), t_i^+)$$

Right after sampling, the region S > M is unreachable. Before sampling, S can be anywhere. For S > M (reachable before the sample), the value comes from the S = M line. You “fold” the solution along the diagonal.

The Similarity Reduction

If the payoff scales as $M \cdot F(S/M)$, we can write $V(S, M, t) = M \cdot W(S/M, t)$ where $\xi = S/M$ and W satisfies:

$$\frac{\partial W}{\partial t} + \frac{1}{2}\sigma^2 \xi^2 \frac{\partial^2 W}{\partial \xi^2} + r\xi\frac{\partial W}{\partial \xi} - rW = 0$$

Three dimensions down to two. Works for floating strike lookbacks but not fixed strike (the strike E breaks the scaling).

Closed-Form Formulas

With constant volatility, explicit formulas exist. For a floating strike lookback call (payoff = $S - S_{\min}$):

$$V = S \cdot N(d_1) - M \cdot e^{-r(T-t)} N(d_2) - \frac{S\sigma^2}{2r}\left[N(-d_1) - e^{-r(T-t)}\left(\frac{M}{S}\right)^{2r/\sigma^2} N(-d_3)\right]$$

Similar formulas exist for all four types (floating/fixed, call/put). The fixed strike version has two cases depending on whether the current max is above or below E. These are useful for benchmarking but limited in practice because volatility is never constant.

Why So Expensive?

A stock with 20% vol might swing 30-40% peak-to-trough over a year. The lookback captures that entire range. A vanilla at-the-money call might be worth 10% of the stock. The lookback could be 15-20% or more. Discrete sampling reduces cost by missing actual highs and lows.

Lookback features also appear in fixed-income products (interest payments depending on rate maximums) and related contracts like Russian options (perpetual lookbacks, exercise anytime for the discounted maximum to date).

The Takeaway

Lookback options let you benefit from hindsight. Key points:

• Continuous measurement: boundary condition $\partial V / \partial M = 0$ at S = M
• Discrete measurement: jump-condition algorithm with updating rule
• Similarity reduction cuts three dimensions to two for certain payoffs
• Closed-form formulas exist for constant volatility (benchmarking only)
• Expensive because they capture the full price range
• Discrete sampling preferred in practice for lower cost

The framework across Chapters 22-26 is now complete. Tools to classify any exotic, techniques to price barriers, Asians, and lookbacks. Next: contracts where the holder makes decisions during the option’s life.

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