Quantitative Derivatives Suite

Lookback Option Pricing & Extrema Simulation Lab

Goldman-Sosin-Gatto (1979) Floating Strike & Conze-Viswanathan (1991) Fixed Strike Closed Forms with Discrete Monitoring Adjustments

Institutional Presets:

Contract & Market Parameters

$100.00
$100.00
Used in Conze-Viswanathan fixed strike payoff.
$100.00
Historical extremum recorded since inception. Default is $S$ for new issues.
0.50 yrs (6 mo)
20.0%
5.0%
0.0%
Adjusts continuous extrema for daily 252-step discrete sampling bias.
Lookback Value
$11.95
Goldman-Sosin-Gatto (1979)
Vanilla Black-Scholes
$6.89
Standard European Option
Lookback Premium
+$5.06
+73.5% over vanilla
Expected Extremum
$88.24
E[S_min] at expiration

Institutional Pricing & Extrema Decomposition

Floating Strike Call
Component Lookback Model Vanilla European Intrinsic Payoff Variance / Markup
Option Theoretical Price $11.95 $6.89 $0.00 +73.5%
Cost of Carry ($b = r - q$) +5.00% Discrete Monitoring Bias -0.45 ($11.50)
Extremum Watermark State S_min = $100.00 (At the Money) Time Value Share 100.0%
Valuation curve vs. underlying spot price comparing Lookback Option, Black-Scholes Vanilla, and Intrinsic Payoff.

Lookback Option Sensitivities (Greeks)

Delta ($\Delta$) +0.784
$\partial V / \partial S$: Sensitivity to $1 spot change.
Gamma ($\Gamma$) +0.016
$\partial^2 V / \partial S^2$: Curvature / rate of delta change.
Vega ($\mathcal{V}$) +0.428
$\partial V / \partial \sigma$: Dollar change per 1% vol shift.
Theta ($\Theta$) -0.041
$-\partial V / \partial T$: Calendar day time-decay drag.
Rho ($\rho$) +0.112
$\partial V / \partial r$: Dollar change per 100 bps rate shift.
Extremum Delta ($\Delta_{\text{ext}}$) -0.485
$\partial V / \partial S_{\min}$: Sensitivity to historical watermark reset.

Spot ($S$) vs. Volatility ($\sigma$)

Maturity ($T$) vs. Extremum ($S_{\text{ext}}$)

Quantitative Mechanics of Lookback Options

1. Floating Strike: "Buy at Low, Sell at High"

Formulated by M.B. Goldman, H.B. Sosin, and M.A. Gatto (1979), floating strike lookbacks allow the investor to acquire the asset at the lowest observed price during the option's life (Call: payoff = $S_T - S_{\min}$) or sell at the highest observed price (Put: payoff = $S_{\max} - S_T$).

Because the terminal spot price $S_T$ is mathematically guaranteed to be greater than or equal to the minimum price $S_{\min}$, a floating-strike call never expires worthless unless the underlying asset remains strictly flat ($S_T = S_{\min}$). It acts as a perfect market timing vehicle.

2. Fixed Strike: Conze & Viswanathan (1991) Model

Derived by Antoine Conze and R. Viswanathan (1991), fixed strike lookbacks maintain a pre-specified strike $K$ while replacing terminal spot with the running extremum. The call payoff is $\max(S_{\max} - K, 0)$, and the put payoff is $\max(K - S_{\min}, 0)$.

Fixed-strike lookbacks provide complete downside protection while capturing the single best peak price during the holding period. Corporate executives and M&A acquirers frequently structure structured notes around fixed-strike lookbacks to guarantee top-of-cycle exit multiples.

3. The Reflection Principle & Extrema Drift

Under Geometric Brownian Motion $dS_t = b S_t dt + \sigma S_t dW_t$, the joint probability distribution of the terminal price $S_T$ and the running maximum $M_T = \max_{0 \le t \le T} S_t$ is derived using the Reflection Principle of Brownian motion.

The analytical formulas introduce power terms $(S/H)^{-\lambda}$ where $\lambda = \frac{2b}{\sigma^2}$. When carrying cost $b = r - q \ne 0$, the drift rotates the path probability density, pulling the expected extremum away from the driftless martingale path.

4. Broadie-Glasserman-Kou Discrete Monitoring Bias

Continuous analytical models assume the asset price is monitored every millisecond. Real-world OTC and exchange contracts fix extrema once daily at the official market close ($N = 252$ trading days).

Discrete monitoring under-samples peaks and over-samples troughs. By Broadie, Glasserman & Kou (1999), the continuous formula accurately prices discrete contracts when the extremum boundary $H$ is shifted by $H_{\text{eff}} = H e^{\pm \beta_1 \sigma \sqrt{\Delta t}}$, where $\beta_1 = -\frac{\zeta(1/2)}{\sqrt{2\pi}} \approx 0.5826$.

Lookback Option Mastery Quiz

Score: 0 / 5

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Frequently Asked Questions

A lookback option is an exotic, path-dependent derivative whose payoff depends on the minimum or maximum price achieved by the underlying asset over the entire lifespan of the contract, rather than merely its price at expiration.

A floating strike lookback option sets the strike at the extreme asset price observed during the contract: a call pays S(T) - S_min (buying at the absolute low), while a put pays S_max - S(T) (selling at the absolute high). A fixed strike lookback option has a pre-determined strike K: a call pays max(S_max - K, 0), while a put pays max(K - S_min, 0).

Lookback options eliminate timing regret by guaranteeing the holder optimal execution relative to historical extremes. A floating strike call is guaranteed to finish in-the-money or at-the-money unless the asset remains strictly flat. Because the maximum is always greater than or equal to terminal spot, the option captures the entire peak-to-trough range, requiring substantially larger option premiums (often 2x to 3x standard vanillas).

Classical analytical formulas assume continuous observation of the asset price. In practice, lookbacks are sampled at discrete intervals (e.g., daily market close). Discrete sampling systematically under-samples the continuous maximum and over-samples the minimum. The Broadie-Glasserman-Kou (1999) correction applies an analytical factor exp(±0.5826 * sigma * sqrt(dt)) to correct for discrete observation bias.

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