References for Deriving the Black-Scholes Lookback Option Formula

Summary

The document asks for a clear derivation of an explicit time-zero price formula for a lookback option in the Black-Scholes framework. The questioner supplies a formula and says they have studied a textbook derivation but found its calculations unsatisfactory. The focus is therefore on finding an accessible source that develops the pricing result, rather than on applying the formula to market data or evaluating a trading strategy.

The replies point to a detailed textbook derivation and to an earlier foundational paper on path-dependent options. They provide references rather than working through the mathematics, so the document does not explain the assumptions behind the displayed formula, clarify its notation, or compare derivation methods. Readers seeking the result itself should treat this as a bibliography lead and consult the cited material; the discussion alone is insufficient to verify or reproduce the derivation.

Key ideas

• The question concerns deriving a lookback option price at time zero under Black-Scholes assumptions.
• The document includes an explicit pricing expression but does not derive or validate it.
• A reply recommends a detailed textbook treatment of the derivation.
• Another reply points to foundational work on path-dependent lookback options.
• The discussion supplies references rather than explaining the formula's assumptions or steps.

Tags

# Lookback option explicit formula using Black Scholes


# Lookback option explicit formula using Black Scholes












I would like to compute the time-0-price for a lookback option using Black Scholes formula, the explicit formula is given by

$$S_0[(\frac{2r+\sigma^2}{2r})\Phi((\frac{2r+\sigma^2}{2\sigma/\sqrt{T}}))-e^{-rT}((\frac{2r-\sigma^2}{2\sigma/\sqrt{T}}))-\frac{\sigma^2}{2r}]$$

I know how to get to this price in theory, I looked into the book "Methods in Financial Modelling" by Musiela which is cited by wikipedia for a derivation but I am not very happy in the way he is doing all the calculations, do you have any reference for a nice clear derivation of this formula? It would be very helpful.

## Answer by DoubleTrouble (score 2)

https://quant.stackexchange.com/a/10034

The derivation on page 238 in Martingale Methods in Financial Modelling by Marek Musiela, Marek Rutkowski is very detailed and you won't find anything better.

Don't hesitate to ask about details here if there are something that you do not understand!

## Answer by sets (score 1)

https://quant.stackexchange.com/a/10019

The original framework for lookback options was devolped by Goldman, Sosin & Satto in Path Dependent Options: "Buy at the Low, Sell at the High".

You can probably find some additional details there.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.