Pricing a Maximum-to-Minimum Stock Move as Lookback Options

Summary

The document considers a European payoff equal to the difference between the highest and lowest observed stock prices over a discrete period. Under its stated multi-step binomial model, the stock is assumed to be a martingale and the risk-free bond has return factor one. The payoff can also be viewed as the largest absolute price change between any two observation times in the period.

The response identifies this range payoff as the sum of a floating-strike lookback call and a floating-strike lookback put, suggesting that established lookback-option pricing methods can be used. This decomposition gives a useful conceptual route to valuing the payoff, rather than requiring it to be treated as an entirely new contract. The response does not provide formulas, show the decomposition algebraically, or work through a binomial-tree valuation. Applying it still requires matching the lookback conventions and observation schedule to the contract’s discrete monitoring assumptions.

Key ideas

• The payoff is the maximum observed stock price minus the minimum observed stock price over the period.
• The same payoff represents the largest absolute price change between two observation times.
• The answer decomposes the range payoff into a floating-strike lookback call and a floating-strike lookback put.
• Lookback pricing methods can provide a starting point for valuation.
• The response does not specify a discrete-tree calculation or contract conventions in detail.

Tags

# Maximal increase payoff


# Maximal increase payoff












I am interested in the following problem.

We have a Multi-Step Binomial Model with discrete time $T=1,\dots,n$. We also assume that the stock $S_t$ is a martingale and there is a risk-free bond with $r=1$.

How to evaluate the price of the European option $$X_n=\max_{1\le i \le n}S_i-\min_{1\le j \le n}S_j$$ $$=\max_{1\le i < j \le n}|S_i-S_j|?$$

Here $X_n$ represents the biggest change of the stock price during the given time period. Is this option well known or maybe can it be rewritten as a combination of well known payoffs? How should one proceed in order to find it's price?

## Answer by Valometrics.com (score 1)

https://quant.stackexchange.com/a/59996

This is the sum of look back call and lookback put with floating strike. You can then price it using the formulas in wikipedia:

https://en.wikipedia.org/wiki/Lookback_option

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.