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Risk-Neutral Valuation of a First-Drop Claim in a Binomial Model

Article Quant Q&A · Author: Fatima
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Summary

The document considers a claim that pays one unit when a stock first falls in a binomial model. It assumes an infinite sequence of periods and the no-arbitrage condition that the down factor is below the risk-free growth factor, which is below the up factor. Under that condition, it gives the risk-neutral probability of a down move.

The answer values the claim by summing the discounted risk-neutral payoff across possible first-drop periods up to a finite horizon of N periods. Each term accounts for consecutive up moves before the first down move. The excerpt provides the valuation expression but does not derive it, clarify the convention used for the risk-free factor in the expression, or take a limit to produce an infinite-horizon price. Readers should verify those conventions when applying the formula.

Key ideas

  • The claim pays one unit at the first downward stock move.
  • The no-arbitrage ordering places the risk-free growth factor between the down and up factors.
  • Risk-neutral valuation weights each possible first-drop time by its probability and discounts the payoff.
  • The stated sum is for a finite horizon, and the excerpt does not derive an infinite-horizon limit.

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Derivatives pricingStatistics
Full text
# Infinite Binomial Pricing no arbitrage


# Infinite Binomial Pricing no arbitrage












How to price a contract that pays only 1 at the first stock price drop? The stock follows an infinite binomial with no arbitrage $d<R<u$ condition.

So the probability of the price going down is $p = (u-R)/(u-d)$

## Answer by JohnDoe (score 1, accepted)

https://quant.stackexchange.com/a/45230

Risk neutral valuation tells you to discount the expected payoff in the risk free world. Therefore, it should be $\sum_{i=0}^{N-1}R^{i+1}(1-p)^i p$, where $N$ is the number of periods.

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.