Pricing a Square-Root Payoff in a One-Step Binomial Model

Summary

The document asks how to price a European-style exotic payoff equal to the square root of the stock price at the end of a one-step binomial period. Its answer uses the risk-neutral probability implied by the current stock price and the two possible future prices. With a zero risk-free rate, that probability is the fraction that makes the current price equal to the probability-weighted average of the up and down outcomes. The exotic payoff is then valued as the probability-weighted average of its two possible terminal values.

The method illustrates risk-neutral valuation for a simple binomial model, but the response does not use the quoted call and put prices, nor explain how those prices might constrain arbitrage-free values. It also assumes a zero risk-free rate and gives no explicit discounting step. Applying the approach beyond those assumptions requires the risk-free rate and a consistent one-step model; the prompt's option prices alone are not analyzed.

Key ideas

• In a one-step binomial model, risk-neutral probabilities make the discounted expected stock price equal its current value.
• With a zero risk-free rate, the up-state probability follows from the current price and the two possible terminal prices.
• A nonlinear payoff can be priced by taking its risk-neutral expected terminal payoff.
• A nonzero risk-free rate requires discounting the expected payoff.
• The response does not use the call and put prices supplied in the question.

Tags

# Arbitrage strategy using binomial tree


# Arbitrage strategy using binomial tree












Suppose that we have a one step binomial tree model for a company. Lets say that the time per step is T, and that price of the stock can go up to $p_1$ or go down to $p_2$. Suppose a T-month European call on the company, with strike price $X$ trades at price $Y$ while a T-month European put with strike price $W$ has a price of $Z$. How can we find the arbitrage-free price of an exotic option with payoff equal to the square root of the stock price at T=1/4?

## Answer by stackoverblown (score 1)

https://quant.stackexchange.com/a/54834

Let's $p_0$ be the initial stock price, $q$ be the risk free probability of $p_0$ ends up at $p_1$, assuming risk free rate is 0 then $$ p_0 = q p_1 + (1-q) p_2$$ so $$ q = \frac{(p_0 - p_2)}{(p_1-p_2)}$$.

So the exotic price after one step is $$ q \sqrt{p_1} + (1-q) \sqrt{p_2} $$.

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