Pricing a One-Period Call with Risk-Neutral Probabilities

Summary

The document asks how to price a one-year call when a stock can move to either of two values, given a risk-free rate and equal real-world probabilities. It uses a one-step binomial model: calculate the call payoff in each state, form a replicating hedge, or equivalently find the risk-neutral probability that makes the stock’s expected growth match the risk-free rate, then discount the expected payoff.

The response explains that the probabilities in the question do not determine the no-arbitrage price; risk-neutral probabilities do. The calculation shown in the attempt produces a price of 16⅔, although its line for the down-state payoff mistakenly writes 30 where the payoff is zero. The later hedge calculation uses zero and is consistent with the stated price. The material is limited to a single-period example and does not discuss transaction costs, early exercise, or assumptions behind the binomial model.

Key ideas

• A call’s payoff is zero when the terminal stock price is below its exercise price.
• The no-arbitrage price can be found by discounting the payoff’s risk-neutral expectation.
• Real-world up and down probabilities do not set the option price in this binomial model.
• A replicating stock-and-bond position provides an equivalent pricing method.

Tags

# Pricing call option


# Pricing call option












Question: The price of a stock is 100. With equal probabilities, it either goes up to 130 or down to 70. What is the price of a 1 year call option with exercise price 100. Risk free rate is 5%.

Attempt: I use black scholes.



- $Cu=max(0,130-100)=30. Cd=max(0,70-100)=30$

Now, $$HedgeRatio = (30-0)/(130-70)=1/2$$ $$B=(Cd-Sd \cdot HedgeRatio)/(1+rf)=(0-70/2)/(1.05)=-(33+1/3)$$ So the price of the call option is $$C_0=S_0*Hedgeratio+B= 100/2 -(33+1/3)=16+2/3$$ but it is wrong.

## Answer by AFK (score 3)

https://quant.stackexchange.com/a/17791

I don't know the BS formula you are trying to use.

The price is the expected value of the discounted payoff under the risk neutral probability measure (I.e. Under which S is a martingale)

So the you need to compute the risk neutral probabilities for S to go up or down. The probabilities given in the problem have no impact. They are just there to trick the candidate.

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