Pricing a One-Step Call with Risk-Neutral Valuation

Summary

The document presents a one-period binomial example for valuing a call option. The underlying security is worth 100 initially and will be worth either 110 or 95 at expiry; the call has a strike of 105. Assuming cash retains its value over the period, the option pays 5 in the up state and zero in the down state.

The answer derives the risk-neutral probability by setting the expected underlying price change to zero, giving an up-state probability of one third. Weighting the two possible option payoffs by those probabilities gives a value of 5/3. The response identifies replication as another route to the same valuation. This is a deliberately simplified setup: it assumes a single step and effectively zero interest over the horizon, so it does not cover discounting, dividends, or more complex price dynamics.

Key ideas

• A one-step binomial model represents the underlying price with an up state and a down state.
• The call pays only in the up state because the down-state price is below the strike.
• The risk-neutral up probability is obtained by making the expected underlying price change zero under the stated cash assumption.
• The option value is the probability-weighted expected payoff, discounted under the assumed zero-interest setup.
• Replication and risk-neutral valuation are alternative approaches to this simple example.

Tags

# Determining price of Option interview question


# Determining price of Option interview question












I'm not too sure what the answer is to this. You have a call option on a security worth 100 now that will either be worth 110 or 95 dollars at a future date. The strike of this option is 105. What is an estimated value of this call option?

Thanks

## Answer by Mark Joshi (score 9, accepted)

https://quant.stackexchange.com/a/35998

I would think you are supposed to assume that cash is worth 1 at all times. There is miniscule interest across a day in any case. They are testing if you can do a one-step binomial tree.

You can then either price by replication or risk-neutral valuation. The RN probability of an up-move is $q$ such that $$ 10 q -5(1-q) =0. $$ So $q=1/3$ so the price is $$ \frac{1}{3} \times (110-105) + \frac{2}{3} \times 0 = \frac{5}{3}. $$

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.