Risk-Neutral Pricing of a One-Period At-the-Money Call
Summary
This example explains how to price a one-period call option when a stock has two possible future values. The call pays nothing in the down state and has a positive payoff in the up state. Its price is found by weighting those payoffs with risk-neutral probabilities, which can be inferred from the requirement that the stock’s discounted expected future value equals its current price.
The document checks the stated probabilities against that no-arbitrage condition, assuming zero interest rates, and obtains the option value from the up-state payoff. The key distinction is between physical probabilities and risk-neutral probabilities: the latter are used for pricing under the model. The example is deliberately simple and does not address transaction costs, nonzero rates, early exercise, or more complex market dynamics. It also includes interview advice that is personal commentary rather than part of the pricing method.
Key ideas
- A call’s payoff is determined separately in each possible future state.
- Risk-neutral probabilities make the discounted expected stock price match its current price.
- The option price is the risk-neutral expected payoff, discounted to today.
- The calculation assumes a one-period model with zero interest rates.
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Full text
# Quant Interview - Options pricing # Quant Interview - Options pricing I am fairly new to all this, merely read the first few chapters of "The Concepts and Practice of Mathematical Finance". I recently had a job interview and was asked a simple question about options pricing: > Given that a stock today is worth 100 and tomorrow has a 20% chance of being worth 70, and would be worth 107.5 otherwise, what is the fair price of an at-the-money call option? I answered, using the risk-neutral approach, that the option was worth 3.75, however, the interviewer told me I am wrong, as I need to weigh the probabilities, i.e the option should be worth 0.8* 7.5 =6. This is precisely what Joshi argues is wrong due to the possibility of hedging all the risk away. Am I missing something? ## Answer by Jan Stuller (score 3, accepted) https://quant.stackexchange.com/a/69435 Call option gives the right to buy the stock. ATM call option struck at 100 would be worth zero in the lower state and would be worth 7.5 units of money in the upper state (you can buy at 100, when the stock is worth 107.5). As @emot points out, the risk-neutral probability is given at 80% for the upper state, so the option is worth 0.8 * 7.5 = 6. You can check that the probabilities given are risk-neutral by focusing on the stock alone: the stock value today has to equal the risk-neutral expected stock price in the future states discounted to today. You can use high-school maths to compute the risk-neutral probabilities yourself using this technique; denoting risk-neutral probability of an up-move with $p$ and assuming rates are zero: $$p*107.5 + (1-p)70=100 \rightarrow p = 0.8$$ Which then gives the option price of 6 as discussed above. PS: if you argue with the interviewer, it'll not only screw up that one interview, but the feedback might also disqualify you from future opportunities at the firm. In all honesty, the example you gave is so basic that it's a waste of your own time (as well as the firm's time) for you to have applied in the first place. You should first spend time on the basics before applying.
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