# Help reconciling incorrect reasoning in options pricing brain teaser

I'm trying to reconcile an interesting brain teaser I was recently posed and I need help understanding the flaw in the reasoning.

The problem states there is an asset which after an announcement has an 80% probability of going to 100 and a 20% probability of going to 50. What is the value of an at the money call option?

The argument I was given is that the current asset price must be 90 because 90=100*.8+50*.2 and the call option value will either be 10 or 0. Then the argument tries to state the value of the option should be 10*.8+0*.2 = 8.

I know it is incorrect to use the real world probability as options are obviously priced using the risk neutral probability measure however the problem is posed in such a way that the numbers work out. The correct option price is 8 but this only works when the current asset price is 90.

Someone tried to tell me the option value depends on the probabilities which I know is not correct. What is the ultimate flaw in this reasoning? Is the flaw arguing that the asset price must be 90?

I couldn't believe my ears when a season portfolio manager was trying to tell me option prices depend on the probability of the underlying price movements and he acted confused when I tried to explain risk neutral valuation.

Any insight is appreciated.

• I believe @StackG's answer is the correct one from a pricing perspective. Another possibility is as follows. You write that the interviewer was a portfolio manager, so maybe he was thinking what the value "to you", as an investor, should be. If your beliefs are that there is an 80% chance the stock will go up to 100 and 20% down to 50, then you can define a break-even option price based on your personal expectation. – Daneel Olivaw Oct 17 '20 at 12:38
• "The correct option price is 8": where does this come from? You need the risk-free rate value to determine the price from a binomial model (unless we are saying it is null). As @StackG has explained, $(S,C)=(90,8)$ is an admissible solution. Maybe the PM was taking the view that the current stock price must be equal to future discounted cash-flows under the real-world measure, and he is assuming there is no time-preference so you merely have to weight the future prices by their probabilities to find the current stock price. Then the "model" would be fully specified. – Daneel Olivaw Oct 17 '20 at 12:44

Assuming that the only things that can happen on the period are $$100$$ and $$50$$, and we can buy a stock and a call option with strike $$90$$, even without knowing the probabilities of these moves we can relate the price of the stock $$S$$ and the option $$C$$

If we buy $$0.2 S$$ and sell one call option $$C$$, we have a portfolio that will be worth $$10$$ in either end-state, so it must also be worth $$10$$ now (or else we have an arbitrage).

So we can be sure that $$0.2 S - C = 10$$, so if we have a value of $$S$$ provided by the market, this will uniquely fix the value of $$C$$. You can see that the PM's suggestions $$S = 90$$ and $$C = 8$$ satisfy this.

However, the actual value of $$S$$ doesn't have to be $$90$$, and this is where the market comes in. Actual investors are risk averse, so might want to pay less than $$90$$ for this stock. Of course, $$90$$ is the price in the real-world measure that leads to $$0$$ expected PnL, but there is no guarantee in a real market that investors will pay this much for it (he should get this - PMs will only buy things if they think they are going to appreciate in value!).

As posed, we have incomplete information to price both $$S$$ and $$C$$. If futures on the stock at expiry were also traded, we could use those prices enforce a price from non-arbitrage.

• "You can see that the PM's suggestions $S=\color{red}{90}$ and $C=8$ satisfy this"? – Daneel Olivaw Oct 17 '20 at 12:43
• Ahh yes of course thanks, changed now – StackG Oct 17 '20 at 13:39

Irrespective of current stock price, the price of the 90 Call option should be 8 given the probability of payoff.

Think, if the probability of price will be at 100 was 100%, then the call price will be 10, again irrespective where the stock price is.