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. Commented Oct 17, 2020 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. Commented Oct 17, 2020 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"? Commented Oct 17, 2020 at 12:43
• Ahh yes of course thanks, changed now Commented Oct 17, 2020 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.