No-Arbitrage Pricing of a Call When the Stock Price Is Unknown

# Help reconciling incorrect reasoning in options pricing brain teaser


# 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.

## Answer by StackG (score 6)

https://quant.stackexchange.com/a/58745

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.

## Answer by pvncad (score 0)

https://quant.stackexchange.com/a/58747

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.