Risk Latte - Option Valuation is Counter-intuitive - Why is that?

Option Valuation is Counter-intuitive - Why is that?

Rahul Bhattacharya
October 13, 2006

Here is a Wall Street interview question, and a profound one as well. Suppose you have $1 million (or $100 doesn't matter) and you want to invest in call options - i.e. buy call options - on either of the two stocks A or B. Stock A, which is currently at $100, is increasing by 20% annualized rate every quarter whereas, Stock B, which is currently at $50, is decreasing by 20% annualized rate every quarter.

Both stocks - A and B - have exactly the same volatility (remember it is a theoretical question) and same expiry (maturity), say, 1 year. They are part of the same economy, so the risk free rate is the same for both and both don't pay any dividends. Further, since you will only buy at the money (ATM) call option therefore the strike price of the option on A and B are also the same, i.e. the spot price of the underlying.

Which call option will you buy - a call on Stock A or a call on Stock B?

If you use a Black-Scholes model to value the option then both the call option on Stock A and the call option on Stock B will have the same value (and the same price). Therefore, in a Black-Scholes world you should be indifferent between these two options. Is that logic correct?

If you knew no Black-Scholes math or option pricing theory and only had the knowledge that a call option pays off if the stock price goes up from the current level (at which you bought the option) then of course you would buy the call option on Stock A, because the stock price of A is going up (historically speaking) and the probability (howsoever you estimate that measure in your mind) of it going up is higher than Stock B. Stock B is going down by 20% and the likelihood of it going down is higher than A - or so you would reason - so the call option on Stock B is likely to be worthless in a year's time. Hence your decision to buy the call option on Stock A.

But you know the Black-Scholes math and option pricing theory and you know all about probabilities and therefore you conclude that you don't care which call option you buy. Because you know from all your knowledge about financial theory of options that option prices and values don't depend on the direction of the stock price. And notion itself is counter-intuitive, isn't it? Ask a broker, ask your dad, ask the anchor or expert commentators on financial TV and they will mostly likely say, why of course, we will buy a call option on stock A because the stock has been going up and its chances of going up are certainly more than stock B. They might even tell you what a silly question this is!

We asked a sample of 10 traders this question and seven out of ten replied, literally without even batting an eyelid, that they would buy the call option on Stock A. And then we asked "why"? What about Black-Scholes? What about risk neutral probabilities?

We quote the response of two of them here:

Trader 1: "the fact that stock A is going up by 20% and stock B is going down by 20% means that the process is not risk neutral. And if risk neutrality is violated then Black-Scholes doesn't hold and the probabilities of being in the money are not risk neutral probabilities any more."

Trader 2: "if I delta hedge on stock A (after buying a call on stock A) then there is a very good chance that I'll retire in a year's time."

This is at the heart of option pricing. The notion of risk neutral pricing and the fact that in a Black-Scholes world all assets diffuse at a risk free rate. But in the real world the drift is very real - as opposed to risk neutral - and the whole process of diffusion becomes a matter of debate.

Try asking this question to your trader friends and see what response you get.

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