What?! Option value doesn't depend on the likelihood of the underlying going up or down?

(post from old site)

WHAT?!

yes, I'm saying, @t0, these options are worth the same:

• Call option with strike price 100, expiring @t1, on underlying A, which is at $100 now
• Call option with strike price 100, expiring @t1, on underlying B, which is at $100 now

 even though 

• underlying A has 99% chance of closing at $110 and 1% at $90 on expiry
• underlying B has 1% chance of closing at $110 and 99% at $90 on expiry

Seriously, wouldn't you wanna put your $ on the call option (A) that has a 99% chance expiring in the $, instead of the one (B) only has 1% chance of expiring in the money?

HOW?!  How are those options worth the same now?!!

2 words:  replicating portfolio

Let's have a portfolio such that it'll replicate the cash flow of the option by investing in the underlying and the cash account, assuming risk free rate = 1%, such that

@t1, the portfolio is worth

• 110x + 1.01y    when underlying =110
• 90x + 1.01y    when underlying = 90

So, 

• 110x + 1.01y = 10   (the option expiring in the $)
• 90x + 1.01y = 0   (the option expiring out of $)

Solving the equations,

• x =0.5
• y = -44.55

meaning

• long half unit of the underlying, and
• short (ie borrow) 44.55 in the cash account at risk free rate

With this portfolio, you'll end up with 

• $10 if the underlying closes at $110 when option expires
• $0 if the underlying closes at $90 when option expires

exactly replicating the option CF.

So, what's this portfolio worth @t0?

0.5*100-44.55(1) = $5.45

2 things to note:

1. both options, A & B, are worth $5.45, disregarding the different probability of up move and down move.
2. if either option is priced higher / lower, you can arbitrage with the replicating portfolio.

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with all that said, think about it..

underlying A is not really worth 100 @t0, given the prob & prices @t1...

underlying B is not really worth 100 @t0, given the prob & prices @t1...