# fair price of call option in this problem

The price of an asset can move to only two values – USD 102 and USD 98 – over the next month. The probability of a price rise is 99%, while the probability of a fall is 1%. The annual simple risk-free interest rate is 12%. What is the value of a one-month call option, on one unit of the asset, struck at USD 100?

According to me fair price is 0.99*(102-(100/1.01)) which is around 2.7. But the correct answer is from one of the options 1.96 or 2.2. What am I doing wrong ?

• Is this homework? – RonJohn Oct 8 '17 at 11:24
• @RonJohn no I took some online course the person asked this question in lecture. Did not tell the correct answer. – sashas Oct 8 '17 at 11:34

The (expected) future value of the option is

FV = 0.99*(102-100) + 0.01*(0) = 1.98


as you have a 99% chance of making \$2 and a 1% of getting no money.

Now with a positive interest rate you must discount this future value. So, the present value of this option must be less than the future value of 1.98. Given the two options 1.96 must be the answer.

Please double check how simple interest rates work, but if I remember correctly you can just divide so

PV = FV/( 1 + (0.12/12) ) = 1.98/(1.01) ~ 1.96

• the interest correction should not be applied though on the whole price ? like in black scholes only strike price is corrected to interest rate, If I follow the same logic here I get the above I said. – sashas Oct 10 '17 at 6:47
• Anytime you change from one time frame to another time frame you must discount the full value (whole price). Above I do the value calculation in the future time (at expiry you get either 2 or 0 'future' dollars which is an expected value of 1.98 'future' dollars) and then I discount that to a present value. – rhaskett Oct 11 '17 at 16:21
• In Black-Scholes, the spot price is a present price so does not need to be discounted, while in the above calculation the value of the asset at expiry (102) and the strike price (100) are both future values and will both eventually need to be discounted. – rhaskett Oct 11 '17 at 16:41
• You can actually derive Black-Scholes like I do above by performing the calculation in the future time and then discounting that future value you get this alternative formulation. en.wikipedia.org/wiki/Black–Scholes_model#Alternative_formulation – rhaskett Oct 11 '17 at 16:43