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21-270: Things to Keep in Mind #1

An early common mistake made by people learning mathematical finance is to think that the value of an option can always be found by using it's pricing formula at expiration.

For example, consider a European call option, currently selling for $8, struck at $50 with expiration T = 1 on a stock with current price S_{0} = $60. As you know, if the stock price has fallen to S_{1} = $55 in one year, we can calculate the terminal value of the call as:

C_{1} = C_{1}(S_{1}) = (S_{1 }- K)^{+} = ($55 - $50)^{+} = $5When asked for the time 0 value of the call, C_{0}, many students will argue that:C_{0} = C_{0}(S_{0}) = (S_{0 }- K)^{+} = ($60 - $50)^{+} = $10This is incorrect. By the value of an option, we simply mean its price; in this example, C_{0} = $8. For our purposes price and value are synonymous; we leave any distinction in the meaning of these terms to the Economists.Valuing (or pricing) the call between time 0 and time T is a substantially more difficult problem and is one of the aims of modern mathematical finance. These valuations are covered in 21-370 and 21-420.

## 2 comments:

Isn't the value of a call defined as S-K? Because by purchasing a call you're betting that the stock will rise. You can purchase at the strike price and then immediately sell at the higher price.

Thanks Anonymous. I originally wrote the example for a put and failed to change my numbers. Hopefully it's correct now.

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