I have seen these two terms for put-call parity:

  1. S + P = C +Ke^(-rT)
  2. S + P = C + K/(1+r)^T

  1. Which one is the correct formula?
  2. When to use each of them?
  3. What is the diffrence?

Well, the first one is based on the "Pert" formula for continuously-compounded present value, while the second one is the periodically-compounded variant.

Typically, the continuously-compounded models represent the ideal; as the compounding period of time-valued money shrinks towards zero, and the discount rate (or interest rate if positive) stays constant over the time period examined, the periodic equation's results approach that of the continuously-compounded equation. Those two assumptions (a constant rate and continuous balance adjustment from interest) that allow simplification to the continuous form are usually incorrect in real-world finance; virtually all financial institutions accrue interest monthly, for a variety of reasons including simpler bookkeeping and less money paid or owed in interest. They also, unless prohibited by contract, accrue this interest based on a rate that can change daily or even more granularly based on what financial markets are doing. Most often, the calculation is periodic based on the "average daily balance" and an agreed rate that, if variable, is based on the "average daily rate" over the previous observed period.

So, you should use the first form for fast calculation of a rough value based on estimated variables. You should use the second form when you have accurate periodic information on the variables involved. Stated alternately, use the first form to predict the future, use the second form in retrospect to the past.

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