Page 114 - DMGT513_DERIVATIVES_AND_RISK_MANAGEMENT

Unit 8: Option Pricing

The value of Portfolio A on expiration date is shown in Table 8.4 and that of Portfolio B is shown Notes
in Table 8.5.
Table 8.4: Portfolio A

Value on the Expiration Date
Action Today S <=k S >k
*
*
Buy one call 0 S -k
*
Total 0 S -k
*

Table 8.5: Portfolio B

Value on the Expiration Date

Action Today S <=k S >k
*
*
Buy one share S * S *
Buy one put k- S * 0
Borrow the PV of k and d -k -k
Total 0 S -k
*

Since the portfolios always have the same final value, they must have the same current value.
Again, this is the rule of no arbitrage. Note that this arrangement of the portfolios is slightly
different from the case with no dividends. In the no dividends case, we had the stock and a put
in portfolio A. In the dividends case, we have just the call in portfolio A. But clearly, we could
have constructed the no dividends case with just a call in the portfolio A – it would have no
impact on the result. Further note, that as the result of borrowing the present value of both the
dividend and the exercise price, we only payoff the exercise price. The reason for this is that if
you get the dividend payment before expiration, then you use it to reduce your total debt. In
fact, you use it to exactly pay off that part of the debt that is related to the dividend part of the
borrowing.

We can express the put-call parity relation as:
c = S + p – PV(k) – PV(d)
where PV(k) is the present value of the exercise price and PV(d) is the present value of the
dividend.
Before discussing the details of various option pricing models, we must understand the upper
and lower limits of both call and put options. These are discussed below.

This put-call parity can be further explained by the help of suitable diagrams by comparing the
expiration value of two portfolios i.e., 1) The call option and an amount of cash equal to the
present value of the strike price ; and 2) The put option and the underlying assets.

Source: A.N. Sridhar, pg. 144- diagrams on put-call parity.

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