Put-call Parity Theory.

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Background

Put-call parity is an important principle in options pricing first identified by Hans Stoll in his paper, The Relation Between Put and Call Prices, in 1969.

Put-call Parity Theory is based on theory of one price. As per this theory of one price, if two assets are expected to have same value on a future date, they should have same cost today.

Example

Suppose we have two portfolios.

(1) Buy 1 BTC for $9,000, also buy European put ( Expiry date - 1 year ) of 1 BTC at a strike price of $10,500.

(2) Buy 1 European call ( Expiry date - 1 year ) at a strike price of $10,500 and 1 Bond carrying Risk free return. ( The amount to be invested in Bond should be such that on the date the Call expires, the bond value including interest should be equal to the strike price ).

On maturity

Case I :- Spot price of BTC is $ 10,000. ( Less than strike price ).

Value of Portfolio (1)

= Value of put on maturity + Spot price

= (10,500-10,000) + 10,000

= $10,500

Value of Portfolio (2)

= Value of call + Maturity amount of Bond

= 0 + 10,500

= $10,500

Case II :- Spot price of BTC is $11,000 ( Higher than strike price ).

Value of Portfolio (1)

= Value of put on maturity + Spot price

= 0 + 11,000

= $11,000

Value of Portfolio (2)

= Value of call + Maturity amount of Bond

= ( 11,000-10,500 ) + 10,500

= $11,000

Case III :- Spot price of BTC is $10,500 ( Equal to the strike price ).

Value of Portfolio (1)

= Value of put on maturity + Spot price

= 0 + 10,500

= $10,500

Value of Portfolio (2)

= Value of call + Maturity amount of Bond

= 0 + 10,500

= $10,500

Both portfolios will have same value on maturity whether the spot price is less than or equal to or greater than the strike price.

Hence, both the portfolios should have same cost today (Put-call Parity Theory).

Cost of portfolio (1)

= Spot price + Put premium

Cost of portfolio (2)

= Call premium + Value of Bond

= Call premium + Present Value of Bond

Hence, combining (1) and (2)

Conclusion

Spot price + put premium = Call premium + Present Value of Bond

Thank you for reading.

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