Can someone please explain how they derive these and how this works? I dont understand why they start combing long and short positions on options combined with purchasing bonds?
In the Synethic Options & Put-Call Parity, what we really need to learn here is to determine the following values: c PV of X (or X/(1+RFR)) S p and given the fallowing the same values. For me, the best way to understand this is much more solving a common college algebra problem by determining a certain variables given a series of variables. Let say we don’t know the Long price (or Selling price of a Call option) but we know the values of excercise price of put (X), the underlying price (S), the RFR, the put price §, and knowing that the Put-Call Parity is --> c+X/(1+RFR) = S+p. Applying some algebra, we can derive the formula for Call Price as --> c = S+p-X/(1+RFR). The c = S+p-X/(1+RFR) can be read as “the Call price is equal to its synthetic value or by buying an underlying and put then issue a bond”. Hope this will helps.
p-c parity: c-p=s-pv(k) so c = s-pv(k)+p where c=call price, p=put price,k=strike price,s=stock price The payoff of a call expiration is max(0,s-k)…if k>s you’ll get 0 & if s>k you’ll get s-k. If you go long stock, long put, short bonds, your costs are -s-p+pv(k). Your payoffs are when k>s are: payoff from stock = s payoff from bonds = -k payoff from put = -s+k This is equal to 0, which is identical to the payoff of the call when k>s. When s>k, the payoffs are: payoff from stock= s payoff from bonds = -k payoff from put = 0 (since k>s we let it expire) The payoff is therefore s-k, which is identical to the payoff of the call when s>k. The law of one price says something like if the payoffs from two assets are the same then the price of those assets has to be the same. Since the payoff of the portfolio long stock, long put, short bonds is the same as the payoff of a call option, the portfolio is a synthetic call option & its price must be the same as the price of the call, otherwise there is an arb. opportunity.
to add to what mh7 and alix12 said, if two assets have same cash flows their price should be identical. If you consider a fudiciary call C + K*e(-rt) //call + purely discount bond position // and protective put P + S, you will see that their pay offs are identical and equal to max(S, K). Therefore, their price should be the same. C + K*e(-rt) = P + S //partity you can re-arrange terms to get formulas for C, P, S or bond (Ke(-rt)). For example, C = P + S - K*e(-rt). Now you can replicate cash flows of a call if you buy P, S and sell bond and , therefore, create a synthetic call position. If parity doesn’t hold and actual C < P + S -K*e(-rt) you can buy actual call and sell synthetic call -> arbitrage profit.