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Chapter 14: Trading strategies

DenisAmbrosov

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Quote: 'put-call parity shows that the price exposure from writing a covered call is the same as the exposure from writing a naked put.'. Put-call parity is c+Ke -rt = p+S0. We know that S0 - c is a covered call and -S0 + c is naked put but what about Ke-rt. I know that we don't have it in the covered call equation. But we deduce the naked put from put-call parity but put-call parity involves Ke-rt. Where is the term Ke-rt then?
 

David Harper CFA FRM

David Harper CFA FRM
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Hi @DenisAmbrosov (btw, is your quoted sentence from GARP's chapter or our note, out of curiosity? ... because it's a sweet comparison, to note that payoff shape of write naked put ~= write covered call). Put-call parity is awesome. I like to start with your c+K*exp(-rT) = p+S0, which to me is "call + cash = protective put." As you say, let's solve for covered call, so we've got +S0 -c = +K*exp(-rT) -p. The plus (+) is long and the minus (-) is short. On the left, we have long stock and short call; on the right we have long (aka, invest) cash and short put. The +PV(K) is just to buy the discounted strike price in cash; i.e., to invest at the risk-free rate. Negative -K*exp(-rT) would be to borrow cash (short cash).

Numbers help me. Arbitrarily, I'll use S0 = K = $20.00, σ = 42%, Rf = 2.0%, T = 1.0 year. BSM returns: PV(cash) = $19.60, p = $3.10, c = $3.50. Using S0 -c = +K*exp(-rT) -p --> 20.00 -3.50 = 19.60 -3.10 = 16.50. How to interpret? It will cost us net $16.50 either way. On the left (covered call), to receive 3.50 and buy the $20.00 stock; on the right (naked put), to receive 3.10 and invest $19.60 in cash. Hope that's helpful,
 
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DenisAmbrosov

New Member
Subscriber
Hi @DenisAmbrosov (btw, is your quoted sentence from GARP's chapter or our note, out of curiosity? ... because it's a sweet comparison, to note that payoff shape of write naked put ~= write covered call). Put-call parity is awesome. I like to start with your c+K*exp(-rT) = p+S0, which to me is "call + cash = protective put." As you say, let's solve for covered call, so we've got +S0 -c = +K*exp(-rT) -p. The plus (+) is long and the minus (-) is short. On the left, we have long stock and short call; on the right we have long (aka, invest) cash and short put. The +PV(K) is just to buy the discounted strike price in cash; i.e., to invest at the risk-free rate. Negative -K*exp(-rT) would be to borrow cash (short cash).

Numbers help me. Arbitrarily, I'll use S0 = K = $20.00, σ = 42%, Rf = 2.0%, T = 1.0 year. BSM returns: PV(cash) = $19.60, p = $3.10, c = $3.50. Using S0 -c = +K*exp(-rT) -p --> 20.00 -3.50 = 19.60 -3.10 = 16.50. How to interpret? It will cost us net $16.50 either way. On the left (covered call), to receive 3.50 and buy the $20.00 stock; on the right (naked put), to receive 3.10 and invest $19.60 in cash. Hope that's helpful,
The quoted sentence is from your note. I just was confused with K*exp(-rt). So in the quote above, we can say, using put-call parity +S0 -c = +K*exp(-rT) -p. So, +S0-c = covered call = K*exp(-rt) - p = naked put?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
@DenisAmbrosov In this context, sure we can. A naked put is uncovered. In the p-c parity context, in addition to the naked put, we are just investing at the risk-free rate, which does not alter the payoff function (curve) yet satisfies the equality.
 

DenisAmbrosov

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@DenisAmbrosov In this context, sure we can. A naked put is uncovered. In the p-c parity context, in addition to the naked put, we are just investing at the risk-free rate, which does not alter the payoff function (curve) yet satisfies the equality.
Thank you very much. The key sentence for me: 'In the p-c parity context, in addition to the naked put, we are just investing at the risk-free rate, which does not alter the payoff function (curve) yet satisfies the equality.'
 
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