Title LM09 Option Replication Using Put–Call Parity IFT Notes.pdf ✅

LM09 Option Replication Using Put–Call Parity 2025 Level I Notes © IFT. All rights reserved 3 Interpretation:  As you can see, the downside risk is limited to the premium paid for the put. If the underlying price falls below X, it can be still be sold at X by exercising the put option.  If the underlying rises above X, then it can be sold at the market price and the put option expires worthless. Fiduciary Call: A fiduciary call is a long call plus a risk-free bond. The cost of this strategy is: c0+ X (1+r)T The diagram below shows the payoff for a fiduciary call: Therefore, according to put-call parity: c0+ X (1+r)T = p0 + S0 In other words, under put–call parity, at t = 0 the price of the long call plus the risk-free asset must equal the price of the long underlying asset plus the long put. The table below shows how the fiduciary call is equal to the protective put under two possible scenarios: when the stock is above the exercise price (call is in the money) and the stock is below the exercise price (put is in the money). Outcome at time T when:  Put expires in the money (ST < X) Call expires in the money (ST > = X) Protective put Asset ST ST Long puts X-ST 0 Total X ST Fiduciary call Long call 0 ST – X Risk-free bond X X Total X ST
LM09 Option Replication Using Put–Call Parity 2025 Level I Notes © IFT. All rights reserved 4 Fiduciary Call Protective Put Constituents Long call + risk-free bond Long put + stock Equation c0+ X (1+r)T p0+S0 Payoff at T if call expires in the money (ST > = X) ST ST Payoff at T if put expires in the money (ST < X) X X If, at time 0, the fiduciary call is not priced the same as the protective put, then there is an arbitrage opportunity. Example: Put-Call Parity (This is based on Example 1 from the curriculum.) A stock currently trades at INR295 per share. An investor is considering the purchase of a six-month put on this stock at an exercise price of INR265. A six-month call option with the same exercise price trades in the market at INR59. What should the investor expect to pay for the put if the relevant risk-free rate is 4%? Solution: According to put call parity: c0+ X (1+r)T = p0 + S0 59 + 265 (1+0.04)0.5 = p0 + 295 59 + 259.85 = p0 + 295 p0 = INR23.85 The investor should expect to pay a six-month put option premium of p0 = INR23.85 3. Option Strategies Based on Put–Call Parity The put-call parity relationship can be rearranged in the following ways: Synthetic call: c0 = p0 + S0 – X (1+r)T Synthetic bond: X (1+r)T = p0 + S0 – c0 Synthetic stock: S0 = c0+ X (1+r)T – p0 Synthetic put: p0 = c0+ X (1+r)T - S0 These equations allow us to replicate an instrument by using the other three instruments.