CFA Level I · CFA Level I Exam

Option Replication Using Put-Call Parity: formula sheet

Full chapter guide

Key formulas

Put-call parity: c + X/(1+r)^T = p + S0; European options, same strike X and expiry T, underlying pays no income. r is the annual risk-free rate, T in years.
Call price: c = p + S0 − X/(1+r)^T; Long call = long put + long underlying + borrowing the present value of X.
Put price: p = c − S0 + X/(1+r)^T; Long put = long call + lend the present value of X + short underlying.
Underlying price: S0 = c − p + X/(1+r)^T; Long underlying = long call + long bond + short put.
Risk-free bond: X/(1+r)^T = p + S0 − c; Long bond = long put + long underlying + short call.
Continuous compounding version: c + X·e^(−rT) = p + S0; Use only if the question gives a continuously compounded rate.
Fiduciary call payoff at expiry: max(S_T, X) = max(S_T − X, 0) + X; Call payoff plus the bond's face value X.
Protective put payoff at expiry: max(S_T, X) = S_T + max(X − S_T, 0); Asset value plus put payoff. Floor is X.
Put-call parity (European options, no income on asset): c + X ÷ (1 + r)^T = S₀ + p; Fiduciary call cost on the left, protective put cost on the right. Same X and T for both options.
Present value of the bond: PV(X) = X ÷ (1 + r)^T; Use the risk-free rate r with the same compounding as T. T is in years.
Rearranged for call or put: c = S₀ + p − X ÷ (1 + r)^T; p = c + X ÷ (1 + r)^T − S₀; Use to find a missing option price.
Put-call parity (European): c + X ÷ (1 + r)^T = S + p; Same underlying, strike and expiry. With continuous compounding, PV(X) = X × e^(−rT). Assumes no dividends or carry on the underlying.
Synthetic long call: c = p + S − X ÷ (1 + r)^T; Buy put, buy stock, borrow PV(X).
Synthetic long put: p = c − S + X ÷ (1 + r)^T; Buy call, short stock, lend PV(X) by buying the bond.
Synthetic long underlying: S = c − p + X ÷ (1 + r)^T; Buy call, sell put, buy the bond.
Synthetic risk-free bond: X ÷ (1 + r)^T = S + p − c; Long stock, long put and short call at the same strike is a riskless (conversion-type) position that pays X at expiry.
Short positions: Reverse every sign of the long replication; For example, short call = −p − S + PV(X).
Put-call parity (no dividends): c + X ÷ (1 + r)^T = p + S0; European options, same X and T, same underlying. Left side is the fiduciary call, right side is the protective put.
Parity with discrete dividends: c + X ÷ (1 + r)^T = p + S0 − PV(D); PV(D) is the present value of dividends paid before expiry. Dividends lower the stock leg.
Forward-based parity: c + X ÷ (1 + r)^T = p + F0(T) ÷ (1 + r)^T; Equivalent to c − p = [F0(T) − X] ÷ (1 + r)^T. Use when a forward price is given.
Continuous version: c + X e^(−rT) = p + S0 e^(−qT); q is the continuous dividend yield. Use only if the question gives continuous rates.
Arbitrage profit: Profit today = |Fiduciary call value − Protective put value|; Per unit of the position, received at time 0 and riskless.

Quick revision

Put-call parity (European, no income): c + X/(1+r)^T = p + S0.
Fiduciary call = long call + zero-coupon bond with face value X.
Protective put = long put + long underlying.
Both portfolios pay max(S_T, X) at expiry.
Parity needs the same underlying, strike and expiry, and European exercise.
Synthetic call = long put + long underlying + borrow PV of X (short the bond).
Synthetic put = long call + short underlying + long bond with PV of X.
Synthetic underlying = long call + short put + long bond.
Synthetic bond = long put + long underlying + short call.
Arbitrage: sell the more expensive portfolio, buy the cheaper, and keep the difference today.
Discount X at the risk-free rate for the option's time to expiry in years.
If the underlying pays income, the parity equation is adjusted, so follow the question's wording.

Common mistakes

Using X instead of its present value Fix: Always compute X/(1+r)^T before touching the other terms.
Using months as T Fix: Convert to years: 3 months is 0.25, 6 months is 0.5.
Using X instead of its present value X ÷ (1 + r)^T in the parity equation. Fix: Payoffs use X at expiry. Costs today use PV(X). Always discount.
Thinking the fiduciary call and protective put have different payoffs. Fix: Both equal max(S_T, X). Check both states, above and below X.
Treating the bond term as the stock price instead of PV(X). Fix: Always write the bond as X ÷ (1 + r)^T, not as S.
Getting the sign wrong on the bond when replicating a call. Fix: Isolate the call: c = p + S − PV(X). The minus means borrow, not lend.
Using X instead of PV(X) in the comparison. Fix: Always write X ÷ (1 + r)^T as the bond leg. Without it the gap is wrong.
Reversing the trade direction. Fix: Sell whichever side has the higher value and buy the lower side. Then check that time 0 cash is positive.

Exam tips

Memorise the pairing: fiduciary call (call + bond) equals protective put (put + stock). Everything else follows from this.
Replication questions use words like 'synthetic'. Rearrange the formula and read off the signs.
Arbitrage items ask which side to buy. Buy the cheaper portfolio, sell the dearer one.
Convert T to years and use the given rate form. Do not mix annual and continuous compounding.
Use elimination: with three options, check signs and bounds first, then calculate only if needed.
Know the payoff max(S_T, X) for both portfolios. Questions often ask which portfolio has a floor at X.
Always discount X with the risk-free rate. Options that use X undiscounted are traps.
Parity holds for European options with the same strike and expiry only. Watch for mismatches.