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CFA Level I · CFA Level I Exam

Option Replication Using Put-Call Parity for CFA Level I

Put-call parity says that for European options on the same underlying, strike and expiry, a fiduciary call (call plus a zero-coupon bond paying the strike) equals a protective put (put plus the underlying). So c + X/(1+r)^T = p + S0. Rearrange it to replicate any one instrument, or to spot arbitrage.

What this chapter covers

This chapter shows that calls, puts, the underlying and a risk-free bond are linked by one equation. For European options with the same strike X and expiry T on the same underlying, c + X/(1+r)^T = p + S0. Here c is the call price, p the put price, S0 the spot price and r the risk-free rate. If the underlying pays no income, this is the form you use. If it pays income or has carrying costs, the exam will tell you how to adjust.

Once you can read that equation as two portfolios with the same payoff, you can rearrange it. Move terms and you get a synthetic call, synthetic put, synthetic underlying or synthetic bond. Each synthetic position is built by buying some pieces and selling others. The signs are where most marks are won or lost.

The chapter sits inside Derivatives and Risk Management. It builds on option payoffs, moneyness and the no-arbitrage idea from forward and futures pricing. It also uses present value from Quantitative Methods and the risk-free discount rate. Later option pricing material, such as the binomial model, relies on the same replication logic.

Questions here are short and calculation-light, which suits 90 seconds per question and a three-option format. You often need only one rearrangement and one present value. Derivatives and Risk Management carries a moderate topic weight, and put-call parity is a clean, repeatable source of marks once you trust the signs. It also trains the no-arbitrage thinking that appears in forwards, futures and swaps, so effort here pays back across the topic.

Option Replication Using Put-Call Parity: topics in the order to study them

  1. 1Fiduciary Call and Protective PutStart with the two portfolios, because their payoffs at expiry are what make parity true.
  2. 2Put-Call Parity for European OptionsOnce you know both portfolios, the equation c + X/(1+r)^T = p + S0 follows and is easy to remember.
  3. 3Replicating Options, Underlying and Bond PositionsRearranging the equation gives each synthetic instrument, so you need the equation first.
  4. 4Arbitrage When Put-Call Parity Is ViolatedArbitrage uses replication: sell the expensive side, buy the cheap side, and lock in the gap. Study it last.

How to prepare Option Replication Using Put-Call Parity

Aim to understand the payoff logic first, then drill the rearrangements until the signs are automatic.

  1. Draw the payoff at expiry of a fiduciary call and a protective put in two cases, S_T above X and S_T below X. Confirm both equal max(S_T, X).
  2. Write c + X/(1+r)^T = p + S0 from memory. Say in words which side is the fiduciary call and which is the protective put.
  3. Rearrange it four ways: c = p + S0 − X/(1+r)^T, p = c − S0 + X/(1+r)^T, S0 = c − p + X/(1+r)^T, and the bond X/(1+r)^T = p + S0 − c. Label each as long or short pieces.
  4. Practise present value with the BA II Plus: enter N = T in years, I/Y = r in percent, FV = X, then CPT PV. Check that the sign of the PV is the opposite of FV, then use the absolute value.
  5. Work numeric examples. For instance, with S0 = $50, X = $50, r = 4% and T = 1 year, X/(1+r)^T = $48.08. If c = $5.00, parity implies p = 5.00 − 50 + 48.08 = $3.08.
  6. Practise arbitrage: compare the two sides, sell the higher-priced portfolio, buy the lower, and check that the net payoff at expiry is zero and today's cash flow is positive.
  7. Finish with timed sets of single-step questions. Eliminate options by checking the direction of the mispricing before you calculate.

Common mistakes in Option Replication Using Put-Call Parity

  • Using X instead of its present value X/(1+r)^T

    Fix: Always compute the bond leg as X/(1+r)^T first. Then rearrange the equation.

  • Getting the sign of a synthetic leg wrong

    Fix: Write each synthetic position as a list of long and short pieces, then check the payoff at expiry in both states.

  • Mixing up fiduciary call and protective put

    Fix: Remember that the fiduciary call contains the call and the bond, while the protective put contains the put and the underlying.

  • Applying parity to American options or to options with different strikes or expiries

    Fix: State the conditions to yourself every time: European, same underlying, same strike, same expiry.

  • Using the wrong time period or rate

    Fix: Convert time to years, use the risk-free rate, and keep the rate and time consistent.

  • Doing arbitrage in the wrong direction

    Fix: Sell the portfolio with the higher current value and buy the lower. Then check that the payoffs at expiry cancel.

Last-day revision: Option Replication Using Put-Call Parity

  • 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.

Option Replication Using Put-Call Parity practice questions

Option Replication Using Put-Call Parity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Option Replication Using Put-Call Parity: frequently asked questions

What is put-call parity in simple words?

It says a call plus a risk-free bond that pays the strike at expiry has the same payoff as a put plus the underlying. Because the payoffs match, today's prices must match too, or an arbitrage exists. It holds for European options with the same strike and expiry on the same underlying.

How do I create a synthetic put?

Rearrange the equation to p = c − S0 + X/(1+r)^T. So buy the call, short the underlying, and buy the zero-coupon bond with present value X/(1+r)^T. Check it by comparing payoffs at expiry in both price states.

Does put-call parity work for American options?

The exact equality is for European options. American options can be exercised early, which breaks the equal-payoff argument. Level I questions on parity assume European options unless stated otherwise.

How do I calculate the present value on the BA II Plus?

Enter the number of years in N, the risk-free rate in percent in I/Y, the strike in FV and 0 in PMT. Then press CPT and PV. The answer is shown as a negative number, so take its absolute value for the bond leg.