Option Replication Using Put-Call Parity

50 questions available

Put-Call Parity Fundamentals5 min
Put-Call Parity establishes a no-arbitrage relationship for European options. The fundamental equation is Stock (S) + Put (P) = Bond (B) + Call (C). The 'Bond' component represents the present value of the strike price (X) discounted at the risk-free rate (RFR), calculated as X / (1 + RFR)^T. A portfolio consisting of a long stock and a long put is known as a Protective Put. A portfolio consisting of a long call and a risk-free bond (zero-coupon bond paying X at maturity) is known as a Fiduciary Call. Since both portfolios guarantee a payoff of at least X at maturity and have unlimited upside potential equal to the stock price, they must be priced identically today to prevent riskless profit (arbitrage).

Key Points

  • Formula: S + P = B + C
  • Mnemonic: Sip Pepsi = Be Cool
  • Protective Put: Long Stock + Long Put
  • Fiduciary Call: Long Bond + Long Call
  • Applies only to European options
Synthetic Construction and Replication6 min
Replication involves creating the cash flow profile of one financial instrument using a combination of others. By algebraically rearranging the Put-Call Parity equation, traders can define synthetic equivalents. A positive sign indicates a long position (buying), and a negative sign indicates a short position (selling/borrowing). For instance, a Synthetic Long Call is derived as C = S + P - B, meaning one buys the stock, buys the put, and shorts the bond (borrows money). Conversely, a Synthetic Long Bond is B = S + P - C. These synthetic relationships are crucial for identifying arbitrage opportunities when market prices deviate from theoretical parity values.

Key Points

  • Synthetic Call: S + P - B
  • Synthetic Put: B + C - S
  • Synthetic Stock: B + C - P
  • Synthetic Bond: S + P - C
  • + means Long, - means Short
Put-Call Forward Parity5 min
When dealing with forward contracts rather than the spot asset, the parity equation is adjusted. Since the Forward Price (F) is the future value of the Spot Price (S), or S = F / (1 + RFR)^T, we can substitute the spot price in the standard equation. The Put-Call Forward Parity equation becomes: (F / (1 + RFR)^T) + P = (X / (1 + RFR)^T) + C. This can be simplified to show that the difference between the discounted forward price and the discounted strike price, plus the put, equals the call. This relationship is vital for pricing options on futures or when the underlying asset is not physically held.

Key Points

  • Spot price is replaced by PV of Forward Price
  • Equation: PV(F) + P = PV(X) + C
  • Used for options on forwards/futures
  • Maintains the no-arbitrage condition
Firm Value and Solvency Analysis4 min
The principles of option pricing can be applied to corporate capital structure. A firm's equity can be modeled as a call option on the total value of the firm's assets (V_T) with a strike price equal to the face value of its debt (D). If the firm is solvent (V_T > D) at maturity, shareholders exercise their option, pay off the debt, and keep the residual (V_T - D). If the firm is insolvent (V_T < D), shareholders let their 'option' expire worthless (Limited Liability), and debtholders take control of the assets (V_T). Thus, the value of debt is the firm value minus the value of the call option (equity).

Key Points

  • Equity = Call Option on Firm Assets
  • Strike Price = Face Value of Debt
  • Solvency Payoff (Equity): V_T - D
  • Insolvency Payoff (Equity): 0
  • Bondholders bear downside risk in insolvency

Questions

Question 1

According to the Put-Call Parity equation, which portfolio is equivalent to a Fiduciary Call?

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Question 2

What represents 'B' in the mnemonic 'Sip Pepsi = Be Cool' (S + P = B + C)?

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Question 3

Which of the following correctly describes a 'Protective Put'?

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Question 4

If you want to create a Synthetic Call option, which positions should you take?

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Question 5

The Put-Call Parity relationship applies strictly to which type of options?

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Question 6

Using the Put-Call Parity, what is the formula for a Synthetic Put?

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Question 7

In a synthetic position, what does a negative sign (-) typically indicate?

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Question 8

Calculate the price of a European Call option if: Stock = 100, Put = 5, PV of Strike (Bond) = 90.

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Question 9

Calculate the price of the underlying Stock if: Call = 12, Put = 7, PV of Strike = 50.

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Question 10

What constitutes a 'Fiduciary Call'?

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Question 11

In Put-Call Forward Parity, the spot price (S) is replaced by which expression?

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Question 12

If a firm is considered insolvent (Value of Assets < Debt), what is the payoff to shareholders?

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Question 13

Viewed as an option, a firm's equity is equivalent to:

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Question 14

What represents the 'Strike Price' in the analogy where Equity is a Call Option on the firm?

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Question 15

If risk-free rates (RFR) increase, what is the impact on the value of a Call Option based on the parity logic shown?

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Question 16

Calculate the value of a Bond (PV of Strike) in a parity arbitrage scenario if: Stock = 50, Put = 3, Call = 8.

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Question 17

A Synthetic Long Bond is created by:

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Question 18

If the market price of a Call is 10, but the synthetic call (S + P - B) is calculated at 12, what arbitrage action should be taken?

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Question 19

In the Put-Call Forward Parity equation, if the Forward Price is F, what is the formula for the Synthetic Call?

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Question 20

What is the payoff to debtholders if a firm is insolvent (Value of Assets < Debt)?

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Question 21

Calculate the PV of the Strike Price (Bond) if Strike = 105, RFR = 5 percent, and time to maturity = 1 year.

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Question 22

If Stock = 80, Strike = 80, RFR = 10 percent, Time = 1 year, and Call = 10, what is the Put price? (Assume X=80 is face value).

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Question 23

A portfolio of Long Stock + Long Put is used to limit downside risk. This is known as:

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Question 24

Identify the incorrect rearrangement of the Put-Call Parity formula (S + P = B + C).

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Question 25

If a Put option is priced at 5, Stock is 50, PV of Strike is 48, and Call is priced at 6, does an arbitrage opportunity exist?

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Question 26

In the arbitrage scenario where S+P = 55 and B+C = 54, what is the correct strategy?

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Question 27

A 'Synthetic Stock' position is constructed by:

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Question 28

If the Spot Price (S) is 100 and the Risk-Free Rate is 5 percent (T=1), what is the Forward Price (F) used in parity?

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Question 29

In the Merton corporate bond model, the value of the firm's debt is equal to:

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Question 30

If a Call is 10, Put is 10, and PV of Strike is 100, what must the Stock price be for parity to hold?

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Question 31

Which component in Put-Call Parity accounts for the time value of money related to the strike price?

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Question 32

The equation 'Stock + Put = Bond + Call' implies that:

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Question 33

Using Put-Call Forward Parity, if F = 105, X = 105, RFR = 5 percent (T=1), and C = 5, what is the value of P?

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Question 34

If Risk-Free Rate increases, what happens to the price of a Put option (holding other factors constant)?

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Question 35

Which position allows an investor to borrow money at the risk-free rate synthetically?

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Question 36

Calculate C if S=50, P=2, X=50, RFR=0 percent (T=1).

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Question 37

In the firm value analogy, if a company has Assets of 100 and Debt of 80, the 'option' is:

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Question 38

If implied volatility increases, what generally happens to both Call and Put prices?

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Question 39

To create a 'Synthetic Short Put', what positions are required?

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Question 40

Which condition allows for the substitution of S with PV(F) in parity equations?

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Question 41

If the equation S + P = B + C is violated, what is the immediate implication?

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Question 42

Calculate the Synthetic Call price if Forward Price = 105, Strike = 100, RFR = 5 percent (T=1), Put = 3.

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Question 43

What is the primary reason American options might not fit the 'Sip Pepsi = Be Cool' formula exactly?

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Question 44

For a solvent firm, the value of Debt (D) plus the value of Equity (E) equals:

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Question 45

When rearranging the formula to S = B + C - P, what does the term 'B' represent in an investment context?

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Question 46

Which strategy mimics owning the stock using derivatives?

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Question 47

If Spot = 100, Call = 5, Put = 5, and Strike = 100, what is the implied Risk-Free Rate?

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Question 48

The combination of a Long Call and a Short Put with the same strike and maturity creates a position similar to:

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Question 49

Put-Call Parity is effectively an application of:

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Question 50

If a firm is solvent, shareholders receive:

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