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Option Replication Using Put-Call Parity

Derivatives

Put-Call Parity

Learning Outcome Statement:

explain put–call parity for European options

Summary:

Put-call parity is a fundamental principle in options pricing which shows that the price of a call option and a put option with the same strike price and expiration date, when combined with the stock price, must be equal to the price of a risk-free bond that pays the strike price at expiration. This relationship is used to derive synthetic positions and arbitrage opportunities.

Key Concepts:

Put-Call Parity Formula

The put-call parity formula establishes a relationship between the prices of European put and call options with the same strike price and expiration. It states that buying a stock and a put option (protective put) should cost the same as buying a call option and a risk-free bond that pays the strike price at expiration (fiduciary call).

Synthetic Positions

Synthetic positions use put-call parity to replicate the payoff of a position using other financial instruments. For example, a synthetic long stock can be created by buying a call and selling a put with the same strike price and expiration, funded by the purchase of a risk-free bond.

Arbitrage Opportunities

If the put-call parity is not held due to mispricing in the market, arbitrage opportunities arise allowing risk-free profits. Arbitrageurs can exploit these discrepancies by constructing portfolios that will have zero net cost at inception and positive value in the future.

Formulas:

Put-Call Parity

S0+p0=c0+X(1+r)−TS_0 + p_0 = c_0 + X(1 + r)^{-T}

This formula shows the relationship between the prices of puts and calls of the same strike price and expiration, indicating that the cost of a protective put should equal the cost of a fiduciary call.

Variables:
S0S_0:
current stock price
p0p_0:
price of the put option
c0c_0:
price of the call option
XX:
strike price
rr:
risk-free interest rate
TT:
time to expiration in years
Units: currency

Option Strategies Based on Put-Call Parity

Learning Outcome Statement:

explain put–call parity for European options

Summary:

The learning outcome focuses on explaining the put-call parity concept for European options, which establishes a foundational relationship between the prices of calls, puts, the underlying asset, and risk-free bonds. This relationship is crucial for replicating and pricing derivative positions and creating riskless arbitrage opportunities.

Key Concepts:

Put-Call Parity Equation

The put-call parity equation is a fundamental relationship in European options that helps in identifying mispricing between options and their underlying assets, which can lead to arbitrage opportunities. It is represented as S0 + p0 = c0 + X(1 + r)^(-T).

Arbitrage Opportunity

Arbitrage involves taking advantage of price differences in different markets or forms to earn a riskless profit. In the context of put-call parity, if the actual prices of options do not align with the parity equation, an investor can execute trades to exploit this mispricing and secure a guaranteed profit.

Replicating Portfolios

Under put-call parity, certain positions in options can be replicated by other positions involving different combinations of calls, puts, the underlying asset, and risk-free bonds. This concept is used to understand and establish equivalent investment positions that have the same payoff structures.

Covered Call Strategy

A covered call strategy involves holding a long position in the underlying asset while selling a call option on the same asset. This strategy is used to generate income through the premiums received from selling the call options, with the underlying asset serving as a cover in case the option is exercised.

Formulas:

Put-Call Parity

S0+p0=c0+X(1+r)−TS_0 + p_0 = c_0 + X(1 + r)^{-T}

This formula shows the relationship that must hold between the prices of calls, puts, the underlying asset, and the discounted exercise price adjusted for the risk-free rate, to prevent arbitrage opportunities.

Variables:
S0S_0:
Current price of the underlying asset
p0p_0:
Current price of the put option
c0c_0:
Current price of the call option
XX:
Exercise price of the options
rr:
Risk-free interest rate
TT:
Time to expiration of the options
Units: currency units

No-Arbitrage Call Price

c0=S0−X(1+r)−T+p0c_0 = S_0 - X(1 + r)^{-T} + p_0

This formula is derived from the put-call parity and is used to calculate the no-arbitrage price of a call option given the prices of the put option, the underlying asset, and the risk-free bond.

Variables:
S0S_0:
Current price of the underlying asset
p0p_0:
Current price of the put option
c0c_0:
Current price of the call option
XX:
Exercise price of the options
rr:
Risk-free interest rate
TT:
Time to expiration of the options
Units: currency units

Put-Call Forward Parity and Option Applications

Learning Outcome Statement:

Explain put-call forward parity for European options

Summary:

Put-call forward parity is a financial principle that links the prices of European put and call options with the same strike price and expiration date to the price of the underlying asset and a forward contract on that asset. This relationship is crucial for pricing options and creating synthetic positions that mimic other financial instruments.

Key Concepts:

Covered Call Strategy

A covered call strategy involves holding a long position in the underlying asset while simultaneously writing (selling) a call option on the same asset. This strategy is used to generate additional income from the option premium, with the trade-off being a capped potential upside.

Put-Call Forward Parity

Put-call forward parity is an extension of the put-call parity concept, incorporating forward contracts. It states that the cost of a synthetic protective put (buying a forward contract and a put option, while investing in a risk-free bond) should equal the cost of buying a call option and investing in a risk-free bond that pays the strike price at maturity.

Synthetic Protective Put

A synthetic protective put is created by purchasing a forward contract on the underlying asset, buying a put option, and investing in a risk-free bond with a face value equal to the forward price. This position is designed to replicate the payoff of a protective put, providing downside protection while allowing for upside potential.

Formulas:

Put-Call Forward Parity Equation

F0(T)(1+r)−T+p0=c0+X(1+r)−TF_0(T)(1 + r)^{-T} + p_0 = c_0 + X(1 + r)^{-T}

This formula establishes the relationship between the prices of puts and calls with the same strike price and maturity, incorporating the forward price of the underlying asset and the risk-free rate. It is used to determine mispricing between these instruments and to create arbitrage opportunities.

Variables:
F0(T)F_0(T):
Forward price of the underlying asset at time T
rr:
Risk-free rate
TT:
Time to maturity
p0p_0:
Current price of the put option
c0c_0:
Current price of the call option
XX:
Strike price of the options
Units: currency

Put-Call Forward Parity

Learning Outcome Statement:

Explain put-call forward parity for European options

Summary:

Put-call forward parity is a fundamental concept in options pricing, which establishes a relationship between the prices of puts, calls, forwards, and bonds under certain conditions. It extends the basic put-call parity by incorporating forward contracts and risk-free bonds to replicate the payoffs of protective puts and fiduciary calls, thereby demonstrating the equivalence of these positions under no-arbitrage conditions.

Key Concepts:

Covered Call Strategy

A covered call strategy involves holding a long position in the underlying asset and selling a call option on the same asset. This strategy is used to generate additional income from the option premium.

Put-Call Forward Parity

Put-call forward parity is an extension of the put-call parity concept for European options, incorporating forward contracts and risk-free bonds to demonstrate the equivalence of certain option strategies. It shows how a synthetic protective put (combining a forward contract, a risk-free bond, and a put option) has equivalent payoffs to a protective put and a fiduciary call under no-arbitrage conditions.

Synthetic Protective Put

A synthetic protective put is created by purchasing a forward contract and a risk-free bond, along with a put option. This strategy replicates the payoff of holding the underlying asset directly and buying a put option, providing downside protection.

Fiduciary Call

A fiduciary call involves buying a call option and a risk-free bond that pays the strike price at maturity. This strategy is designed to mimic the payoff of a protective put, providing a way to participate in potential upside while limiting downside risk.

Formulas:

Put-Call Forward Parity

F0(T)(1+r)−T+p0=c0+X(1+r)−TF_0(T)(1 + r)^{-T} + p_0 = c_0 + X(1 + r)^{-T}

This formula establishes the relationship between the prices of puts, calls, forwards, and risk-free bonds, ensuring no-arbitrage conditions are met.

Variables:
F0(T)F_0(T):
Forward price of the underlying asset at time T
rr:
Risk-free rate
TT:
Time to maturity
p0p_0:
Price of the put option at inception
c0c_0:
Price of the call option at inception
XX:
Strike price of the options
Units: Currency units

Rearranged Put-Call Forward Parity

p0−c0=[X−F0(T)](1+r)−Tp_0 - c_0 = [X - F_0(T)](1 + r)^{-T}

This rearranged form of the put-call forward parity formula shows the relationship between the net cost of a put and a call option, equating it to the discounted difference between the strike price and the forward price.

Variables:
p0p_0:
Price of the put option at inception
c0c_0:
Price of the call option at inception
XX:
Strike price of the options
F0(T)F_0(T):
Forward price of the underlying asset at time T
rr:
Risk-free rate
TT:
Time to maturity
Units: Currency units

Option Put-Call Parity Applications: Firm Value

Learning Outcome Statement:

Explain put-call parity for European options and put-call forward parity for European options.

Summary:

The learning outcome statement focuses on explaining the put-call parity and put-call forward parity for European options, particularly in the context of firm value. It explores how these relationships can be used to model the value of a firm by considering the financial claims of equity and debt holders. The content also delves into the implications of these parity relationships for option trading strategies and risk management.

Key Concepts:

Put-Call Parity

Put-call parity is a fundamental principle in options pricing, which establishes a riskless, arbitrage-free relationship between the prices of European put and call options with the same strike price and expiration date. It helps in determining the fair price of options.

Firm Value Modeling

The firm's value is modeled as the sum of the present value of its debt and equity. This relationship is akin to the put-call parity where the firm's assets are seen as a portfolio of options held by shareholders and debt holders.

Debtholder and Shareholder Claims

In the context of firm value, debtholders have a priority claim on the firm's assets up to the face value of the debt, similar to holding a risk-free bond combined with a sold put option. Shareholders, on the other hand, hold a residual claim on the firm's assets, akin to holding a call option.

Formulas:

Put-Call Forward Parity

p0−c0=(X−F0(T))(1+r)−Tp_0 - c_0 = (X - F_0(T))(1 + r)^{-T}

This formula relates the prices of put and call options to the forward price of the underlying asset, adjusted for the risk-free rate over the option's time to maturity.

Variables:
p0p_0:
Current put option price
c0c_0:
Current call option price
XX:
Exercise price
F0(T)F_0(T):
Forward price of the asset at time T
rr:
Risk-free rate
TT:
Time to maturity
Units: Currency

Firm Value at Time 0

V0=c0+PV(D)−p0V_0 = c_0 + PV(D) - p_0

This formula captures the value of the firm's assets at time 0 from the perspective of shareholders and debtholders, incorporating the value of options held by each.

Variables:
V0V_0:
Market value of the firm's assets at time 0
c0c_0:
Value of a call option on the firm's assets
PV(D)PV(D):
Present value of the firm's debt
p0p_0:
Value of a put option on the firm's assets
Units: Currency
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