derivatives 9: Option Replication Using Put–Call Parity

What is put–call parity?

A no-arbitrage relationship linking the prices of European calls, puts, the underlying, and a risk-free bond.

Why does put–call parity hold?

Because two portfolios with identical payoffs at maturity must have identical prices today (law of one price).

what is a fiduciary call portfolio?

combines a call with a risk-free bond that guarantees the strike payment.

• Long call option

• Long risk-free bond paying X at maturity

what is a protective put portfolio?

put protects downside risk of holding the underlying.

• Long underlying

• Long put option

when is the put or call exercised for

1. protective put

• underlying asset

• put option

2. fiduciary call

• call option

• risk free asset

Portfolio Position	Put Exercised (ST < X)	No Exercise (ST = X)	Call Exercised (ST > X)
Protective Put:
Underlying Asset	ST	ST	ST
Put Option	X – ST	0	0
Total:	X	ST (= X)	ST
Fiduciary Call:
Call Option	0	0	ST – X
Risk-Free Asset	X	X	X
Total:	X	X (= ST )	ST

Put–call parity equation

$$S_0+p_0=c_0+X(1_{}+r)^{_{-T}}$$

underlying asset + put = call + risk free asset

What condition indicates arbitrage in put–call parity?

$$S_0+p_0\ne c_0+X(1_{}+r)^{_{-T}}$$

what to do if S_0+p_0>c_0+X(1_{}+r)^{_{-T}}

• Sell stock

• Sell put

• Buy call

• Buy bond

How can a long put be replicated

• Buy call

• Buy bond (PV of strike)

• Short underlying

how to replicate the

1. underlying

2. risk free bond

3. call option

4. put option

Position	Underlying S 0	Risk-Free Bond X(1 + r)– T	Call Option c 0	Put Option p 0
Underlying	—	Long	Long	Short
Risk-free bond	Long	—	Short	Long
Call option	Long	Short	—	Long
Put option	Short	Long	Long	—

Put–call forward parity formula for synthetic put?

$$F_0​(T)(1+r)^{−T}+p_0​=c_0​+X(1+r)^{−T}$$

where $F_0(T)(1+r)^{-T}$ = PV of the forward price

Spot price $S_0$ is replaced by a synthetic stock = forward + bond.

What is a synthetic protective put?

• Long forward

• Long risk-free bond (PV of forward price)

• Long put

Basic firm value identity?

$$V_0 = E_0 +PV(D)$$

V₀ = firm value
E₀ = equity value
D = face value of zero-coupon debt

when is a firm solvent?

V_T > D

i.e: value of the firm (V_T ) exceeds the face value of the debt, or V_T > D

• Debtholders receive D and are repaid in full.

• Shareholders receive the residual: E_T = V_T – D

when is a firm insolvent?

V_{T}<D

value of the firm (V_T ) is below the face value of the debt, or V_T < D

• Debtholders have a priority claim on assets and receive V_T < D.

• Shareholders receive the residual, E_T = 0.

shareholder and debtholder payoff formulas

• shareholder payoff is max(0, V_T – D)

  • unlimited upside, limited downside

• debtholder payoff is min(V_T , D).

  • limited upside and downside

how is a shareholder described in terms of options

• long position in the underlying firm’s assets (V_T )

• Long call on firm value with strike D.

how is a debtholder described in terms of options

• long position in a risk-free bond (D) and have

• Short put on firm value (V_T ) with an exercise price of D.

Firm-value version of put–call parity?

rearrange previous formula

$$V_0​+p_0​=c_0​+PV(D)$$