Lesson 1: Put-Call ParityACTS 4302

Natalia A. Humphreys

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Acknowledgement

This work is based on the material in ASM MFE Study manual forExam MFE/Exam 3F. Financial Economics (7th Edition), 2009, by

Abraham Weishaus.

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Put - Call parity (PCP)

Put - Call parity gives a relationship between the premium of a calland the premium of a put.

For now, we’ll consider only the European options.

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PCP - Derivation

Suppose we bought a European call option and sold a Europeanput option, both having the same underlying asset St , the samestrike K , and the same time to expiry T . We would then pay

C (K ,T )− P(K ,T )

BUT: The result can be achieved without using options at all!

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PCP - Derivation

At time T one of the options will be exercised. In either case, wepay K and receive the underlying asset:

I If ST > K , exercise the call option that we bought. Pay Kand receive the asset.

I If ST < K , exercise the put option that the counterpartybought from us. We will pay K and receive the asset.

I If ST = K , it does not matter whether we have K or theunderlying asset.

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PCP - Derivation

There are two ways to receive ST at time T .

1. Buy a call option and sell a put option at time 0, and pay K attime T .

2. Enter a forward agreement to buy ST , and at time T pay F0,T ,the price of forward.

By the principle of ”no arbitrage”, the two ways must cost thesame. Discounting to time 0, we have:

C (K ,T )− P(K ,T ) + Ke−rT = F0,T e−rT

OrC (K ,T )− P(K ,T ) = F0,T e

−rT − Ke−rT

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PCP - Non-dividend paying stock

Recall that for non-dividend paying stock F0,T = S0erT . Thus, the

PCP becomes

C (K ,T )− P(K ,T ) = S0 − Ke−rT

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PCP - Non-dividend paying stock - Example 1.1

A non-dividend paying stock has a price of 40. A European calloption allows buying the stock for 45 at the end of 9 months. Thecontinuously compounded risk-free rate is 5%. The premium of thecall option is 2.84.

Determine the premium of a European put option allowing sellingthe stock for 45 at the end of 9 months.

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PCP - Non-dividend paying stock - Example 1.1 - Solution

Solution. In this problem

S0 = 40, K = 45, C (K ,T ) = 2.84, T = 0.75, r = 5%

By the PCP for non-dividend paying stock, we have:

P(K ,T ) = C (K ,T )− S0 + Ke−rT =

= 2.84− 40 + 45 · e−0.05·0.75 = 6.1838 ≈ 6.18

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PCP - Dividend paying stock with discrete dividends

Recall that for discrete dividend paying stockF0,T = S0e

rT − CumValue(Div). Thus, the PCP becomes

C (K ,T )− P(K ,T ) = F0,T e−rT − Ke−rT ⇔

C (K ,T )− P(K ,T ) = S0 − PV(Div)− Ke−rT

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PCP - Dividend paying stock with discrete dividends -Example 1.2

A stock price is 45. The stock will pay a dividend of 1 after 2months. A European put option with a strike of 42 and an expirydate of 3 months has a premium of 2.71. The continuouslycompounded risk-free rate is 5%.

Determine the premium of a European call option on the stockwith the same strike and expiry.

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PCP - Dividend paying stock with discrete dividends -Example 1.2 - Solution

Solution. In this problem

S0 = 45, K = 42, P(K ,T ) = 2.71, T = 0.25, r = 5%,

Div = 1 at time t =1

6

By the PCP for discrete-dividend paying stock, we have:

C (K ,T ) = P(K ,T ) + S0 − PV(Div)− Ke−rT =

= 2.71 + 45− 1 · e−0.05· 16 − 42 · e−0.05·0.25 =

= 47.71− 0.9917− 41.4783 = 5.24

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PCP - Dividend paying stock with discrete dividends -Example 1.3

A stock price is 50. The stock will pay a dividend of 2 after 4months. A European call option with a strike of 50 and an expirydate of 6 months has a premium of 1.62. The continuouslycompounded risk-free rate is 4%.

Determine the premium of a European put option on the stockwith the same strike and expiry.

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PCP - Dividend paying stock with discrete dividends -Example 1.3 - Solution

Solution. In this problem

S0 = 50, K = 50, C (K ,T ) = 1.62, T = 0.5, r = 4%,

Div = 2 at time t =1

3

By the PCP for discrete-dividend paying stock, we have:

P(K ,T ) = C (K ,T )− S0 + PV(Div) + Ke−rT =

= 1.62− 50 + 2 · e−0.04· 13 + 50 · e−0.04·0.5 =

= −48.38 + 1.9735 + 49.01 = 2.60

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PCP - Dividend paying stock with continuous dividends

Let δ be a dividend rate.Recall that for continuous dividend paying stock F0,T = S0e

(r−δ)T .Thus, the PCP becomes

C (K ,T )− P(K ,T ) = F0,T e−rT − Ke−rT ⇔

C (K ,T )− P(K ,T ) = S0e−δT − Ke−rT

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PCP - Dividend paying stock with continuous dividends -Example 1.4

You are given:

(i) A stock’s price is 40.

(ii) The continuously compounded risk-free rate is 8%.

(iii) The stock’s continuous dividend rate is 2%

A European 1-year call option with a strike of 50 costs 2.34.

Determine the premium of a European 1-year put option on thestock with a strike of 50.

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PCP - Dividend paying stock with continuous dividends -Example 1.4 - Solution

Solution. In this problem

S0 = 40, K = 50, C (K ,T ) = 2.34, T = 1, r = 8%,

δ = 2%

By the PCP for continuous-dividend paying stock, we have:

P(K ,T ) = C (K ,T )− S0e−δT + Ke−rT =

= 2.34− 40 · e−0.02·1 + 50 · e−0.08·1 =

= 2.34− 39.21 + 46.16 = 9.29

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PCP - Dividend paying stock with continuous dividends -Example 1.5

You are given:

(i) A stock’s price is 57.

(ii) The continuously compounded risk-free rate is 5%.

(iii) The stock’s continuous dividend rate is 3%

A European 3-month put option with a strike of 55 costs 4.46.

Determine the premium of a European 3-month call option on thestock with a strike of 55.

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PCP - Dividend paying stock with continuous dividends -Example 1.5 - Solution

Solution. In this problem

S0 = 57, K = 55, P(K ,T ) = 4.46, T = 0.25, r = 5%,

δ = 3%

By the PCP for continuous-dividend paying stock, we have:

C (K ,T ) = P(K ,T ) + S0e−δT − Ke−rT =

= 4.46 + 57 · e−0.03·0.25 − 55 · e−0.05·0.25 =

= 4.46 + 56.5741− 54.3168 = 6.72

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Currency options - Puts and Calls notation

I C (x0,K ,T ) - a call option on currency with spot exchangerate x0 to purchase it at exchange rate K at time T

I P(x0,K ,T ) - a put option on currency with spot exchangerate x0 to sell it at exchange rate K at time T

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Currency options - PCP

Recall:

I General formula for a PCPC (S ,K ,T )− P(S ,K ,T ) = (F0,T − K )e−rT

I rf is the ”foreign” risk-free rate for the currency which isplaying the role of a stock

I rd is the ”domestic” risk-free rate which is playing the role ofcash

I Price of a forward expressed in domestic currency to deliverforeign currency at x0 exchange rate: F0,T = x0e

(rd−rf )T

Then PCP becomes:

C (x0,K ,T )− P(x0,K ,T ) = x0e−rf T − Ke−rdT

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Currency options - Example 1.6

You are given:

I The spot exchange rate for dollars to pounds is 1.4$/£.

I The continuously compounded risk-free rate for dollars is 5%.

I The continuously compounded risk-free rate for pounds is 8%.

A 9-month European put option allows selling £1 at the rate of$1.50/£. A 9-month dollar denominated call option with the samestrike costs $0.0223.

Determine the premium of the 9-month dollar denominated putoption.

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Currency options - Example 1.6. Solution.

Solution. We are buying pounds. Thus, pounds play the role ofstock or foreign currency.We pay for pounds with dollars. Thus, dollars play the role of cashor domestic currency. Therefore,

rf = 0.08, rd = 0.05, x0 = 1.4, K = 1.5,

C (x0,K ,T ) = 0.0223, T = 0.75

By the PCP for currency options,

P(x0,K ,T ) = C (x0,K ,T )− x0e−rf T + Ke−rdT =

= 0.0223− 1.4 · e−0.08·0.75 + 1.5 · e−0.05·0.75 =

= 0.0223− 1.3185 + 1.4448 = 0.1486

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Currency options - Example 1.7

You are given:

I The spot exchange rate for yen to dollars is 90U/$.

I The continuously compounded risk-free rate for dollars is 5%.

I The continuously compounded risk-free rate for yen is 1%.

A 6-month yen-denominated European call option has a strike of90U/$ and costs U3.25.

Determine the premium of a 6-month yen denominated Europeanput option having a strike of 90U/$.

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Currency options - Example 1.7. Solution.

Solution. We are buying dollars. Thus, dollars play the role ofstock or foreign currency.We pay for pounds with yen. Thus, yen play the role of cash ordomestic currency. Therefore,

rf = 0.05, rd = 0.01, x0 = 90, K = 90,

C (x0,K ,T ) = 3.25, T = 0.5

By the PCP for currency options,

P(x0,K ,T ) = C (x0,K ,T )− x0e−rf T + Ke−rdT =

= 3.25− 90 · e−0.05·0.5 + 90 · e−0.01·0.5 =

= 3.25− 87.7789 + 89.5511 = 5.0232

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Exchange options - Introduction

So far we’ve discussed receiving (call) or giving (put) stock inreturn for cash.

GENERALIZE

An option to receive a stock in return for a different stock.

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Exchange options - Notation

I St - the value of the underlying asset, the one for which theoption is written

I Qt - the price of the strike asset, the one which is paid

I Ft,T (S) - a forward agreement to purchase asset S at time T

I FPt,T (S) - prepaid forward: FP

t,T (S) = e−r(T−t)Ft,T (S)

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Exchange options - Calls and Puts

I C (St ,Qt ,T − t) a call option written at time t which lets thepurchaser elect to receive ST in return for QT at time T , i.e.to receive max(0, ST − QT ).

I P(St ,Qt ,T − t) a put option written at time t which lets thepurchaser elect to give ST in return for QT at time T , i.e. toreceive max(0,QT − ST ).

Then the Put-Call Parity is

C (St ,Qt ,T − t)− P(St ,Qt ,T − t) = FPt,T (St)− FP

t,T (Qt)

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Exchange options - Calls and Puts -Final expression

If

I δS is continuous dividend rate of stock S and

I δQ is continuous dividend rate of stock Q

Then the Put-Call Parity is

C (St ,Qt ,T − t)− P(St ,Qt ,T − t) = Ste−δS (T−t) − Qte

−δQ(T−t)

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Exchange options - Example 1.8

A European call option allows one to purchase 2 shares of Stock Bwith 1 share of stock A at the end of a year. You are given:

I The continuously compounded risk-free rate is 5%.

I Stock A pays dividends at a continuous rate of 2%.

I Stock B pays dividends at a continuous rate of 4%.

I The current price for Stock A is 70.

I The current price for Stock B is 30.

A European put option which allows one to sell 2 shares of StockB for 1 share of Stock A costs 11.50.

Determine the premium of the European call option mentionedabove, which allows one to purchase 2 shares of Stock B for 1share of Stock A.

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Exchange options - Example 1.8. Solution.

Solution. Risk-free rate in these type of problems is irrelevant. Letus figure out what plays the role of S and what plays the role of Qin our problem.Since we are purchasing 2 shares of Stock B, they play the role ofstock S in the PCP for exchange options. Note S = 2 · 30 = 60.Since we use 1 share of Stock A to pay for 2 shares of Stock B, itplays the role of stock Q in the PCP for exchange options. NoteQ = 70.Further, δS = 0.04, δQ = 0.02, T = 1 and

C (St ,Qt ,T − t) = P(St ,Qt ,T − t) + FPt,T (S)− FP

t,T (Q) =

= P(St ,Qt ,T − t) + Ste−δST − Qte

−δQT =

= 11.50 + 60 · e−0.04·1 − 70 · e−0.02·1 =

= 11.50 + 57.6474− 68.6139 = 0.5335 ≈ 0.53

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Exchange options - The mirror image

The definitions of calls and puts are mirror images. A call topurchase Q for S is the same as a put to sell S for Q:

C (Qt , St ,T − t) = P(St ,Qt ,T − t)

Thus, PCP for exchange options

C (St ,Qt ,T − t)− P(St ,Qt ,T − t) = Ste−δS (T−t) − Qte

−δQ(T−t)

⇔C (St ,Qt ,T − t)− C (Qt ,St ,T − t) = Ste

−δS (T−t) − Qte−δQ(T−t)

Thus, in the previous example, a call which allows one to purchase2 shares of Stock B for 1 share of Stock A is the same as the putthat allows one to sell 1 share of Stock A for 2 shares of Stock B.

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Put-Call connection in different currency units

Let us look at currency options in light of exchange options we’vejust discussed.

A call to purchase pounds with dollars

⇔

A put to sell dollars for pounds

BUT: the units are different. Let us see how to translate the units.

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Put-Call connection in different currency units

Call-Put relationship in ”domestic” currency: If K is the strikeprice and C (x0,K ,T ) is the call price in domestic currency, then

KPd

(1

x0,

1

K,T

)= Cd(x0,K ,T )

Ex. If x0 =$3/£ and the strike price is K = $2, then a $ -denominated call to buy 1£ for $2 is the same as a $ -denominated put to sell $2 for 1£ or 2 $ denominated puts to sell$1 for 1/2£.Call-Put relationship in ”foreign” and ”domestic” currency:

Kx0Pf

(1

x0,

1

K,T

)= Cd(x0,K ,T )

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Put-Call connection in different currency units. Example1.9.

The spot exchange rate for dollars into euros is $1.05/e. A6-month dollar denominated call option to buy one euro at strikeprice $1.1/e costs $0.04.

Determine the premium of the corresponding euro-denominated putoption to sell one dollar for euros at the corresponding strike price.

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Put-Call connection in different currency units. Example1.9. Solution.

Solution. Here the domestic currency = dollars and the foreigncurrency = euros. We have:

x0 = 1.05, T = 0.5, K = 1.1, Cd(x0,K ,T ) = 0.04

Using Call-Put relationship in ”foreign” and ”domestic” currency:

Kx0Pf

(1

x0,

1

K,T

)= Cd(x0,K ,T )⇔

Pf

(1

x0,

1

K,T

)=

1

Kx0Cd(x0,K ,T )

we obtain:

Pf =1

1.05 · 1.10.04 = 0.03463 ≈ 0.035

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Put-Call connection in different currency units. Example1.10.

The spot rate for yen denominated in pounds sterling is £0.005/U.A 3-month pound-denominated put option has strike price£0.0048/U and costs £0.0002.

Determine the premium in yen for an equivalent 3-monthyen-denominated call option with a strike of U2081

3 .

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Put-Call connection in different currency units. Example1.10. Solution.

Solution. Here the domestic currency = pounds and the foreigncurrency = yen. We have:

x0 = 0.005, T = 0.25, K = 0.0048, Pd(x0,K ,T ) = 0.0002

Note that 1/K = 20813 . We need to find Cf .

Note that Call-Put relationship in ”foreign” and ”domestic”currency:

Pf

(1

x0,

1

K,T

)=

1

Kx0Cd (x0,K ,T )

can be re-written as

Cf

(1

x0,

1

K,T

)=

1

Kx0Pd (x0,K ,T )

Hence,

Cf =1

0.0048 · 0.005· 0.0002 = 8

1

3≈ 8.33

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