Econ 141 (Winter, 2016) - Lecture 6

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.

Today’s Class

Finish Put-Call ParityStocks with dividendsExamples from Chapters 3 and 9

Chapter 3(continued)


Synthetic Forwards• A synthetic long forward is

created when you buy a call and sell a put on the same underlying asset, with the same strike price and time to expiration

• Example: buy the $1,000-strike S&R call and sell the $1,000-strike S&R put, each with 6 months to expiration

• At expiration, we pay the strike price ($1000) to own the asset (see page 69)

-$95.68 profit

+$75.68 profit

+$1,000 index price

+$1,020 index price

Figure 3.6(compare to Figure 2.10)

lecture 4 review


Synthetic Forwards (continued)• Purchase a forward contract

• at t=0 we pay zero premium• at expiration (t),we pay the forward price (F)• So the present value of the cost is = PV(F0,t)

• Purchase a synthetic forward• at t=0, as shown on the previous page, to buy a synthetic forward

requires that we pay the net option premium (i.e. the difference between premiums for the purchased call and the written put)

• at expiration (t), we pay the strike price (K)•So the present value of the cost is = Call(K,t) – Put(K,t) + PV(K)

lecture 4 review

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved. 3-

Put-Call Parity• The net cost of buying the index using options must equal

the net cost of buying the index using a forward contract, thus

Call(K,t) – Put(K,t) = PV(F0,t – K)

• Call(K,t) and Put(K,t) denote the premiums of options with strike price K and time t until expiration

• PV(F0,t) is the present value of the forward price• PV(K) is the present value of the strike price

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Examples of Put-Call Parity (1/4)• 3.2 Suppose you short the S&R Index for $1000, short a 1000-strike put

which has a premium of $74.201. The risk-free rate is 2%.–Construct a table in the format of Table 3.1, which summarizes the payoff and profit of this position. (how do your numbers compare to those in Table 3.1?)

–Verify that your numbers matches those in Figure 3.5

Figure 3.5

lecture 4 review

Chapter 9Put-Call Parity


IBM Option Quotes

• What trends do you see?• Are October options always > June options for a given K?• Does C always decrease with increasing K?• Does P always increase with increasing K?• Do Premiums always change by less than the change in K?


A quick note about continuous compounding

• First see Appendix in back of the book, page 854 …

• … from this understanding, and our original Put-Call Parity formula: Call(K,t) – Put(K,t) = PV(F0,t – K), we can now write the following for continuous compounding

• PV(F0,t) = e-rT F0,T = S0 – PV0,T (Div)• PV(K) = e-rT K • and the Parity relationship at t=0 becomes …• Call(K,t) – Put(K,t) = S0– e-rT K


Call – Put = S0– e-rT K• Example 9.1

– Price of a non-dividend-paying stock = $40 with r=8%– option strike price = $40– time to expiration = 3 months– European call = $2.78– European put = $1.99

• $2.78 - $1.99 = $40 – $40e-0.08x0.25

– Yes, this is consistent with the Put-Call Parity Equation

Why is the Call $0.79 more expensive than the Put?

Examples of Put-Call Parity (2/4)


Why is the Call $0.79 more expensive than the Put?

– Both an Outright purchase of Stock and the synthetic purchase (i.e. buy call, sell put) gives you 1 share on day 91

– With method 2, we own the stock economically, but don’t pay fully until day 91– This deferral of payment is worth 3 month’s interest on $40 to us– (3 months interest on $40) = $40e+0.08x0.25 - $40= $0.81– PV(3 months interest on $40) = e-0.08x0.25$0.81 = $0.79– The option premiums differ by interest on the deferred payment for the stock

Compare Cash Flow Timing


What happens if we have dividends?

• The holder of the Forward contract or option does not receive the dividends because they do not yet hold the underlying asset itself

• If the underlying asset is a stock and PV0,T(Div) is the present value of the dividends payable over the life of the option, then

• PV(F0,t) = e-rT F0,T = S0 – PV0,T (Div) and the Parity relationship at t=0 becomes …Call – Put = S0– PV0,T (Div) – e-rT K


Call – Put = S0– PV0,T (Div) – e-rT KExample 9.1

• Price of a non-dividend-paying stock: $40, r=8%, option strike price: $40, time to expiration: 3 months, European call premium = $2.78, European put premium = $1.99. Then, by Put-Call Parity we can write:

• $2.78 - $1.99 = $40 – $40e-0.08x0.25

Example 9.2 Same as 9.1, but now the stock pays $5 dividend just before expiration• We are told the call premium = $0.74, and the put premium = $4.85.• Thus, by Put-Call Parity we can write:• $0.74 - $4.85=($40 – $5e-0.08x0.25) – $40e-0.08x0.25

• This is consistent with the Put-Call Parity Equation• The Call price is higher than the Put price by interest on the strike ($0.79) and lower

by the PV of the dividend ($4.90) for a net difference = -$4.11




Problem 9.2• A stock currently sells for $32.00. A 6-month call option

with a strike (K) of $30.00 has a premium of $4.29, and a 6-month put with the same strike has a premium of $2.64. Assume a 4%, continuously compounded, risk-free rate.

• Question - What is the present value of dividends payable over the next 6 months?

• Answer - this is a direct application of Put-Call Parity

Chapter 9 Solutions are now onlineWork through them on your own and ask questions as needed

To create a synthetic stock, using the Put-Call Parity “Recipe” we need to

– Buy the call & Sell the put– Buy T-bills with a value [PV(div) + Ke-rT] (i.e. invest at the risk-free rate)

S0 = Call – Put + PV0,T (Div) + e-rT K

• interpretation: In addition to buying the call and selling the put (which gives a synthetic long) we also need to lend (buy a bond) of an amount = PV(dividends + strike)

Put-Call Parity gives us a Recipe to Create Synthetic Instruments

NOTE - The sign convention is as follows: Long Position (+) and Short Position (-)


To create a synthetic long call:

– Buy the put– Buy the stock– Sell T-bills with a value Ke-rT (borrow at the risk-free rate)

To create a synthetic short call:

– Sell the put– Sell the stock – Buy T-bills with a value Ke-rT (invest at the risk-free rate)

K

K

Synthetic Call


To create a synthetic long put:

– Buy the call– Sell the stock– Buy T-bills with a value Ke-rT (invest at the risk-free rate)

To create a synthetic short put:

– Sell the call– Buy the stock – Sell T-bills with a value Ke-rT (borrow at the risk-free rate)

K

K

Synthetic Put


To create a synthetic T-bill, using the Put-Call Parity “Recipe” we need to

–Sell the call and Buy the put–Buy the stock

S0 - Call + Put = PV0,T (Div) + e-rT K

interpretation: We have purchased the stock and sold the synthetic stock. This produces an instrument with no risk

• The RHS tells us this instrument costs PV of the dividends and strike and pays K + FV(div) at expiration with certainty

• interpretation = This is a synthetic T-bill because it has no risk with a positive payment at expiration

Synthetic T-bills


NOTE - The sign convention is as follows: Long Position (+) and Short Position (-)


Summary of Parity Relationships

For Next Class• Reading

– finish Chapter 9– Chapter 10, Sections 10.1, 10.2

• Practice problems from the book

– continue to check for updates to the Word doc online, more problems are added as we cover new material

Econ 141 (Winter, 2016) - Lecture 6

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