Description

1. (20) One of the more complex options strategies is the so-called Ã¢â‚¬Å“iron condorÃ¢â‚¬Â portfolio, using a combinations

of long and short positions, call and put options, and four different strike prices. In this question we will

consider what would motivate traders to construct such a portfolio and the structure of the payoffs that they

receive from it. Suppose that stock in Hieroglyph Inc. is currently priced at St = 150 per share, with prices of

various options written on this stock given below.

Strike Call Price Put Price

120 35 5

140 20 10

160 10 20

180 5 35

(a) (10) Suppose that you are convinced that this stock is not as volatile as the market believes it to be,

but are unsure of the direction of future price movements. Construct a suitable portfolio to match your

beliefs, and plot the payoff diagram (you may assume a zero risk-free rate).

(b) (10) Now suppose that, while you still believe that extreme price movements in either direction are

unlikely, you wish to limit your downside risk just in case. Modify your portfolio to achieve this, and plot

the payoff diagram.

2. (60) Consider applying the binomial method to price American options. Suppose that stock in the MAGA

Hat Company is priced at S0 = 100 and consider options that are at the money K = 100. The options mature

in T = 1 year, which we divide into n = 3 periods, each of length t = T

n = 1

3 . The risk-free rate is r = 0.08

and MAGA stock has a volatility of = 0.45.

(a) (10) Calculate the magnitude of upticks u and downticks d, as well as the risk-neutral probability p.

(b) (10) Draw the stock price tree and calculate the payoffs to call and put options at each final price.

(c) (10) Using backward induction, calculate the price of European call and put options.

(d) (10) Verify that put-call parity holds for European options.

(e) (10) Find the price of American call and put options. Is it ever worthwhile to exercise early?

(f) (10) Does put-call parity hold for American options?

3. (20) Pick any stock and look up the current option prices. Pay particular attention to the implied volatility,

the volatility which, if plugged into the Black-Scholes pricing formula along with the other known parameters,

would yield the price of the option.

(a) (10) Choose an expiry date, and plot the implied volatility of call options against the strike price, ignoring

any implied volatilities of 0. What do you see, and what might explain this?

(b) (10) Now select at least three more expiry dates, and repeat the same plot for each. What happens to

implied volatility as time to maturity increases?

AS.640.440 Financial Economics

Assignment 4 – Options

1. (20) One of the more complex options strategies is the so-called Ã¢â‚¬Å“iron condorÃ¢â‚¬Â portfolio, using a combinations

of long and short positions, call and put options, and four different strike prices. In this question we will

consider what would motivate traders to construct such a portfolio and the structure of the payoffs that they

receive from it. Suppose that stock in Hieroglyph Inc. is currently priced at St = 150 per share, with prices of

various options written on this stock given below.

Strike

120

140

160

180

Call Price

35

20

10

5

Put Price

5

10

20

35

(a) (10) Suppose that you are convinced that this stock is not as volatile as the market believes it to be,

but are unsure of the direction of future price movements. Construct a suitable portfolio to match your

beliefs, and plot the payoff diagram (you may assume a zero risk-free rate).

(b) (10) Now suppose that, while you still believe that extreme price movements in either direction are

unlikely, you wish to limit your downside risk just in case. Modify your portfolio to achieve this, and plot

the payoff diagram.

2. (60) Consider applying the binomial method to price American options. Suppose that stock in the MAGA

Hat Company is priced at S0 = 100 and consider options that are at the money K = 100. The options mature

in T = 1 year, which we divide into n = 3 periods, each of length Ã¢Ë†â€ t = Tn = 13 . The risk-free rate is r = 0.08

and MAGA stock has a volatility of ÃÆ’ = 0.45.

(a) (10) Calculate the magnitude of upticks u and downticks d, as well as the risk-neutral probability p.

(b) (10) Draw the stock price tree and calculate the payoffs to call and put options at each final price.

(c) (10) Using backward induction, calculate the price of European call and put options.

(d) (10) Verify that put-call parity holds for European options.

(e) (10) Find the price of American call and put options. Is it ever worthwhile to exercise early?

(f) (10) Does put-call parity hold for American options?

3. (20) Pick any stock and look up the current option prices. Pay particular attention to the implied volatility,

the volatility which, if plugged into the Black-Scholes pricing formula along with the other known parameters,

would yield the price of the option.

(a) (10) Choose an expiry date, and plot the implied volatility of call options against the strike price, ignoring

any implied volatilities of 0. What do you see, and what might explain this?

(b) (10) Now select at least three more expiry dates, and repeat the same plot for each. What happens to

implied volatility as time to maturity increases?

1

AS.440.640 Financial Economics

Lecture 11

Zhou, Nan

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Options

I In contrast to forward and futures contracts, an option gives

the right, but not the obligation, to buy or sell shares at a

specified strike price at a specified maturity date

I Call – right to buy the underlying at strike

I Put – right to sell the underlying at strike

I Types of options

I European – may only be exercised at maturity

I American – may be exercised at any time before maturity

I Bermudan – may be exercised at specific dates before and at

maturity

I Lookback – exercise price is most favorable stock price over

period

I Asian – exercise price is average stock price over period

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Option Prices

I The strike price is an arbitrary value and options are available

at a range of strikes around the current price of the underlying

asset

I Because no one would exercise an option at a loss, they

always yield a non-negative payoff and hence a price must be

paid for this right

I Option prices for AAPL expiring 10/30/20, stock price 119.02

as of 10/18/20

Strike Call

Put

100

19.20 0.23

110

10.25 1.21

120

4.01

4.95

130

1.20 12.20

140

0.44 20.45

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Option Payoffs

I An optionÃ¢â‚¬â„¢s price at maturity is determined by its exercise

value

I Not really much of an Ã¢â‚¬Å“optionÃ¢â‚¬Â involved in a European call exercise if and only if the stock price at maturity exceeds the

strike price K

I Otherwise even if you still wanted to hold the stock youÃ¢â‚¬â„¢d be

better off just buying it at the market price instead of

exercising the option

(

ST Ã¢Ë†â€™ K ST > K

CT =

= (ST Ã¢Ë†â€™ K )+

0

ST < K
I Likewise for a put
(
0
PT =
K Ã¢Ë†â€™ ST
Zhou, Nan
ST > K

= (K Ã¢Ë†â€™ ST )+

ST < K
AS.440.640 Financial Economics Lecture 11
Long Option Payoff Diagram
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
Short Option Payoff Diagram
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
Option Profit Diagrams
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
Option Price Components
I An option always yields a non-negative payoff at maturity realize the gain from any favorable price movements but not
the losses from unfavorable ones
I For example, recall that AppleÃ¢â‚¬â„¢s stock price was St = 119.02
and the price of a call option at K = 110 was Ct = 10.25,
which can be decomposed into two components
I Intrinsic value - This option is in the money, if exercised today
it would yield a payoff of 9.02
I Time value - The remaining 1.23 comes from potential payoff
from future price increases
I On the other hand, the call at K = 130 is out of the money
and has zero intrinsic value, the price of 1.20 entirely reflects
the time value
I How does the volatility of the underlying stock price affect the
option price?
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
Options and Hedging
I The payoff structures of options can be used to create
portfolios that hedge some of the risk in the underlying
I For example, suppose you buy a stock and want to realize the
potential upside but need protection against the possible
downside, how can you achieve this?
I Buy both a share of stock and a put option at a strike price
representing the greatest loss youÃ¢â‚¬â„¢d be willing to accept
I Can exercise the option and limit your losses if price falls
below this strike, but realize the full gain if the price goes up
I Like any kind of insurance policy, this type of strategy has a
cost - the option price
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
Protective Put
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
Put-Call Parity
I Two portfolios with the same payoffs should have the same
price, suggesting a connection between call and put prices
I As always we establish this connection by a no arbitrage
argument
t
T (ST Ã¢â€°Â¥ K ) T (ST < K )
Position
Long stock
Ã¢Ë†â€™St
ST
ST
Long put
Ã¢Ë†â€™Pt
0
K Ã¢Ë†â€™ ST
Short call
Ct
K Ã¢Ë†â€™ ST
0
Ã¢Ë†â€™r
(T
Ã¢Ë†â€™t)
Borrow
Ke
Ã¢Ë†â€™K
Ã¢Ë†â€™K
Ã¢Ë†â€™r
(T
Ã¢Ë†â€™t)
Net
Ct + Ke
0
0
Ã¢Ë†â€™Pt Ã¢Ë†â€™ St
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
Put-Call Parity
I No arbitrage implies the put-call parity equation
Ct Ã¢Ë†â€™ Pt = St Ã¢Ë†â€™ Ke Ã¢Ë†â€™r (T Ã¢Ë†â€™t)
I If these were not equal, how could we exploit the inefficiency?
I As always buy the undervalued and sell the overvalued
I If Ct Ã¢Ë†â€™ Pt < St Ã¢Ë†â€™ Ke Ã¢Ë†â€™r (T Ã¢Ë†â€™t) - long call, short put, short stock,
put Ke Ã¢Ë†â€™r (T Ã¢Ë†â€™t) in the bank
I If Ct Ã¢Ë†â€™ Pt > St Ã¢Ë†â€™ Ke Ã¢Ë†â€™r (T Ã¢Ë†â€™t) – short call, long put, long stock,

borrow Ke Ã¢Ë†â€™r (T Ã¢Ë†â€™t)

I Note that this relation only holds for calls and puts with the

same strike, comparing options with different strikes requires

considering the distribution of the returns to the underlying

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Option Spreads

I By combining various options, traders can create portfolios

with a range of payoff structures

I For example, what if you wanted to limit your exposure to risk

in either direction?

I Buy insurance against downside risk to limit potential losses

I Sell upside risk for some payoff now in return for capping

potential gains

I Two calls with different strikes K 1 < K 2 , note that the option
with the lower strike has a higher price Ct1 > Ct2 , why?

I Bull spread – long call at K 1 , short call at K 2

I Bear spread – short call at K 1 , long call at K 2

I By put-call parity, these spreads can be equivalently

constructed with put options instead

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Vertical Spreads

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Option Pricing

I Option prices are determined by five parameters, four of which

are directly observable and the fifth unobservable

I

I

I

I

I

St – current stock price

K – strike price

T Ã¢Ë†â€™ t – time to expiration

r – risk free rate

ÃÆ’ – volatility of stock price

I As ÃÆ’ is the only input that cannot be directly measured, the

option prices set by the market can be taken as a measure of

the underlying stock volatility

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

One-Period Option Pricing Model

I Start with a basic example, suppose we have a stock whose

price today is S0 = 100

I Next period, the price will either go up by a factor of u = e ÃÆ’ ,

with probability p Ã¢Ë†â€” , or down by a factor of d = u1 , with

probability 1 Ã¢Ë†â€™ p Ã¢Ë†â€”

I If u = 1.25, then we have that

(

125 w/ prob. p Ã¢Ë†â€”

S1 =

80

w/ prob. 1 Ã¢Ë†â€™ p Ã¢Ë†â€”

I Suppose we have a European call option with strike price

K = 100, first consider the exercise value

(

25 w/ prob. p Ã¢Ë†â€”

C1 = (S1 Ã¢Ë†â€™ K )+ =

0

w/ prob. 1 Ã¢Ë†â€™ p Ã¢Ë†â€”

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Replicating Portfolio

I Consider a portfolio where we short the call and long 95 of a

share of the stock, what is our payoff at maturity?

(

69.444 Ã¢Ë†â€™ 25 w/ prob. p Ã¢Ë†â€”

5

S1 Ã¢Ë†â€™ C1 =

= 44.444

9

44.444

w/ prob. 1 Ã¢Ë†â€™ p Ã¢Ë†â€”

I Suppose that e r = 1.111, then we could obtain the same

payoff by simply putting 40 in the bank, so this must be the

price of our portfolio

5

5

S0 Ã¢Ë†â€™ C0 =

S1 Ã¢Ë†â€™ C1 e Ã¢Ë†â€™r = 40

9

9

I This implies that the price of the option must be

5

5

C0 = S0 Ã¢Ë†â€™

S1 Ã¢Ë†â€™ C1 e Ã¢Ë†â€™r = 15.556

9

9

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Hedge Ratio

I Where did we get that 59 from the previous slide?

I How much does the exercise value of the call option change

with the stock price?

Ã¢Ë†â€ C =

25 Ã¢Ë†â€™ 0

5

(S0 u Ã¢Ë†â€™ K )+ Ã¢Ë†â€™ (S0 d Ã¢Ë†â€™ K )+

=

=

S0 u Ã¢Ë†â€™ S0 d

125 Ã¢Ë†â€™ 80

9

I By definition, the delta of the stock itself is Ã¢Ë†â€ S = 1, so the

portfolio we constructed has Ã¢Ë†â€ = 0

I A delta neutral portfolio yields the same payoff in both states

and hence has no exposure to price risk, therefore its return

must equal the risk-free rate

Ã¢Ë†â€ C S0 Ã¢Ë†â€™ C0 = (Ã¢Ë†â€ C S1 Ã¢Ë†â€™ C1 )e Ã¢Ë†â€™r

I If the above equality didnÃ¢â‚¬â„¢t hold, what arbitrage opportunity

would be possible?

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Put-Call Delta Parity

I Now consider pricing a put option with K = 100

(

0

w/ prob. p Ã¢Ë†â€”

+

P1 = (K Ã¢Ë†â€™ S1 ) =

20 w/ prob. 1 Ã¢Ë†â€™ p Ã¢Ë†â€”

I Compute the hedge ratio

Ã¢Ë†â€ P =

4

0 Ã¢Ë†â€™ 20

= Ã¢Ë†â€™ = Ã¢Ë†â€ C Ã¢Ë†â€™ 1

125 Ã¢Ë†â€™ 80

9

I Not a coincidence that Ã¢Ë†â€ C Ã¢Ë†â€™ Ã¢Ë†â€ P = 1, what is the payoff to a

portfolio of a long call and short put?

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Pricing a Put Option

I Once again constructing a delta neutral portfolio, we compute

the price of a put option

(

55.556

w/ prob. p Ã¢Ë†â€”

P1 Ã¢Ë†â€™ Ã¢Ë†â€ P S1 =

= 55.556

20 + 35.556 w/ prob. 1 Ã¢Ë†â€™ p Ã¢Ë†â€”

P0 = (P1 Ã¢Ë†â€™ Ã¢Ë†â€ P S1 )e Ã¢Ë†â€™r + Ã¢Ë†â€ P S0 = 5.556

I Equivalently we may solve for the price using put-call parity

P0 = C0 Ã¢Ë†â€™ (S0 Ã¢Ë†â€™ Ke Ã¢Ë†â€™r ) = 15.556 Ã¢Ë†â€™ 100(1 Ã¢Ë†â€™ 0.9) = 5.556

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Risk Neutral Pricing

I LetÃ¢â‚¬â„¢s pretend that investors are risk neutral, what is the

perceived probability p of an uptick?

I Find the value for p such that investing in the stock has no

risk premium

S0 = E [S1 e Ã¢Ë†â€™r ] = [p(S0 u) + (1 Ã¢Ë†â€™ p)(S0 d)]e Ã¢Ë†â€™r

er Ã¢Ë†â€™ d

Ã¢â€°Ë† 0.691

p=

uÃ¢Ë†â€™d

I What is the price of an option under these probabilities?

C0 = E [(S1 Ã¢Ë†â€™ K )+ e Ã¢Ë†â€™r ] = p(S0 u Ã¢Ë†â€™ K )e Ã¢Ë†â€™r Ã¢â€°Ë† 15.556

P0 = E [(K Ã¢Ë†â€™ S1 )+ e Ã¢Ë†â€™r ] = (1 Ã¢Ë†â€™ p)(K Ã¢Ë†â€™ S0 d)e Ã¢Ë†â€™r Ã¢â€°Ë† 5.556

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Risk Neutral Probabilities

I Using the risk neutral pricing method, we obtain the same call

and put option prices as before

I Note that we do not actually expect investors to be risk

neutral, what if instead they demand some return ÃŽÂ± > r ?

I Implies that probability of an uptick is some other value p 0

such that

S0 = E [S1 e Ã¢Ë†â€™ÃŽÂ± ] = [p 0 (S0 u) + (1 Ã¢Ë†â€™ p 0 )(S0 d)]e Ã¢Ë†â€™ÃŽÂ±

I If ÃŽÂ± = log 1.2, then

p0 =

eÃŽÂ± Ã¢Ë†â€™ d

Ã¢â€°Ë† 0.889

uÃ¢Ë†â€™d

I If ÃŽÂ± is the discount rate used for the underlying stock, what is

the discount rate for the option?

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Replicating Portfolio

I A call option can be replicated with a portfolio consisting of

Ã¢Ë†â€ C shares of stock and the risk free asset

Ã¢Ë†â€ C S0 Ã¢Ë†â€™ C0 = (Ã¢Ë†â€ C S1 Ã¢Ë†â€™ C1 )e Ã¢Ë†â€™r Ã¢â€°Â¡ BC

C0 = Ã¢Ë†â€ C S0 Ã¢Ë†â€™ BC

I Letting ÃŽÂ³ be the expected return for the call option, we have

that

eÃŽÂ³ =

E [C1 ]

Ã¢Ë†â€ C E [S1 ] Ã¢Ë†â€™ BC e r

Ã¢Ë†â€ C S0 e ÃŽÂ± Ã¢Ë†â€™ BC e r

=

=

C0

Ã¢Ë†â€ C S0 Ã¢Ë†â€™ BC

Ã¢Ë†â€ C S0 Ã¢Ë†â€™ BC

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Replicating Portfolio

I In this example we have Ã¢Ë†â€ C =

that

eÃŽÂ³ =

5

ÃŽÂ±

r

9 (100)e Ã¢Ë†â€™ 40e

5

9 (100) Ã¢Ë†â€™ 40

=

5

9

and BC = 40, this implies

25 6 18 10

10

Ã¢Ë†â€™

=

Ã¢â€°Ë† 1.429

7 5

7 9

7

I The option price is given by

C0 = E [(S1 Ã¢Ë†â€™ K )+ e Ã¢Ë†â€™ÃŽÂ³ ] = p 0 (S0 u Ã¢Ë†â€™ K )e Ã¢Ë†â€™ÃŽÂ³ = 15.556

I Obtain the same option price no matter what probabilities we

used

I Advantage of risk neutral probabilities is that we have

ÃŽÂ³ = ÃŽÂ± = r , so that all assets here have the same expected

return

Zhou, Nan

AS.440.640 Financial Economics Lecture 11

Week 11 Script

1. This week weÃ¢â‚¬â„¢ll start talking about options.

2. If you remember from last week, we talked about how forwards and futures allow you to buy an

asset in the future at a price determined today. And while locking in the price today might be useful

if you expect the price to go up, what if the price actually went down? Then because a futures

contract obliges you to follow through, youÃ¢â‚¬â„¢d be stuck paying more. What you really want is an

asset that gave you the option of buying if you wanted to, but didnÃ¢â‚¬â„¢t oblige you to if the price moved

the other way. To go over some of the basic terminology, a call option gives you the right to buy

and a put option gives you the right to sell some underlying asset at a predetermined price, known

as the strike price. There are also a variety of different types of options. European options can only

be exercised at the specified maturity date, while American options can be exercised any time

before maturity. Note that these labels donÃ¢â‚¬â„¢t actually reflect different standards across continents,

because both of types are traded everywhere. Beyond these, there are also various types of exotic

options, a Bermudan option is somewhat in between American and European options, just like

Bermuda itself is between America and Europe, because it gives a set of dates on which you can

exercise the option. A lookback option looks back throughout the life of option and finds the most

favorable price at which to exercise, while an Asian option takes the average stock price over the

period.

3. Unlike a futures contract, the strike price is arbitrary and doesnÃ¢â‚¬â„¢t need to be related to the current

price of the underlying. Since you would never willingly exercise an option for a loss, the payoff

could be positive but is never negative, there is some inherent value to holding it and so, unlike a

futures contract, you need to pay some price up front to buy one, even if you donÃ¢â‚¬â„¢t end up

exercising it. If we look at a couple of different options for Apple stock, you can see that call options

are worth more when the strike price is low, while put options are worth more when the stock price

is high. A call option with a strike of 100 is quite valuable because itÃ¢â‚¬â„¢s unlikely the stock price will dip

below 100 so youÃ¢â‚¬â„¢re almost guaranteed to be exercising it, whereas a call with a strike of 140 is

worth very little, since thereÃ¢â‚¬â„¢s very little chance the stock price will go that high so you probably

wonÃ¢â‚¬â„¢t get an opportunity to exercise it.

4. Despite what the name says, options donÃ¢â‚¬â„¢t give you much a meaningful choice, the right thing to do

is always to exercise when youÃ¢â‚¬â„¢re in the money, and decline when youÃ¢â‚¬â„¢re out of the money. So for a

call option, you exercise if the stock price at maturity S_T is higher than the strike price K, otherwise,

even if you wanted to hold the stock youÃ¢â‚¬â„¢d be better off buying it in the spot market instead. So

your payoff is S_T-K if S_T>K, or 0 if itÃ¢â‚¬â„¢s not, and we can write this as (S_T-K)+, an operation that

means we take the value inside the parentheses if itÃ¢â‚¬â„¢s positive, and 0 if itÃ¢â‚¬â„¢s not. On the other hand,

for a put, you want to exercise if S_T

Purchase answer to see full

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