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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 ÃÆ’ = 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
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
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
Zhou, Nan
AS.440.640 Financial Economics Lecture 11
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
ÃÆ’ – volatility of stock price
I As ÃÆ’ 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 ÃÆ’ ,
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
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