BREHIMA KOUMARE (20-342-267)

DESPOINA TSAKIRIDOU (19-309-582)

ROLAND DE PURY VARGAS (15-311-608) Geneva, October 26, 2021
ZILDETE LANDIM CRAUSAZ (18-308-528)

University of Geneva Prof. BORIS NIKOLOV

GSEM Asst. HONGZHE SHAN
Asst. YE ZHANG

ADVANCED CORPORATE FINANCE

PROBLEM SET 3

TABLE OF CONTENT

page

Problem 1 2

Problem 2 6

Problem 3 6

Problem 4 7

Problem 5 8

Problem 6 9
 PROBLEM 1

1. You own a call option on Intuit stock with a strike price of K = $40. The
option will expire in exactly three months’ time (T = 3/12) .
1.a. If the stock is trading at $55 in three months (XT = $55), the payoff of
the call will be

max (XT − K , 0) = max (55 − 40 , 0) = max (15 , 0) = 15 $

1.b. If the stock is trading at $35 in three months (XT = $35), the payoff of
the call will be

max (XT − K , 0) = max (35 − 40 , 0) = max (−5 , 0) = 0 $

1.c. The payoff of the long call at expiration is given by the formula

Cash F low = max (XT − 40 , 0)

With R we obtain the graph below :

2
 2. Assume that you have shorted the call option with the above characteristics.
2.a. If the stock is trading at $55 in three months (XT = $55) , then you will
owe

− max (XT − K , 0) = − max (55 − 40 , 0) = − max (15 , 0) = −15 $

2.b. If the stock is trading at $35 in three months (XT = $35), then you will
owe

− max (XT − K , 0) = − max (35 − 40 , 0) = − max (−5 , 0) = 0 $

2.c. The payoff of the short call at expiration is given by the formula

Cash F low = − max (XT − 40 , 0)

We obtain the following payoff diagram with R :

3. You own a put option on Ford stock with a strike price of K = $10 . The
option will expire in exactly six months’ time (T = 6/12) .

3
 3.a. If the stock is trading at XT = $8 in six months, then the payoff of the
put will be

max (K − XT , 0) = max (10 − 8 , 0) = max (2 , 0) = 2 $

3.b. If the stock is trading at XT = $23 in six months, then the payoff of the
put will be

max (K − XT , 0) = max (10 − 23 , 0) = max (−13 , 0) = 0 $

3.c. The payoff of this long put at expiration is given by the formula

Cash F low = max (10 − XT , 0)

We obtain the following payoff diagram with R :

4. Assume that you have shorted the put with the above characteristics.

4
 4.a. If the stock is trading at $8 in six months (XT = $8) , then the payoff of
the put at expiration is

− max (K − XT , 0) = − max (10 − 8 , 0) = − max (2 , 0) = −2 $

4.b. If the stock is trading at $23 in six months (XT = $23) , then the payoff
of the put at expiration is

− max (K − XT , 0) = − max (10 − 23 , 0) = − max (−13 , 0) = 0 $

4.c. The payoff of this long put at expiration is given by the formula

Cash F low = − max (10 − XT , 0)

We obtain the following payoff diagram on R :

5
 PROBLEM 2

According to the call/put parity relation, we have

Pt + St = Ct + K × (1 + r)−(T −t)

with
Pt = $2.10 ; St = $33 ; K = $35 ; r = 10% ; T − t = 1
Hence, the price Ct of a one year european call option on Dynamic’s stock is
given by the formula

Ct = Pt + St − K × (1 + r)−(T −t)
361
= 2.10 + 33 − 35 × (1 + 0.1)−1 = ' 3.282 $
110

PROBLEM 3

We have

Ct − Pt − St + K × (1 + r)−(T −t) > 0 = 7 − 3.33 − 20 + 18 × (1 + 0.08)−1

More precisely,

Ct − Pt − St + K × (1 + r)−(T −t) = 7 − 3.33 − 20 + 18 × (1 + 0.08)−1 ' 0.3367

Thus, the call/put parity relation is not satisfied. It may be that the call is
overpriced, or that the put is underpriced.
Anyway we can take advantage of this arbitrage opportunity by constituting a
portfolio, short the call option, long the put option, long in the underlying stock,
short a zero coupon risk free debt with a 1 year maturity and face value K = 18.
The present value of this debt is 18 × (1 + 0.08)−1 ' $16.67.
With this portfolio we get an immediate gain of 0.3367 $, with zero cash flow
at maturity.

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 PROBLEM 4

By definition, a butterfly spread is a portfolio that is long two call options

with different strike prices K1 and K2 , and short two call options with a strike
price equal to K3 = (K1 + K2 ) /2, all these options having the same underlying.
The payoff of a butterfly spread is given by the formula:

Butterf ly Spread P ayof f =

= max (XT − K1 , 0) + max (XT − K2 , 0) − 2 × max (XT − K3 , 0)

But, for all real numbers X and K, we have

max (X − K , 0) = X − K + max (0 , K − X) = X − K + max (K − X , 0)

Thus

Butterf ly Spread P ayof f =

= XT − K1 + max (K1 − XT , 0) + XT − K2 + max (K2 − XT , 0)
−2 × (XT − K3 + max (K3 − XT , 0))
= 2K3 − K1 − K2 + max (K1 − XT , 0) + max (K2 − XT , 0)
−2 max (K3 − XT , 0)
= 2 K1 +K
2
2
− K1 − K2 + max (K1 − XT , 0) + max (K2 − XT , 0)
−2 max (K3 − XT , 0)
= max (K1 − XT , 0) + max (K2 − XT , 0) − 2 max (K3 − XT , 0)

This result proves that a butterfly spread is equivalent to a portfolio that is

long two put options with different strike prices K1 and K2 , and short two put
options with a strike price equal to K3 = (K1 + K2 ) /2, all these options having
the same underlying.
By generalizing, we can also say that a butterfly spread is a portfolio that is
long two put options with different strike prices K1 and K2 , K1 < K2 , and short
two put options with a strike price equal to K3 , K1 < K3 < K2 , all these options
having the same underlying.

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 PROBLEM 5

Consider the following portfolio: long position in the underlying asset, short
position in a call with strike price K2 , long position in a put with strike price K1 ,
K1 < K2 . The notations T and XT are the expiration date and the underlying
asset price at T . The payoff of this portfolio is given by the formula:
Collar = XT − max (XT − K2 , 0) + max (K1 − XT , 0)

 K1 if XT ≤ K1
= XT if K1 ≤ XT ≤ K2
K2 if XT ≥ K2


The buyer of such a portfolio takes a long position in the underlying asset.
He protects this position on the downside by buying a put. But buying a put is
costly, and he chooses to finance this purchase by selling a call on the underlying
asset, giving up some of the upside potential.
The Collar payoff profile can be drawn with the R package. For K1 = 20 and
K2 = 60, we obtain the following graph :

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 PROBLEM 6
a) The present value of the Wesley’s Equity is
E0 = $25 × 20 million = 500 million $
The present value of the Wesley’s debt is
D0 = (Debt/Equity ratio) × E0 = 0.5 × 500 = 250 million $
The payoff of the Wesley’s debt at maturity in 5 years is
D5 = (1 + rD )5 × D0 = (1 + 0.1)5 × D0 = 402.6275 million $
Thus, the payoff of the Wesley’s equity in 5 years will be
E = max V5L − D5 , 0

where V5L is the Wesley’s assets value in 5 years.

Consequently, Wesley’s equity can be described as a call option on the Wesley’s
assets, with a 5 year maturity and strike price D5 = 402.6275 million $. The
underlying of this call option is the Wesley’s assets, which market present value
is
V0L = E0 + D0 = 500 + 250 million $ = 750 million $
b) The payoff of the debt in 5 years is

D5 if V5L ≥ D5 L


payof f of the debt = = min V , D5
V5L if V5L < D5 5

V5L + min D5 − V5L , 0 = V5L − max V5L − D5 , 0

=
We deduce that Wesley’s debt can be described as a portfolio, long the Wesley’s
assets and short the firm’s equity call option in a).
c) The payoff of the debt in 5 years can also be expressed as follows :

D5 if V5L ≥ D5 L


payof f of the debt = = min V , D5
V5L if V5L < D5 5

D5 + min V5L − D5 , 0 = D5 − max D5 − V5L , 0

=
Thus Wesley’s debt can be described as a portfolio, long a zero coupon risk
free debt with a 5 year maturity and face value D5 , and short a put on Wesley’s
assets with a 5 year maturity and strike price D5 = 402.6275 million $.