Past exam questions

723g28Financial economics

2012

2

Example: from past exam

• The current price of the stock of AstraZeneca is 300 kr. During each twelve-month period it will either rise by 25 % or fall by 20 %. The interest rate is 3 % a year. Assume no dividends during the life of the option.

• Calculate the value of a two-year American put option on AstraZeneca with an exercise price of 250 kr.

3

Option valuation

• S=300 u=25% d=-20% rf =3%, no dividend,• Value of a 2 year American put option at strike

price of 250?

300

375

240300

468,75

192

P=23%/45%=51,1111%1-p=48,8889%

Put at strike =250

0

0

58

0

10

27,529

13,066

The value of the call is 13,07$, 27,53> intrinsic value 10, don’t exercise!

4

Svar:

p = 51,1 % (1-p) = 48,9% P1 levande = 27,53 P1 död = 10 P1 levande > S1 död P0 = 13,07 kr

5

S=300, Strike price=250. rf=0,03, u=25%, d=-20%

375

240300

192

• The value of the share in 3 periods

300

468,75585,93

375

240

153,6

6

Obs: American put can not have less than intrinsic value, option should be exercised at t+2!

0

0

10

96,4

0

4,7465

50,72

26,43

2,25313,66

K-S=10K-S=58

tt+1 t+2

• S=300 u=25% d=-20% rf =3%, no dividend,• Value of a 3 year American put option at strike

price of 250?

What about a 3 year american option value? The same method:

t+3

7

Value of an American call with 3 period

335,94

0

0

125

226,0315

62,028

0

141,604

103,6197

The last period value S-K, K=250

Use (p*Cu+(1-p)*Cd)/(1+r)=C(t-1)

Note, the american call can’t not be lower than the intrinsic value, otherwise it is an exercise point.

S-K=218,75S-K= 125

S-K=50

31,703

Exempel 2: en amerikansk säljoption utan utdelning

Du är innehavare av en 1-årig amerikansk säljoption utan utdelning under löptiden. Aktiens marknadsvärde är idag 100 kr och den kan under varje 6-månadersperiod antingen falla med 10 % eller stiga med 11,1 %. Säljoptionens lösenpris är 102 kr och den riskfria 6-månadersräntan är 5 %.

Vad är din amerikanska säljoption utan utdelning värd idag?

Put option at strike price =102• P=(5%+10%)/(11,1%+10%)=71,09%• 1-p=28,91%• Start from last period: K-S• (2*71,09%+21*28,91%)/1,05=7,136• Etc.

100111,1

90100

81

123,43

0

2

21

7,1360

0,55062,3376

K-S=12

American put option can not be lower than the intrinsic value. Option should be exercised!

Exam questions:Value the following options:

A European call option written on SKF A selling for 145 kr. The exercise price is 140 kr. The stock’s yearly volatility is 30 %. The option matures in 6 months. The risk-free yearly interest rate is 3 %. A European put option written on the same stock at the same time, with the same EX and expiration date. What is the time value of the call option?

Call option price=?

S=145, K=140, rf=3%, σ=30%T=1/2Get column value=P0/PV(K) Row value= σ* table value= C0/P0

Row value= 30%*=0,212 Column value=145/(140/(1,03)1/2 )=1,05125=1,05 C0/table value=P0 (table value=(10,9+9,88)/2)=10,39% C0=P0/table value= 145*10,39%= 15,07

• Put option value is:

• P=c+PV(K)-S=15,07+140/(1+0,03)1/2 -145=8,0

• Time value of the call=call premium-(intrinsic value) =15,07-(145-140)=10,07

Black & Scholes metod

• En ”genväg” för att beräkna en köpoptions värde med hjälp av Black & Scholes metod:– Beräkna radvärde = σ * √ t– Beräkna kolumnvärde = P0 / PV(X)– Se i tabell 6 för att utläsa optionens värde i procent av underliggande

tillgång (tabellvärde)– Köpoptionens värde (C0) = tabellvärde * P0

– Du får ut säjoptionens värde genom Put-Call parity

14

Options on Financial Assets

Executive Stock Options

Warrants

Convertible Bonds

Callable Bonds

15

Replicating the call

• The price of the SCA A stock is 100 kr. During the next year the price may either rise by 33 % or fall by 25 %. The yearly interest rate is 3 %. You have an European one-year call option on SCA A with an exercise price of 120 kr.

• Use the replicating-portfolio method to value this call option!

Strike=120, rf=3%• ∆=(13-0)/(58)=22,41%, • B= -∆*Sd/(1+0,03)=-22,44%*75/1,03=-16,34∆*Su+B(1+r)=13 ∆=(13-0)/(133-75)=22,41%

∆*Sd+B(1+r)=0 B=-16,34 C=100*22,41%-16,34=6,07

100

133

75

C=13

0

NPV of lease agreement

NLT needs a new forklift. It can either buy it for 2 500 000 kr or lease it. The lease terms require NLT to make 3 annual payments of 1 000 000 kr. The lessor can depreciate the forklift for tax purposes over 3 years. NLT can borrow at 6 %. NLT and the lessor pays tax at 30 %. What is the NPV of the lease for NLT? Is it possible to create a financial lease that has a positive value for both the lessor and NLT? Explain!

The depreciation tax shield is a forgone benefit of leasing, cash outflow.

t0 t1 t2 t3

IKF 2,5 SA (foregone tax benefit of depreciation) -0,25 -0,25 -0,25

SA = 2,5/3 * 0,3 = 0,25

LA (rental) -1 -1 -1

SL (skatte avdrag) 0,3 0,3 0,3 After tax leasing payment: -0,7

Summa 1,8 -0,95 -0,95 -0,25

Värde leasingavtal 1,8 -0,9117 -0,8783 -0,2210 -0,211

Svar: Leasingavtalet är ej finansiellt lönsamt för leasetagaren NLT

Diskonteringsränta = 6 % * 0,7 = 4,2 %

Discounting the after tax lease payment and the lost depreciation tax shield+ the benefit of renting the machine

Alternatively,Spread the cost of 2,5 million over the three years.

• 2,5=a*(1/r*(1-1/(1+r)^3)• a=2,5/2,765=0,9042 (Max value for the

company)• 0,25 yearly forgone benefit of depreciation, • 0,7294 yearly cost+0,25>0,9042 n• it is a negative NPV lease contract.