Past exam questions
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Transcript of Past exam questions
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.
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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!
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Svar:
p = 51,1 % (1-p) = 48,9% P1 levande = 27,53 P1 död = 10 P1 levande > S1 död P0 = 13,07 kr
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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
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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
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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
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Options on Financial Assets
Executive Stock Options
Warrants
Convertible Bonds
Callable Bonds
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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.