VALUATION OF REAL OPTIONS – EXERCISES
REAL OPTIONS IN DECISION MAKING
Lauri Frank,
LTY 2012
EXERCISE 1
Call option
Asset value = 300
Exercise price = 300
u = 1,2
r = 0,1
t=2
1. Calculate the development of the asset’s value
2. Calculate the value of the option at maturity
3. Calculate the present value of the option using the portfolio method
EXERCISE 2
Possibility to upscale (e.g. add resources to a project)
Asset value = 200
Exercise price = 10
Added resources = 10%
u = 1,75
r = 0,1
t=2
1. Calculate the development of the asset’s value
2. Calculate the value of the option at maturity
3. Calculate the present value of the option using the portfolio method
EXERCISE 3
Possibility to abandon / exit (e.g. sell a project)
Asset value = 200
Compensation for exit = 175
u = 1,75
r = 0,1
t=2
1. Calculate the development of the asset’s value
2. Calculate the value of the option at maturity
3. Calculate the present value of the option using the portfolio method
EXERCISE 4
Calculate the value of an american put option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 200
Compensation = 175
Up movement = 1.75
Risk free interest rate = 10 %
Time to maturity = 2 years
EXERCISE 5
Calculate the value of an european call option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 1000
Strike price at time period 1 = 1250
Strike price at time period 2 = 1250
Up movement = 1.5
Risk free interest rate = 10 %
Time to maturity = 2 years
EXERCISE 6
Calculate the value of an european call option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 1000
Strike price at time period 1 = 1250
Strike price at time period 2 = 1250
Annual dividend = 10 % of asset value
Up movement = 1.5
Risk free interest rate = 10 %
Time to maturity = 2 years
EXERCISE 7
Calculate the value of an american call option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable
Underlying asset value = 75
Strike price at time period 1 = 100
Strike price at time period 2 = 175
Up movement = 1.9
Risk free interest rate = 10 %
Time to maturity = 2 years
EXERCISE 8
Calculate the value of an european call option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 75
Strike price at time period 2 = 130
Up movement at time period 1 = 1.5
Up movement at time period 2 = 1.8
Risk free interest rate = 10 %
Time to maturity = 2 years
EXERCISE 9
Calculate the value of an american call option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 100
Strike price at time period 1 = 110
Strike price at time period 2 = 150
Up movement at time period 1 = 1.2
Up movement at time period 2 = 1.4
Risk free interest rate at time period 1 = 10 %
Risk free interest rate at time period 2 = 20 %
Time to maturity = 2 years
EXERCISE 10
Calculate the value of an european put option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 105
Compensation at time period 2 =80
Up movement at time period 1 = 1.5
Up movement at time period 2 = 1.8
Risk free interest rate = 10 %
Time to maturity = 2 years
EXERCISE 11
Calculate the value of an american put option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 20
Compensation at time period 1 = 10
Compensation at time period 2 = 20
Up movement at time period 1 = 1.2
Up movement at time period 2 = 1.4
Risk free interest rate at time period 1 = 20 %
Risk free interest rate at time period 2 = 10 %
Time to maturity = 2 years
EXERCISE 12
Calculate the value of an european call option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 75
Strike price at time period 3 = 130
Up movement = 1.5
Risk free interest rate = 10 %
Time to maturity = 3 years
EXERCISE 13
Calculate the value of an american call option using the binomial method and
information given below. Point out the situations in which exercising the option is
profitable.
Underlying asset value = 100
Strike price at time period 1 = 110
Strike price at time period 2 = 150
Strike price at time period 3 = 150
Up movement = 1.5
Risk free interest rate = 10 %
Time to maturity = 3 years
EXERCISE 14
Calculate the value of the option using the Black & Scholes formula:
S = 100
K = 100
T-t = 4
r=6%
volatility = 30 %
EXERCISE 15
Calculate the value of the option using the Black & Scholes formula:
S = 200
K = 50
T-t = 9
r = 3,5 %
volatility = 50 %
EXERCISE 16
A company produces gizmos. After two years (at the third year) the company has the
possibility to dismiss its employees and shut down the production. Selling the
production machinery yields 175 m€ to the company. The value of producing gizmos
is currently 200 m€. Experts have evaluated that the production value can rise 38% or
27.5% per annum. The risk free interest rate is 10%.
a) What type of option is represented in the exercise?
b) Point out the situations in which shutting down the production is profitable.
c) What is the value of the possibility to shut down the production?
EXERCISE 17
A consumer electronics store makes a two year lease contract for its business
premises. The annual rent to be paid at the beginning of every year is 12 000 € which
is the current market rate. The volatility of market rents is 20% and the risk free
market interest rate is 4%. Every years market rent is determined at the beginning of
the year. After two years the store has a possibility to continue the lease contract by
one year with the same price.
a) What type of option is represented in the exercise?
b) Point out the situations in which continuing the lease agreement is profitable.
c) What is the value of the possibility to continue the lease agreement?
d) Calculate the NPV of rental costs. How does the possibility to continue the
lease agreement affect the rental costs?
EXERCISE 18
Make up real life stories for exercises 1-15.
ATTACHEMENT
FORMULAE AND CUMULATIVE NORMAL DISTRIBUTION
Vt
N
V0 = ∑
t =0
t
(1 + r )
Vt = e− rδ t ⎡⎣ wfu + (1 − w) f d ⎤⎦
(e
w=
rδ t
−d)
(u − d )
d=
1
u
uCRR = eσ
Δt
, dCRR =
PBS = -SN (−d1 ) − Ke
CBS = SN (d1 ) − Ke
1
= e −σ
u
− r (T −t )
− r (T − t )
Δt
, Δt =
T
N
N ( −d 2 )
N ( d2 )
⎡ ⎛ σ 2 ⎞ ⎤
ln ( S / K ) + ⎢ r + ⎜
⎟ ⎥ (T − t )
⎝ 2 ⎠ ⎦
⎣
d1 =
σ T −t
⎡ ⎛ σ 2 ⎞ ⎤
ln ( S / K ) + ⎢ r − ⎜
⎟ ⎥ (T − t )
⎝ 2 ⎠ ⎦
⎣
d2 =
= d1 − σ T − t
σ T −t
Kumulatiivinen normaalijakauma
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
2,0
2,1
2,2
2,3
2,4
2,5
2,6
2,7
2,8
2,9
3,0
0,00
0,500
0,540
0,579
0,618
0,655
0,691
0,726
0,758
0,788
0,816
0,841
0,864
0,885
0,903
0,919
0,933
0,945
0,955
0,964
0,971
0,977
0,982
0,986
0,989
0,992
0,994
0,995
0,997
0,997
0,998
0,999
0,01
0,504
0,544
0,583
0,622
0,659
0,695
0,729
0,761
0,791
0,819
0,844
0,867
0,887
0,905
0,921
0,934
0,946
0,956
0,965
0,972
0,978
0,983
0,986
0,990
0,992
0,994
0,995
0,997
0,998
0,998
0,999
0,02
0,508
0,548
0,587
0,626
0,663
0,698
0,732
0,764
0,794
0,821
0,846
0,869
0,889
0,907
0,922
0,936
0,947
0,957
0,966
0,973
0,978
0,983
0,987
0,990
0,992
0,994
0,996
0,997
0,998
0,998
0,999
0,03
0,512
0,552
0,591
0,629
0,666
0,702
0,736
0,767
0,797
0,824
0,848
0,871
0,891
0,908
0,924
0,937
0,948
0,958
0,966
0,973
0,979
0,983
0,987
0,990
0,992
0,994
0,996
0,997
0,998
0,998
0,999
0,04
0,516
0,556
0,595
0,633
0,670
0,705
0,739
0,770
0,800
0,826
0,851
0,873
0,893
0,910
0,925
0,938
0,949
0,959
0,967
0,974
0,979
0,984
0,987
0,990
0,993
0,994
0,996
0,997
0,998
0,998
0,999
0,05
0,520
0,560
0,599
0,637
0,674
0,709
0,742
0,773
0,802
0,829
0,853
0,875
0,894
0,911
0,926
0,939
0,951
0,960
0,968
0,974
0,980
0,984
0,988
0,991
0,993
0,995
0,996
0,997
0,998
0,998
0,999
0,06
0,524
0,564
0,603
0,641
0,677
0,712
0,745
0,776
0,805
0,831
0,855
0,877
0,896
0,913
0,928
0,941
0,952
0,961
0,969
0,975
0,980
0,985
0,988
0,991
0,993
0,995
0,996
0,997
0,998
0,998
0,999
0,07
0,528
0,567
0,606
0,644
0,681
0,716
0,749
0,779
0,808
0,834
0,858
0,879
0,898
0,915
0,929
0,942
0,953
0,962
0,969
0,976
0,981
0,985
0,988
0,991
0,993
0,995
0,996
0,997
0,998
0,999
0,999
0,08
0,532
0,571
0,610
0,648
0,684
0,719
0,752
0,782
0,811
0,836
0,860
0,881
0,900
0,916
0,931
0,943
0,954
0,962
0,970
0,976
0,981
0,985
0,989
0,991
0,993
0,995
0,996
0,997
0,998
0,999
0,999
0,09
0,536
0,575
0,614
0,652
0,688
0,722
0,755
0,785
0,813
0,839
0,862
0,883
0,901
0,918
0,932
0,944
0,954
0,963
0,971
0,977
0,982
0,986
0,989
0,992
0,994
0,995
0,996
0,997
0,998
0,999
0,999