Prof. Dr. Karl Ludwig Keiber
Lehrstuhl für Betriebswirtschaftslehre,
insb. Finance
Corporate Finance
SS 2021 — Tutorial #6
Problem 1
Actual date is t = 0. A company is engaged in a project with uncertain future value and has the
permanent option to abandon this project. In case of abandonment the company retains a salvage value
which amounts to I = 800. The option to abandon expires at T = 3. The value of the project without
ωφ
the option to abandon obeys Vt+1
= ω · Vtφ where ω ∈ {u, d} and φ denotes the state of nature at date t.
The value of project with the permanent abandonment option is given by
i
h ωφ
}.
F Vtφ = max{I; P V Et F Vt+1
Let πuQ = 0.75, V2ud = 880, V0 = 1000,andhu =
A default free bond exists for investment at all
1.1. i
ωφ
dates. Note that at expiration date P V ET F VT +1 = VTφ .
a. Explain concisely the meaning of the equation P V
h i
= VTφ .
ET F VTωφ
+1
b. Determine (i.) the value of the project with the permanent abandonment option at all dates and
in all states, (ii.) the value of the right to abandon, and (iii.) the optimal abandonment policy.
c. Assume that the abandonment option can be exercised solely at date T = 3. Determine (i.) the
value of the project with the abandonment option, (ii.) the value of the right to abandon, (iii.) the
optimal abandonment policy, and (iv.) the value of the flexibility to abandon at dates t ∈ {0, 1, 2}
additionally.
1
Prof. Dr. Karl Ludwig Keiber
Lehrstuhl für Betriebswirtschaftslehre,
insb. Finance
Corporate Finance
SS 2021 — Tutorial #5
Problem 1
Presume the validity of the Black and Scholes (1973) option pricing model. Actual date is t = 0. The
value of a company is a geometric Brownian motion and V0 = 250. The variance rate of the instantaneous
return of the company’s value is σ2 = 20%. The company is mixed debt–equity financed. The debt’s face
value (zero bond) amounts to F = 200 and its maturity date is T = 2. Finally, the one period risk free
rate of return is RF = 5%.
a. Determine the value of the company’s equity if the company were all equity financed. Explain your
answer.
b. Determine the value of debt if it were risk free.
c. Determine the value of the shareholder’s default put.
d. Determine the value of the equity.
e. Determine the value of the debt.
f. Determine the risk neutral default probability.
Problem 2
ωφ
= ω · Vtφ where ω ∈ {u, d} and φ denotes the state of nature
The value of a levered company obeys Vt+1
at date t. The debt outstanding (zero bond) matures at date T = 3 and has face value F = 600. The
shareholders have the right to repay the debt prematurely. More precisely, the value of equity is given by
i
h ωφ
}
E Vtφ = max{Vtφ − F ; P V Et E Vt+1
prior to maturity (t < T ) and E VTφ = max{VTφ − F ; 0} at maturity date. The company’s per period
cost of capital are 9.25% . Furthermore, actual date is t = 0, V0 = 1000, u = 1.25 = d−1 , and RF = 5%.
a. (i.) Determine the value of the equity at all dates and in all states. (ii.) Determine the shareholders’
optimal default policy. (iii.) Rationalize the shareholders’ optimal default policy.
b. (i.) Determine the per period equity cost of capital at all dates prior to maturity date and in all
states. (ii.) Discuss the equity cost of capital compared to the company’s cost of capital. (iii.)
Explain the state-contingency of the equity cost of capital.
c. Determine (i.) the value of an American–type call option on the company’s equity with expiration
date T = 3 and exercise price X = 400 at all dates and in all states, (ii.) the optimal exercise
policy, and (iii.) the hedge portfolios of equity and call options (i.e. the percentage of equity per
call option) at all dates prior to expiration date and in all states.
1
Cumulative Standard Normal Distribution
The table displays values of the integral
N (z) ≡
Z
for
−∞
z ≥ 0. If z < 0 then N (z) ≡ 1 − N (−z).
1 − N (1.28) = 1 − 0.899727 = 0.100273.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
2
2,1
2,2
2,3
2,4
2,5
2,6
2,7
2,8
2,9
3
0
0,500000
0,539828
0,579260
0,617911
0,655422
0,691462
0,725747
0,758036
0,788145
0,815940
0,841345
0,864334
0,884930
0,903200
0,919243
0,933193
0,945201
0,955435
0,964070
0,971283
0,977250
0,982136
0,986097
0,989276
0,991802
0,993790
0,995339
0,996533
0,997445
0,998134
0,998650
0,01
0,503989
0,543795
0,583166
0,621720
0,659097
0,694974
0,729069
0,761148
0,791030
0,818589
0,843752
0,866500
0,886861
0,904902
0,920730
0,934478
0,946301
0,956367
0,964852
0,971933
0,977784
0,982571
0,986447
0,989556
0,992024
0,993963
0,995473
0,996636
0,997523
0,998193
0,998694
0,02
0,507978
0,547758
0,587064
0,625516
0,662757
0,698468
0,732371
0,764238
0,793892
0,821214
0,846136
0,868643
0,888768
0,906582
0,922196
0,935745
0,947384
0,957284
0,965620
0,972571
0,978308
0,982997
0,986791
0,989830
0,992240
0,994132
0,995604
0,996736
0,997599
0,998250
0,998736
0,03
0,511966
0,551717
0,590954
0,629300
0,666402
0,701944
0,735653
0,767305
0,796731
0,823814
0,848495
0,870762
0,890651
0,908241
0,923641
0,936992
0,948449
0,958185
0,966375
0,973197
0,978822
0,983414
0,987126
0,990097
0,992451
0,994297
0,995731
0,996833
0,997673
0,998305
0,998777
z
1
1 2
√
e− 2 x dx
2π
For instance,
0,04
0,515953
0,555670
0,594835
0,633072
0,670031
0,705401
0,738914
0,770350
0,799546
0,826391
0,850830
0,872857
0,892512
0,909877
0,925066
0,938220
0,949497
0,959070
0,967116
0,973810
0,979325
0,983823
0,987455
0,990358
0,992656
0,994457
0,995855
0,996928
0,997744
0,998359
0,998817
2
N (1.28) = 0.899727
0,05
0,519939
0,559618
0,598706
0,636831
0,673645
0,708840
0,742154
0,773373
0,802337
0,828944
0,853141
0,874928
0,894350
0,911492
0,926471
0,939429
0,950529
0,959941
0,967843
0,974412
0,979818
0,984222
0,987776
0,990613
0,992857
0,994614
0,995975
0,997020
0,997814
0,998411
0,998856
0,06
0,523922
0,563559
0,602568
0,640576
0,677242
0,712260
0,745373
0,776373
0,805105
0,831472
0,855428
0,876976
0,896165
0,913085
0,927855
0,940620
0,951543
0,960796
0,968557
0,975002
0,980301
0,984614
0,988089
0,990863
0,993053
0,994766
0,996093
0,997110
0,997882
0,998462
0,998893
and
0,07
0,527903
0,567495
0,606420
0,644309
0,680822
0,715661
0,748571
0,779350
0,807850
0,833977
0,857690
0,879000
0,897958
0,914657
0,929219
0,941792
0,952540
0,961636
0,969258
0,975581
0,980774
0,984997
0,988396
0,991106
0,993244
0,994915
0,996207
0,997197
0,997948
0,998511
0,998930
N (−1.28) =
0,08
0,531881
0,571424
0,610261
0,648027
0,684386
0,719043
0,751748
0,782305
0,810570
0,836457
0,859929
0,881000
0,899727
0,916207
0,930563
0,942947
0,953521
0,962462
0,969946
0,976148
0,981237
0,985371
0,988696
0,991344
0,993431
0,995060
0,996319
0,997282
0,998012
0,998559
0,998965
0,09
0,535856
0,575345
0,614092
0,651732
0,687933
0,722405
0,754903
0,785236
0,813267
0,838913
0,862143
0,882977
0,901475
0,917736
0,931888
0,944083
0,954486
0,963273
0,970621
0,976705
0,981691
0,985738
0,988989
0,991576
0,993613
0,995201
0,996427
0,997365
0,998074
0,998605
0,998999
Prof. Dr. Karl Ludwig Keiber
Lehrstuhl für Betriebswirtschaftslehre,
insb. Finance
Corporate Finance
SS 2021 — Tutorial #4
Problem 1
be the price process of a share of stock under the risk neutral probability
Let dS̃t = rSt dt + σSt dz̃
√t
measure where dz̃t = ε̃t dt is a standard Wiener process. That is ε̃t ∼ N (0, 1) and z0 = 0. Note that
the total differential of any twice differentiable function H(St , t) is
dH =
∂H
1 ∂2H 2
∂H
dS̃t +
dS̃ .
dt +
∂t
∂St
2 ∂ 2 St t
a. Determine the probability distribution of ln S̃t by calculating the integral 0t d ln Sτ and specify
both the mean E[ln S̃t ] and the variance Var[ln S̃t ]. Hint: Calculate d ln Sτ at first. Note: dtk = 0
if k > 1.
R
b. Determine the date t value of a European–type call option on a share of stock with strike price X
and maturity date T according to the risk neutral valuation procedure. Hint: Exploit the martingale
property of any price process under the risk neutral probability measure. Note: If x̃ ∼ N µ, σ2
then E[ex̃] = eµ+0.5σ . Remark: The solution is the Black and Scholes (1973) model.
2
Problem 2
Consider a European–type call option on a share of stock with strike price X = 40 and maturity date
The actual price of a share of stock is S0 = 40. The variance rate of the stock’s return is σ2 = 0.04.
The continuously compounded risk free rate of return is 0.05.
T = 2.
a. Determine the price of the option at date t = 0 according to the CRR binomial model where (i.)
∆t = 1 and (ii.) ∆t = 0.5. Furthermore, u = d−1 .
b. Determine the price of the option at date t = 0 according to the Black and Scholes (1973) model.
1
Cumulative Standard Normal Distribution
The table displays values of the integral
N (z) ≡
Z
for
−∞
z ≥ 0. If z < 0 then N (z) ≡ 1 − N (−z).
1 − N (1.28) = 1 − 0.899727 = 0.100273.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
1,1
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
2
2,1
2,2
2,3
2,4
2,5
2,6
2,7
2,8
2,9
3
0
0,500000
0,539828
0,579260
0,617911
0,655422
0,691462
0,725747
0,758036
0,788145
0,815940
0,841345
0,864334
0,884930
0,903200
0,919243
0,933193
0,945201
0,955435
0,964070
0,971283
0,977250
0,982136
0,986097
0,989276
0,991802
0,993790
0,995339
0,996533
0,997445
0,998134
0,998650
0,01
0,503989
0,543795
0,583166
0,621720
0,659097
0,694974
0,729069
0,761148
0,791030
0,818589
0,843752
0,866500
0,886861
0,904902
0,920730
0,934478
0,946301
0,956367
0,964852
0,971933
0,977784
0,982571
0,986447
0,989556
0,992024
0,993963
0,995473
0,996636
0,997523
0,998193
0,998694
0,02
0,507978
0,547758
0,587064
0,625516
0,662757
0,698468
0,732371
0,764238
0,793892
0,821214
0,846136
0,868643
0,888768
0,906582
0,922196
0,935745
0,947384
0,957284
0,965620
0,972571
0,978308
0,982997
0,986791
0,989830
0,992240
0,994132
0,995604
0,996736
0,997599
0,998250
0,998736
0,03
0,511966
0,551717
0,590954
0,629300
0,666402
0,701944
0,735653
0,767305
0,796731
0,823814
0,848495
0,870762
0,890651
0,908241
0,923641
0,936992
0,948449
0,958185
0,966375
0,973197
0,978822
0,983414
0,987126
0,990097
0,992451
0,994297
0,995731
0,996833
0,997673
0,998305
0,998777
z
1
1 2
√
e− 2 x dx
2π
For instance,
0,04
0,515953
0,555670
0,594835
0,633072
0,670031
0,705401
0,738914
0,770350
0,799546
0,826391
0,850830
0,872857
0,892512
0,909877
0,925066
0,938220
0,949497
0,959070
0,967116
0,973810
0,979325
0,983823
0,987455
0,990358
0,992656
0,994457
0,995855
0,996928
0,997744
0,998359
0,998817
2
N (1.28) = 0.899727
0,05
0,519939
0,559618
0,598706
0,636831
0,673645
0,708840
0,742154
0,773373
0,802337
0,828944
0,853141
0,874928
0,894350
0,911492
0,926471
0,939429
0,950529
0,959941
0,967843
0,974412
0,979818
0,984222
0,987776
0,990613
0,992857
0,994614
0,995975
0,997020
0,997814
0,998411
0,998856
0,06
0,523922
0,563559
0,602568
0,640576
0,677242
0,712260
0,745373
0,776373
0,805105
0,831472
0,855428
0,876976
0,896165
0,913085
0,927855
0,940620
0,951543
0,960796
0,968557
0,975002
0,980301
0,984614
0,988089
0,990863
0,993053
0,994766
0,996093
0,997110
0,997882
0,998462
0,998893
and
0,07
0,527903
0,567495
0,606420
0,644309
0,680822
0,715661
0,748571
0,779350
0,807850
0,833977
0,857690
0,879000
0,897958
0,914657
0,929219
0,941792
0,952540
0,961636
0,969258
0,975581
0,980774
0,984997
0,988396
0,991106
0,993244
0,994915
0,996207
0,997197
0,997948
0,998511
0,998930
N (−1.28) =
0,08
0,531881
0,571424
0,610261
0,648027
0,684386
0,719043
0,751748
0,782305
0,810570
0,836457
0,859929
0,881000
0,899727
0,916207
0,930563
0,942947
0,953521
0,962462
0,969946
0,976148
0,981237
0,985371
0,988696
0,991344
0,993431
0,995060
0,996319
0,997282
0,998012
0,998559
0,998965
0,09
0,535856
0,575345
0,614092
0,651732
0,687933
0,722405
0,754903
0,785236
0,813267
0,838913
0,862143
0,882977
0,901475
0,917736
0,931888
0,944083
0,954486
0,963273
0,970621
0,976705
0,981691
0,985738
0,988989
0,991576
0,993613
0,995201
0,996427
0,997365
0,998074
0,998605
0,998999
Prof. Dr. Karl Ludwig Keiber
Lehrstuhl für Betriebswirtschaftslehre,
insb. Finance
Corporate Finance
SS 2021 — Tutorial #3
Problem 1
Consider a one–period economy under uncertainty. Let s ∈ S = {1, . . . , n} count the future states of
nature at the end of the period. In particular, n = 3. The payoff matrix of the market securities is

2 1

Ω = 4 3
4 5

1

1 .
4
The price system is p~ = (3.20, 3.10, 2.10)′ . Let ϑ~ = (0, 1, 1)′ represent the payoff of an additional asset.
a. Check the completeness of the capital market by (i.) calculating the determinant of Ω and (ii.)
calculating the rank of Ω.
b. Determine both the replicating portfolio of the risk free security and its arbitrage free price.
c. Determine the arbitrage free prices of the state securities.
d. Determine the one–period risk free rate of return.
e. Determine the risk neutral probability measure Q.
f. Determine the arbitrage free price of the asset with payoff ϑ~ (i.) by calculating the price of the
replicating portfolio and (ii.) according to the risk neutral valuation procedure.
g. Qualify the asset with payoff ϑ~ .
h. Determine the price of a European–type put option on the first market security with strike price
X = 3 and maturity T = 1 (i.) as the price of the replicating portfolio and (ii.) according to the
put–call parity.
1
Problem 2
Consider a multi–period economy under uncertainty. The length of a subperiod is ∆t = 1. The price of
a share of stock at date t = 0 amounts to S0 = 40. The uncertainty of the share price evolves according
to the factors u = 1.2 and d = 0.8 per subperiod. The one–period risk free rate of return amounts to
RF = 10%.
a. Determine the price of a European–type call option on a share of stock with strike price X
and maturity T = 3.
= 48
b. Determine the hedge ratios in all states prior to maturity such that a portfolio consisting of a share
of stock and the according number of call options is risk free instantaneously.
c. Determine the price of a European–type put option on a share of stock with strike price X = 48 and
maturity T = 3 according to both (i.) the risk neutral valuation procedure and (ii.) the put–call
parity.
2
Prof. Dr. Karl Ludwig Keiber
Lehrstuhl für Betriebswirtschaftslehre,
insb. Finance
Corporate Finance
SS 2021 — Tutorial #2
Problem 1
Consider a one–period economy under uncertainty. Let s ∈ S = {1, . . . , n} count the future states of
nature at the end of the period. In particular, n = 2. Two market securities are traded in the securities
market. The market price system is ~p = (S0 , B0 )′ = (50, 100)′ . The payoff matrix of the market securities
is
!
Ω=
uS0
dS0
110
.
110
Furthermore, u = 1.25 and d = 0.8.
a. Check the completeness of the capital market.
b. Determine the one–period risk free rate of return.
c. Determine both the replicating portfolios of the state securities and its prices.
d. Determine the pricing kernel of the economy.
e. Determine the risk neutral probability measure Q.
f. Determine both the replicating portfolio of a European–type call option on the first market security
with strike price X = 55 and maturity T = 1 and the price of the call option.
Problem 2
Consider a one–period economy under uncertainty. Let s ∈ S = {1, . . . , n} count the future states of
nature at the end of the period. In particular, n = 3. The payoff matrix of the market securities is

2 2

4 6 .
7 10

1

Ω = 1
2
a. Check the completeness of the capital market by (i.) calculating the determinant of Ω, (ii.) calculating the rank of Ω, and (iii.) determining the non–trivial solution to the system of equations
Ω~x = ~0.
b. Check whether the price system p~ = (2.50, 3.00, 0.75)′ is arbitrage free.
c. Check whether the price system p~ = (0.50, 2.00, 3.00)′ is arbitrage free.
1
Problem 3
Consider a one–period economy under uncertainty. Let s ∈ S = {1, . . . , n} count the future states of
nature at the end of the period. In particular, n = 3. The payoff matrix of the market securities is

The price system is p~ =
′
5 14
6, 3 , 4
.
1 8

Ω = 1 4
1 6

6

4 .
5
a. Judge the completeness of the capital market by checking whether the market securities’ payoff
vectors are linearly independent.
b. Check whether the above price system is arbitrage free.
c. Determine the one–period risk free rate of return.
d. Determine the risk neutral probability measure Q.
2
Prof. Dr. Karl Ludwig Keiber
Lehrstuhl für Betriebswirtschaftslehre,
insb. Finance
Corporate Finance
SS 2021 — Tutorial #1
Problem 1
Consider a one–period economy under uncertainty. Let s ∈ S = {1, . . . , n} count the future states of
nature at the end of the period. In particular, n = 4. The prices of the pure securities equal p∗s = 11s
and the state probabilities amount to πs = 0.1 · s. The preferences of the representative investor are
represented by negative exponential utility. That is, U (C) = −e−aC where a ∈ R+ . Finally, let W0
denote the aggregate wealth.
a. Determine the one–period risk free rate of return.
b. Determine the optimal consumption C0 and investment decision Qs .
c. Now, W0 = 1000 and a = 2. Determine both the fractions of wealth which are allocated to the
pure securities and the portfolio weights of the risky investment.
Problem 2
Consider a one–period economy under uncertainty. Let s ∈ S = {1, . . . , n} count the future states of
nature at the end of the period. In particular, n = 2. Two market securities are traded in the securities
market. The market price system is p~ = (S0 , B0 )′ . The payoff matrix of the market securities is
Ω=
uS0
dS0
!
B0 (1 + RF )
.
B0 (1 + RF )
a. Determine the pairs (u, d) ∈ R+ × R+ for which the capital market is complete.
b. Determine the replicating portfolios of the state securities.
c. Determine the price system p~∗ .
d. Determine the pairs (u, d) ∈ R+ × R+ for which the securities marekt is arbitrage free.
e. Determine the pricing kernel of the economy.
f. Determine the risk neutral probability measure Q.
g. Determine both the replicating portfolio of a European–type call option on the first market security
with strike price X and maturity T = 1 and the price of the call option.
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