Application of the Black-Scholes Model in Real-
Life Scenarios
1 Question 1: Scenario Analysis
Using the Black-Scholes Model, consider the following scenario:
JOOUST negotiates a call option to defer the payment of Ksh 50 million in
CBA-related salary increments until six months after government funding is se-
cured
Current government funding is uncertain but expected to range between Ksh
45 million and Ksh 55 million. The university’s administration is willing to
pay Ksh 2 million upfront for this option to hedge against the possibility of a
funding shortfall.
Assume the following:
o The current price of the underlying asset (S) is Ksh 45 million.
o The strike price (K) is Ksh 50 million.
o The time to expiration (T) is 6 months
o The risk-free interest rate (r) is 10% per annum.
o The volatility (σ) in government funding availability is 25%
1. Using the Black-Scholes Formula, calculate the fair value of this option.
Show your workings clearly.
Black-Scholes Model Calculation
The Black-Scholes formula for a European call option is given as:
C = S · N (d1 ) − K · e−rT · N (d2 )
Where:
σ2
ln(S/K) + r + 2 T
d1 = √
σ T
√
d2 = d1 − σ T
the given values
S = 45 million Ksh (current price of the underlying asset)
K = 50 million Ksh (strike price)
T = 0.5 years (time to expiration)
r = 0.1 (risk-free interest rate, 10% per annum)
σ = 0.25 (volatility, 25%)
1
, Computation of d1 and d2
Calculate d1 :
σ2
ln(S/K) + r + 2 T
d1 = √
σ T
Substituting the values:
2
ln(45/50) + 0.1 + 0.25
2 · 0.5
d1 = √
0.25 · 0.5
ln(0.9) + (0.1 + 0.03125) · 0.5
d1 =
0.25 · 0.7071
−0.10536 + 0.065625
d1 =
0.17678
−0.039735
d1 = = −0.2247
0.17678
Calculate d2 : √
d2 = d1 − σ T
d2 = −0.2247 − 0.25 · 0.7071
d2 = −0.2247 − 0.17678 = −0.4015
Calculation of the call Option Price
The cumulative standard normal distribution values are:
N (d1 ) = N (−0.2247) = 0.4110
N (d2 ) = N (−0.4015) = 0.3440
Substitute into the Black-Scholes formula:
C = S · N (d1 ) − K · e−rT · N (d2 )
C = 45 × 0.4110 − 50 × e−0.1×0.5 × 0.3440
e−0.05 ≈ 0.9512
Substitute:
C = 45 × 0.4110 − 50 × 0.9512 × 0.3440
C = 18.495 − 16.344
C = 2.151 million Ksh
2
Life Scenarios
1 Question 1: Scenario Analysis
Using the Black-Scholes Model, consider the following scenario:
JOOUST negotiates a call option to defer the payment of Ksh 50 million in
CBA-related salary increments until six months after government funding is se-
cured
Current government funding is uncertain but expected to range between Ksh
45 million and Ksh 55 million. The university’s administration is willing to
pay Ksh 2 million upfront for this option to hedge against the possibility of a
funding shortfall.
Assume the following:
o The current price of the underlying asset (S) is Ksh 45 million.
o The strike price (K) is Ksh 50 million.
o The time to expiration (T) is 6 months
o The risk-free interest rate (r) is 10% per annum.
o The volatility (σ) in government funding availability is 25%
1. Using the Black-Scholes Formula, calculate the fair value of this option.
Show your workings clearly.
Black-Scholes Model Calculation
The Black-Scholes formula for a European call option is given as:
C = S · N (d1 ) − K · e−rT · N (d2 )
Where:
σ2
ln(S/K) + r + 2 T
d1 = √
σ T
√
d2 = d1 − σ T
the given values
S = 45 million Ksh (current price of the underlying asset)
K = 50 million Ksh (strike price)
T = 0.5 years (time to expiration)
r = 0.1 (risk-free interest rate, 10% per annum)
σ = 0.25 (volatility, 25%)
1
, Computation of d1 and d2
Calculate d1 :
σ2
ln(S/K) + r + 2 T
d1 = √
σ T
Substituting the values:
2
ln(45/50) + 0.1 + 0.25
2 · 0.5
d1 = √
0.25 · 0.5
ln(0.9) + (0.1 + 0.03125) · 0.5
d1 =
0.25 · 0.7071
−0.10536 + 0.065625
d1 =
0.17678
−0.039735
d1 = = −0.2247
0.17678
Calculate d2 : √
d2 = d1 − σ T
d2 = −0.2247 − 0.25 · 0.7071
d2 = −0.2247 − 0.17678 = −0.4015
Calculation of the call Option Price
The cumulative standard normal distribution values are:
N (d1 ) = N (−0.2247) = 0.4110
N (d2 ) = N (−0.4015) = 0.3440
Substitute into the Black-Scholes formula:
C = S · N (d1 ) − K · e−rT · N (d2 )
C = 45 × 0.4110 − 50 × e−0.1×0.5 × 0.3440
e−0.05 ≈ 0.9512
Substitute:
C = 45 × 0.4110 − 50 × 0.9512 × 0.3440
C = 18.495 − 16.344
C = 2.151 million Ksh
2
