FIN 301: Options Pricing Homework - Black-Scholes & Binomial

This assignment presents solutions to two options pricing problems. The first question demonstrates the application of the Black-Scholes model to determine the price of an American call option, providing the formula, variable definitions, and detailed calculations of d1, d2, and the final call price. The second question utilizes a two-step binomial tree model to price an option, outlining the calculation of up and down factors, and the subsequent price movements over two periods. The solution includes formulas, intermediate calculations of stock prices at each step, and a visual representation of the binomial tree, offering a comprehensive understanding of both option pricing methodologies. The document references key academic resources on option pricing.

Question 1
To determine the price of the American call option using Black’s Model, we apply the formula
below:
C=SN ( d1 )−K e−r (T −t1−t 2) N ( d2)
Where d1=
ln ( s
k ) +( r−q1−q2+ σ 2
2 ) (T −t1−t2 )
σ √ T −t1−t2
,d2=d1−σ √T −t1−t2
s-current stock price c- call premium k-option striking price
r-option free interest rate N - cumulative standard normal distribution σ - volatility
q-dividend ratio
Evaluating for the values of d1, we get,
d1=
ln ( 20
22 )+ (0.02− 2
20 − 3
20 + 0.152
2 )(1− 1
12 − 7
12 )
0.15 √(1¿ −1
12 − 7
12 )=−1.9425 ¿
d2=−1.0659−0.15 √ 4
12 =−2.0291
N ( d1 ) =1−0.97381=0.02619
N ( d2 )=1−0.97882=0.02118
Solving for the call price, we get
C=20∗0.02619−22 e−0 . 02 ( 4
12 )∗0.02118=0.06094∗100=6.094
Question 2
We first calculate the up or down factors, uor d respectively for each step.
For step one,u=eσ √ t and d=e−σ √t where σ is the volatility
u=e0.18 √ 3
12 =1.0942
d=e−0.18 √ 3
12 =0.9139

We use the values of u and d to calculate the price over the next two periods. The price at the
end of the first period is given by Sup=u∗S, where S is the current price and Sup is the price after
period one in case the price goes up, and Sdown=d∗S where Sdownis the price after period one in
case the price goes down.
We calculate this values to get Sup=1.0942∗1050=1148.91
Sdown=0.9139∗1050=959.595
To calculate the prices after the second period, we use the formulas below
Sup 1=u2∗S Where Sup 1 is the price at the end of period two in case of an increase after the first
period
Sup 2=Sdown 2=u∗d∗S Where:
 Sup 2 is the price at the end of period two in case of a drop after an increase in
period one.
 Sdown 2 is the price at the end of period two in case of an increase after a drop in
period one
Sdown 1=d2∗S Where Sdown 1 is the price at the end of period two in case of a drop after the first
drop.
Calculating these values gives
Sup 1=1.09422∗1050=1257.137
Sdown 1=0.91392∗1050=876.974
Sup 2=Sdown 2=1.0942∗0.9139∗1050=1049.989
The figure below illustrates the two-step tree option pricing with levels at t=1 and t=2 each
period representing an interval of 3 months.
References
Marlow, J. (2001). Option Pricing. Wiley; Pap edition.
Richard j. rendleman, j. a. (1979). Two-state Option Pricing. Journal of Finance 24, 1093-1110.