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Sure! Let's break down the solution to each part of the question step-by-step:
### Part a: European Call Option Valuation using Black-Scholes Model
Given data:
- Current share price ([tex]\( S \)[/tex]) = 4.50
- Strike price ([tex]\( K \)[/tex]) = 3.50
- Risk-free rate ([tex]\( r \)[/tex]) = 0.10
- Time to maturity ([tex]\( T \)[/tex]) = 3/12 years (3 months)
- Volatility ([tex]\( \sigma \)[/tex]) = 0.8
The Black-Scholes formula for the price of a European call option is given by:
[tex]\[ C = S \cdot N(d1) - K \cdot e^{-rT} \cdot N(d2) \][/tex]
where:
[tex]\[ d1 = \frac{\ln(\frac{S}{K}) + (r + \frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}} \][/tex]
[tex]\[ d2 = d1 - \sigma \sqrt{T} \][/tex]
Let's calculate [tex]\( d1 \)[/tex] and [tex]\( d2 \)[/tex] first.
1. [tex]\( d1 \)[/tex]:
[tex]\[ d1 = \frac{\ln(\frac{4.50}{3.50}) + (0.10 + \frac{1}{2} \cdot 0.8^2) \cdot \frac{3}{12}}{0.8 \cdot \sqrt{\frac{3}{12}}} \][/tex]
[tex]\[ d1 \approx \frac{\ln(1.2857) + (0.10 + 0.32) \cdot 0.25}{0.8 \cdot 0.5} \][/tex]
[tex]\[ d1 \approx \frac{0.2513 + 0.42 \cdot 0.25}{0.4} \][/tex]
[tex]\[ d1 \approx \frac{0.2513 + 0.105}{0.4} \][/tex]
[tex]\[ d1 \approx \frac{0.3563}{0.4} \][/tex]
[tex]\[ d1 \approx 0.89075 \][/tex]
2. [tex]\( d2 \)[/tex]:
[tex]\[ d2 = d1 - 0.8 \cdot \sqrt{\frac{3}{12}} \][/tex]
[tex]\[ d2 = 0.89075 - 0.8 \cdot 0.5 \][/tex]
[tex]\[ d2 = 0.89075 - 0.4 \][/tex]
[tex]\[ d2 \approx 0.49075 \][/tex]
Next, we use the cumulative distribution function [tex]\( N \)[/tex] for the standard normal distribution to find [tex]\( N(d1) \)[/tex] and [tex]\( N(d2) \)[/tex].
After obtaining [tex]\( N(d1) \)[/tex] and [tex]\( N(d2) \)[/tex]:
[tex]\[ N(d1) \approx 0.81332 \][/tex]
[tex]\[ N(d2) \approx 0.68895 \][/tex]
Now, we can find the call option price [tex]\( C \)[/tex]:
[tex]\[ C = 4.50 \cdot 0.81332 - 3.50 \cdot e^{-0.10 \cdot 0.25} \cdot 0.68895 \][/tex]
[tex]\[ C \approx 4.50 \cdot 0.81332 - 3.50 \cdot 0.97531 \cdot 0.68895 \][/tex]
[tex]\[ C \approx 3.660 \cdot 3.50 \cdot 0.67143 \][/tex]
[tex]\[ C \approx 3.660 - 2.3485748965711 \][/tex]
[tex]\[ C \approx 1.3113841749290787 \][/tex]
Hence, the current value of the call option is approximately 1.3114.
### Part b: Forward Rate Premium or Discount According to Interest Rate Parity
Given data:
- Annual interest rate in Kenya ([tex]\( r_{\text{Kenya}} \)[/tex]) = 0.18
- Annual interest rate in the USA ([tex]\( r_{\text{USA}} \)[/tex]) = 0.12
According to interest rate parity, the forward rate premium or discount is given by the formula:
[tex]\[ \text{Forward Rate Premium/Discount} = \frac{1 + r_{\text{Kenya}}}{1 + r_{\text{USA}}} - 1 \][/tex]
Substituting the given values:
[tex]\[ \text{Forward Rate Premium/Discount} = \frac{1 + 0.18}{1 + 0.12} - 1 \][/tex]
[tex]\[ \text{Forward Rate Premium/Discount} = \frac{1.18}{1.12} - 1 \][/tex]
[tex]\[ \text{Forward Rate Premium/Discount} \approx 1.0535714285714285 - 1 \][/tex]
[tex]\[ \text{Forward Rate Premium/Discount} \approx 0.05357142857142838 \][/tex]
Interpreting this result:
The forward rate premium or discount implies that the Kenyan Shilling is expected to appreciate by approximately 5.36% relative to the U.S. Dollar over the given period.
Thus, the result suggests a forward rate premium of approximately 5.36%.
### Part a: European Call Option Valuation using Black-Scholes Model
Given data:
- Current share price ([tex]\( S \)[/tex]) = 4.50
- Strike price ([tex]\( K \)[/tex]) = 3.50
- Risk-free rate ([tex]\( r \)[/tex]) = 0.10
- Time to maturity ([tex]\( T \)[/tex]) = 3/12 years (3 months)
- Volatility ([tex]\( \sigma \)[/tex]) = 0.8
The Black-Scholes formula for the price of a European call option is given by:
[tex]\[ C = S \cdot N(d1) - K \cdot e^{-rT} \cdot N(d2) \][/tex]
where:
[tex]\[ d1 = \frac{\ln(\frac{S}{K}) + (r + \frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}} \][/tex]
[tex]\[ d2 = d1 - \sigma \sqrt{T} \][/tex]
Let's calculate [tex]\( d1 \)[/tex] and [tex]\( d2 \)[/tex] first.
1. [tex]\( d1 \)[/tex]:
[tex]\[ d1 = \frac{\ln(\frac{4.50}{3.50}) + (0.10 + \frac{1}{2} \cdot 0.8^2) \cdot \frac{3}{12}}{0.8 \cdot \sqrt{\frac{3}{12}}} \][/tex]
[tex]\[ d1 \approx \frac{\ln(1.2857) + (0.10 + 0.32) \cdot 0.25}{0.8 \cdot 0.5} \][/tex]
[tex]\[ d1 \approx \frac{0.2513 + 0.42 \cdot 0.25}{0.4} \][/tex]
[tex]\[ d1 \approx \frac{0.2513 + 0.105}{0.4} \][/tex]
[tex]\[ d1 \approx \frac{0.3563}{0.4} \][/tex]
[tex]\[ d1 \approx 0.89075 \][/tex]
2. [tex]\( d2 \)[/tex]:
[tex]\[ d2 = d1 - 0.8 \cdot \sqrt{\frac{3}{12}} \][/tex]
[tex]\[ d2 = 0.89075 - 0.8 \cdot 0.5 \][/tex]
[tex]\[ d2 = 0.89075 - 0.4 \][/tex]
[tex]\[ d2 \approx 0.49075 \][/tex]
Next, we use the cumulative distribution function [tex]\( N \)[/tex] for the standard normal distribution to find [tex]\( N(d1) \)[/tex] and [tex]\( N(d2) \)[/tex].
After obtaining [tex]\( N(d1) \)[/tex] and [tex]\( N(d2) \)[/tex]:
[tex]\[ N(d1) \approx 0.81332 \][/tex]
[tex]\[ N(d2) \approx 0.68895 \][/tex]
Now, we can find the call option price [tex]\( C \)[/tex]:
[tex]\[ C = 4.50 \cdot 0.81332 - 3.50 \cdot e^{-0.10 \cdot 0.25} \cdot 0.68895 \][/tex]
[tex]\[ C \approx 4.50 \cdot 0.81332 - 3.50 \cdot 0.97531 \cdot 0.68895 \][/tex]
[tex]\[ C \approx 3.660 \cdot 3.50 \cdot 0.67143 \][/tex]
[tex]\[ C \approx 3.660 - 2.3485748965711 \][/tex]
[tex]\[ C \approx 1.3113841749290787 \][/tex]
Hence, the current value of the call option is approximately 1.3114.
### Part b: Forward Rate Premium or Discount According to Interest Rate Parity
Given data:
- Annual interest rate in Kenya ([tex]\( r_{\text{Kenya}} \)[/tex]) = 0.18
- Annual interest rate in the USA ([tex]\( r_{\text{USA}} \)[/tex]) = 0.12
According to interest rate parity, the forward rate premium or discount is given by the formula:
[tex]\[ \text{Forward Rate Premium/Discount} = \frac{1 + r_{\text{Kenya}}}{1 + r_{\text{USA}}} - 1 \][/tex]
Substituting the given values:
[tex]\[ \text{Forward Rate Premium/Discount} = \frac{1 + 0.18}{1 + 0.12} - 1 \][/tex]
[tex]\[ \text{Forward Rate Premium/Discount} = \frac{1.18}{1.12} - 1 \][/tex]
[tex]\[ \text{Forward Rate Premium/Discount} \approx 1.0535714285714285 - 1 \][/tex]
[tex]\[ \text{Forward Rate Premium/Discount} \approx 0.05357142857142838 \][/tex]
Interpreting this result:
The forward rate premium or discount implies that the Kenyan Shilling is expected to appreciate by approximately 5.36% relative to the U.S. Dollar over the given period.
Thus, the result suggests a forward rate premium of approximately 5.36%.
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