Black and Scholes formula -Option Pricing

 The Black-Scholes formula is a mathematical model used to estimate the theoretical value of a call option or a put option. The formula is:

*Call Option:*

C = SN(d1) - Ke^(-rt)N(d2)

*Put Option:*

P = Ke^(-rt)N(-d2) - SN(-d1)

Where:

- C = call option value

- P = put option value

- S = current stock price

- K = strike price

- r = risk-free interest rate

- t = time to expiration (in years)

- N(d1) and N(d2) = cumulative distribution functions (CDFs) of the standard normal distribution

- d1 and d2 = variables calculated using the following formulas:

d1 = [ln(S/K) + (r + σ^2/2)t] / (σ√t)

d2 = d1 - σ√t

- σ = volatility of the underlying stock

The Black-Scholes formula takes into account the following factors:

1. Current stock price (S)

2. Strike price (K)

3. Risk-free interest rate (r)

4. Time to expiration (t)

5. Volatility of the underlying stock (σ)

The formula provides a theoretical estimate of the option's value, which can be used to make informed investment decisions.

Note: The Black-Scholes formula assumes a European-style option, which can only be exercised at expiration. For American-style options, which can be exercised before expiration, the formula needs to be modified.

Problem:The annual risk free rate is 10%, then the 'r' used in the black and scholes formula should be

Solution:The 'r' used in the Black-Scholes formula represents the risk-free interest rate, which is the annual risk-free rate in this case. Since the annual risk-free rate is given as 10%, the 'r' value used in the Black-Scholes formula should be:

r = 10% = 0.10

Note that the risk-free rate is typically expressed as a decimal, so 10% becomes 0.10.

In the Black-Scholes formula, 'r' is used to calculate the present value of the option's payoff, and it's an important input in determining the theoretical value of a call or put option.

Here's the Black-Scholes formula for reference:

C = SN(d1) - Ke^(-rt)N(d2)

Where:

- C = call option value

- S = stock price

- K = strike price

- r = risk-free interest rate (0.10 in this case)

- t = time to expiration

- N(d1) and N(d2) = cumulative distribution functions

More questions:

1. Calculate the value of a call option with the following parameters:

S = ₹100, K = ₹110, r = 8%, t = 0.5 years, σ = 20%

Sol. d1 = [ln(100/110) + (0.08 + 0.20^2/2)0.5] / (0.20√0.5) = -0.1017

d2 = d1 - 0.20√0.5 = -0.2817

N(d1) = 0.4582, N(d2) = 0.3891

C = 100(0.4582) - 110e^(-0.08*0.5)(0.3891) = ₹4.59

2. Determine the value of a put option with the following parameters:

S = ₹80, K = ₹90, r = 10%, t = 0.25 years, σ = 30%

Sol. d1 = [ln(80/90) + (0.10 + 0.30^2/2)0.25] / (0.30√0.25) = -0.3333

d2 = d1 - 0.30√0.25 = -0.5333

N(-d1) = 0.6331, N(-d2) = 0.7019

P = 90e^(-0.10*0.25)(0.7019) - 80(0.6331) = ₹8.11

3. A call option has a strike price of ₹120 and expires in 6 months. The current stock price is ₹110, and the risk-free rate is 9%. If the volatility is 25%, calculate the option value.

Sol. d1 = [ln(110/120) + (0.09 + 0.25^2/2)0.5] / (0.25√0.5) = -0.0707

d2 = d1 - 0.25√0.5 = -0.2707

N(d1) = 0.4714, N(d2) = 0.3936

C = 110(0.4714) - 120e^(-0.09*0.5)(0.3936) = ₹5.42

4. . A put option has a strike price of ₹100 and expires in 9 months. The current stock price is ₹90, and the risk-free rate is 8%. If the volatility is 20%, calculate the option value.

Sol. d1 = [ln(90/100) + (0.08 + 0.20^2/2)0.75] / (0.20√0.75) = -0.2019

d2 = d1 - 0.20√0.75 = -0.4019

N(-d1) = 0.5881, N(-d2) = 0.6554

P = 100e^(-0.08*0.75)(0.6554) - 90(0.5881) = ₹10.19

5. Calculate the value of a call option with the following parameters:

S = ₹150, K = ₹160, r = 7%, t = 0.75 years, σ = 22%

Sol. d1 = [ln(150/160) + (0.07 + 0.22^2/2)0.75] / (0.22√0.75) = -0.0949

d2 = d1 - 0.22√0.75 = -0.2949

N(d1) = 0.4625, N(d2) = 0.3855

C = 150(0.4625) - 160e^(-0.07*0.75)(0.3855) = ₹6.39

6.  What is the value of a call option with a strike price of ₹100, current stock price of ₹110, risk-free rate of 8%, time to expiration of 0.5 years, and volatility of 20%?

Sol. Using Black-Scholes formula:

d1 = [ln(110/100) + (0.08 + 0.20^2/2)0.5] / (0.20√0.5) = 0.1017

d2 = d1 - 0.20√0.5 = -0.0817

N(d1) = 0.5398, N(d2) = 0.4686

C = 110(0.5398) - 100e^(-0.08*0.5)(0.4686) = ₹6.19

7.  A put option has a strike price of ₹120 and expires in 6 months. The current stock price is ₹100, and the risk-free rate is 9%. If the volatility is 25%, what is the option value?

Sol. Using Black-Scholes formula:

d1 = [ln(100/120) + (0.09 + 0.25^2/2)0.5] / (0.25√0.5) = -0.3333

d2 = d1 - 0.25√0.5 = -0.5333

N(-d1) = 0.6331, N(-d2) = 0.7019

P = 120e^(-0.09*0.5)(0.7019) - 100(0.6331) = ₹14.19

8.  Calculate the value of a call option with a strike price of ₹80, current stock price of ₹90, risk-free rate of 7%, time to expiration of 0.25 years, and volatility of 18%.

Sol. Using Black-Scholes formula:

d1 = [ln(90/80) + (0.07 + 0.18^2/2)0.25] / (0.18√0.25) = 0.2019

d2 = d1 - 0.18√0.25 = 0.0519

N(d1) = 0.5821, N(d2) = 0.5222

C = 90(0.5821) - 80e^(-0.07*0.25)(0.5222) = ₹5.39

9. What is the effect on the option value if the volatility increases from 20% to 25% in the Black-Scholes model?

Sol. The option value increases with an increase in volatility.

10. How does an increase in the risk-free rate affect the value of a call option in the Black-Scholes model?

Sol. The value of a call option decreases with an increase in the risk-free rate.

11. Calculate the value of a put option with a strike price of ₹150, current stock price of ₹160, risk-free rate of 6%, time to expiration of 0.75 years, and volatility of 22%.

Sol. Using Black-Scholes formula:

d1 = [ln(160/150) + (0.06 + 0.22^2/2)0.75] / (0.22√0.75) = 0.0949

d2 = d1 - 0.22√0.75 = -0.2949

N(-d1) = 0.4625, N(-d2) = 0.3855

P = 150e^(-0.06*0.75)(0.3855) - 160(0.4625) = ₹10.39