Question

Consider the Black-Scholes formula for prices of European call and put options with strike K each, maturity T each on a non-dividend-paying stock with price S and volatility σ, with risk-free rate r. The formulas are written in terms of quantities d1 and d2 used to calculate the probabilities normal distribution. If the volatility of the stock becomes large and approaches infinity,

(a) what values do d1 and d2 approach?
(b) what value does the call price approach?

(c) what value does the put price approach?

Answer 1

Solution:

The Black Scholes model is commonly known as the BSM model. It is a mathematical model for pricing options, in particular, this BSM model estimates the variation over time of the financial instruments. It assumes the instruments such as stocks (or) futures will have a normal distribution of prices. Using this assumption in other important variables the equation derives the price of a call option,
also called the Black-Scholes-Merton (BSM) model, it was the first and most widely used model for option pricing.

It is used to calculate the value of options using the current stock price, expected dividend. The option's strike price expected interest rates, time to expiration and the expected volatility.

The Black Scholes model is best-known for the option pricing method. This model's formula is derived, multiplying the stock price by the cumulative standard normal probability distribution function, thereafter, the net present value of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation.

The Black Scholes model tells us the most important concepts in modern financial theory. It was developed by Fischer Black, Robert Merton, and Myron Scholes and this model is still widely used today. It is known as one of the best ways of determining fair prices of options.
This Black Scholes model has required five input variables those are the strike price of an option
the current stock price
the time to expiration
the risk-free rate and
the volatility.

(a)

d1 and d2 approaches the below-mentioned values

here;

K=Strike price

T=maturity time

S=stock price

σ = Volatility

R = Risk-free rate

for value and formula please find the attachment

(b)

Call option price approaches the below-mentioned values

Call option value = current stock price (S) - Strike price (K)

Call option approach value = S - K

(c)

Put option price approaches the below-mentioned values

Put option value = Strike price (K) - current stock price (S)

Put option approach value = K - S