Describe the black and scholes model and it's applicability in the nse
The Black-Scholes model, also known as the Black-Scholes-Merton model, is a mathematical model used to calculate the theoretical price of options. It was developed by economists Fischer Black and Myron Scholes in 1973, with contributions from Robert C. Merton.
The model assumes that markets are efficient and that the underlying asset price follows a geometric Brownian motion with constant volatility. It takes into account various factors, including the strike price, time to expiration, risk-free interest rate, and expected volatility of the underlying asset.
The Black-Scholes model is primarily used for European-style options, which can only be exercised at expiration. American-style options, which can be exercised at any time before expiration, are typically priced using variations of the Black-Scholes model.
To apply the Black-Scholes model in the NSE (National Stock Exchange), you would need the following parameters:
1. Underlying asset price: The current market price of the asset on which the option is based. This value can typically be obtained from an exchange or financial data provider.
2. Strike price: The agreed-upon price at which the option can be bought or sold.
3. Time to expiration: The time left until the option expires. This is typically measured in days or fractions of a year.
4. Risk-free interest rate: The rate of return on a risk-free investment, such as government bonds, with a maturity equal to the time to expiration.
5. Expected volatility: The standard deviation of the returns of the underlying asset. This can be estimated based on historical data or implied from option prices.
Once you have these parameters, you can calculate the theoretical price of the option using the Black-Scholes formula:
C = S × N(d₁) - X × e^(-rt) × N(d₂)
Where:
- C is the theoretical call option price
- S is the current asset price
- X is the strike price
- r is the risk-free interest rate
- t is the time to expiration
- N() represents the cumulative standard normal distribution
- d₁ = [ln(S/X) + (r + σ²/2) × t] / (σ × sqrt(t))
- d₂ = d₁ - σ × sqrt(t)
The Black-Scholes model is widely used in finance and investments to value options, calculate hedge ratios, and analyze various trading strategies. However, it is important to note that the model has certain assumptions and limitations, and actual market prices may deviate from the calculated values due to market dynamics and other factors.