Black-Scholes calculator

This Black-Scholes calculator is a versatile tool designed to help you in calculating the theoretical prices of call and put options.

To use this calculator effectively, you need to input several parameters including the stock price, strike price, time to maturity, volatility, risk-free interest rate, and dividend yield. Once these parameters are provided, the calculator computes the option prices along with various Greeks such as delta, gamma, theta, vega, and rho.

You can also select different options such as call or put, and choose to display specific parameters such as stock price, strike price, and time to maturity on the X-axis of the accompanying figure. Additionally, you can customize the display of Greeks and other metrics to suit your analysis needs. By adjusting the input parameters and observing the corresponding option prices and Greeks, you can gain insights into the potential risks and rewards associated with different options strategies.

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The Black-Scholes formulas

The Black-Scholes formulas are a mathematical model used to calculate the theoretical price of European-style options. They were developed by economists Fischer Black and Myron Scholes in the early 1970s and later expanded upon by Robert Merton. The formulas revolutionized options pricing and are widely used by investors, traders, and financial institutions for valuing stock options and other derivatives.

The Black-Scholes formulas assume that the price of the underlying asset (e.g., stock) follows a random walk with constant volatility, and they provides a way to calculate the fair value of an option based on various factors including the current stock price, the option’s strike price, the time until expiration, the risk-free interest rate, and the volatility of the underlying asset.

The importance of the Black-Scholes formulas lie in their ability to provide a theoretical value for options, which helps investors and traders in making informed decisions about buying, selling, or hedging options. By knowing the fair value of an option, market participants can assess whether an option is overvalued, undervalued, or fairly priced, and adjust their trading strategies accordingly.

The Black-Scholes formula for calculating the price of a European call option is:

C=S_0N(d_1)-Ke^{-rt}N(d_2)

and for a European put option:

P=Ke^{-rt}N(-d_2)-S_0N(-d_1)

where:

• C = Price of the call option
• P = Price of the put option
• S_0​ = Current price of the underlying asset
• K = Strike price of the option
• t = Time until expiration (in years)
• σ = Volatility of the underlying asset
• r = Risk-free interest rate
• N(d) = Cumulative distribution function of the standard normal distribution
• d_1​ and d_2 are calculated as follows:

d_1 = \frac{\ln(S_0/K) + (r + \frac{\sigma^2}{2})t}{\sigma\sqrt{t}}

d_2 = d_1 - \sigma\sqrt{t}

The formulas with dividends

When dividends are included, adjustments need to be made to the Black-Scholes formulas to account for the impact of dividends on the underlying stock price. Here are the adjusted formulas for European call and put options with continuous dividend yield:

For the price of a European call option with dividends:

C=S_0e^{-qt}N(d_1)-Ke^{-rt}N(d_2)

For the price of a European put option with dividends:

P=Ke^{-rt}N(-d_2)-S_0e^{-qt}N(-d_1)

where:

• q = Continuous dividend yield (expressed as a decimal)
• d_1​ and d_2​ are calculated as follows:

d_1 = \frac{\ln(S_0/K) + (r -q + \frac{\sigma^2}{2})t}{\sigma\sqrt{t}}

d_2 = d_1 - \sigma\sqrt{t}

​

These formulas adjust the stock price by subtracting the present value of expected dividends (S_0e^{−qt}) from the current stock price in the call option formula and adding it in the put option formula.

Assumptions and limitations

The Black-Scholes model, while groundbreaking, has several assumptions and limitations:

1. Efficient markets: The model assumes that markets are efficient, meaning that prices reflect all available information. In reality, markets may not always be perfectly efficient, leading to deviations between the model’s predictions and actual market behavior.
2. Constant volatility: The model assumes that volatility remains constant over the life of the option. However, in practice, volatility can fluctuate, especially during times of market stress or significant events, leading to inaccuracies in option pricing.
3. Continuous trading and no transaction costs: The model assumes continuous trading without any transaction costs, which may not be realistic in all markets. Transaction costs such as commissions and bid-ask spreads can impact the profitability of option trading strategies.
4. Risk-free interest rate: The model assumes a constant risk-free interest rate, which may not hold true in real-world scenarios where interest rates fluctuate over time.
5. No dividends: The original Black-Scholes model assumes that the underlying asset does not pay dividends during the option’s life. In cases where the underlying asset pays dividends, the model’s predictions may deviate from reality.
6. European options only: The model is designed specifically for European-style options, which can only be exercised at expiration. It does not directly apply to American-style options, which can be exercised at any time prior to expiration.
7. No consideration of market frictions: The model does not account for factors such as market frictions, liquidity constraints, or institutional trading behavior, which can impact option prices in real-world markets.
8. Normal distribution of returns: The model assumes that returns on the underlying asset follow a lognormal distribution. While this assumption holds for some assets, it may not accurately capture the distribution of returns for all types of assets.
9. Sudden jumps in prices: The model assumes that asset prices move continuously, without sudden jumps or discontinuities. In reality, unexpected events or news announcements can lead to sudden price movements that the model does not account for.
10. Limited predictive power: While the Black-Scholes model provides a theoretical framework for option pricing, it may not always accurately predict actual market prices. Traders and investors often use the model as a guideline but incorporate additional factors and market insights into their decision-making process.