09. Black-Schole's formula

Part 1

Understanding Risk Neutral Probability Measure and European Call Option Pricing

This guide covers calculating the price of a European call option using the risk-neutral measure:

• Risk Neutral Probability Measure: Essential for calculating the expectation of an option's payoff on the expiry date, crucial for determining its current price.

• European Call Option Pricing: The option price on the expiry date is expressed mathematically, discounted to reflect the time value of money.

• Expectation Calculation: Split into two parts:

  • The second part involves the expectation of the event where the expiry date stock price exceeds the strike price, calculated under the risk-neutral measure.

• Probability Under Risk Neutral Measure: Derived by modifying the stock price model, replacing growth rate (Mu) with risk-free rate (R).

• Stock Price Probability Calculation: Utilizes standard normal distribution and expresses conditions as inequalities, making evaluation straightforward.

• Expectation Evaluation: With simplified expressions, the expectation involving stock price is rewritten in terms of a standard normal variable, using probability density functions for integration.

Part 2

Overview of Black-Scholes Formula for Options Pricing

An essential tool in finance for pricing European options: the Black-Scholes formula. Delivers insights into call and put options.

Key Concepts:

• Exponential Functions: Combined and simplified to derive the Black-Scholes formula.

• Definitions:

  • S_0: Current stock price
  • C_0 & P_0: Current prices of call and put options
  • K: Strike price, the price to buy (call) or sell (put) the stock
  • R: Risk-free interest rate (US: SOFR, UK: SONIA)
  • T: Time to option expiry, measured in years

• Variables:

  • D_1 & D_2: Convenience variables for simplicity
  • Phi (Φ): Represents the cumulative distribution function of the normal distribution

Example Application:

• Pricing a Put Option:
  • Example for stock ABC
  • Stock price: Dropping from $100 to a target of $85
  • Time to expiry: 0.5 years, with 30% volatility
  • Calculated put price: $1.99

Conclusion:

• Options more costly with high volatility. Assess before hedging.

Programming Black-Schole's Formula in Python

Programming Black-Scholes Formula in Python

Learn to compute European option pricing using Python by applying the Black-Scholes formula:

• Geometric Brownian Motion (GBM):

  • The price of European options derived under GBM, defined by a stochastic differential equation.
  • Requires calibration for parameters like drift (Mu) and volatility (Sigma).

• European Call Option Basics:

  • Grants a right, not an obligation, to purchase stock at a strike price (K) on expiry date (T).
  • Value represented by C(0), the price at time zero, based on the standard normal cumulative distribution.

• Key Variables in Pricing:

  • d1 and d2, dependent on stock price (S0), strike price (K), risk-free rate (r), time to expiry (T), and volatility (Sigma).

• Practical Python Implementation:

  • Import necessary modules for calculation.
  • Compute critical components like d1 and d2.
  • Demonstrate with a practical example (e.g., pricing options "at the money").
  • Show how changing volatility affects the price.

Conclude with an understanding that higher volatility increases option prices, emphasizing its role in risk assessment.