O-MSF-FIN-521-Option-Contracts-Part-3-transcription

Investment Analysis Option Contracts: Black-Scholes Model

Introduction to Black-Scholes Model

The Black-Scholes Model, established in 1973 by Fischer Black, Myron Scholes, and Robert Merton, is a key framework for options valuation, providing a continuous-time approach to fair pricing, vastly used in modern financial markets.

Key Concepts of the Black-Scholes Model

1. Valuation Model: Based on geometric Brownian motion, allowing a closed-form solution for options pricing.

2. Components of Stock Price Movement:

   • Expected Component: 𝜇[ΔT] (expected rate of return)

   • Noise Component: σ[ΔT]^{1/2} (risk and variability in stock price).

Continuous Compounding

The model applies continuously compounded returns nearing convergence to the exponential function, emphasizing the risk-free rate's significance.

Formula for Call Option Valuation

Call option pricing formula:[C₀ = S N(d₁) - X e^{-RFR T} N(d₂)]Where:

• S = Current stock price

• X = Strike price

• RFR = Risk-free rate

• T = Time to expiration

• N(d₁), N(d₂) = Normal distribution probabilities.

Components d₁ and d₂

Defined as:

• d₁ = (ln(S/X) + (RFR + 0.5σ²)T) / (σ√T)

• d₂ = d₁ - σ√TThis allows for precise options pricing calculations.

Properties of the Black-Scholes Model

Five key variables influence options value:

• Current Security Price (S)

• Exercise Price (X)

• Time to Expiration (T)

• Risk-Free Rate (RFR)

• Security Price Volatility (σ)

Hedge Ratio (Delta)

Delta measures the sensitivity of an option's value to stock price changes, calculated as:

• For calls: N(d₁)

• For puts: N(d₁) - 1

Excel Application

Implementing the model in Excel involves calculating d₁, d₂, and option prices while considering factors like dividend yields.

Factors Affecting Option Values

Various factors impact call and put option values, including security price, exercise price, time to expiration, risk-free rate, volatility, and dividend yield.

Conclusion

Understanding the Black-Scholes model is crucial for effective investment analysis, particularly in options trading, providing a structured approach for valuing options amid market volatility.