Four Derivations of the Black-Scholes Formula

Overview

• Author: Fabrice Douglas Rouah
• Objective: Derive the Black-Scholes formula for European call options in four different ways.
• Formula: [ C(St; K; T) = St\Phi(d1) - e^{-r(T-t)}K\Phi(d2) ]
  • d1: [ d1 = \frac{\ln \frac{St}{K} + \left(r + \frac{\sigma^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}} ]
  • d2: [ d2 = d1 - \sigma \sqrt{T-t} ]
  • ( \Phi(y) ) is the standard normal cumulative distribution function (CDF).

Derivations

1. Straightforward Integration
2. Feynman-Kac Theorem
3. Transformation into Heat Equation
   • Original method used by Black and Scholes.
4. Capital Asset Pricing Model (CAPM)

Black-Scholes Economy

• Assets: Risky stock ( S ) and riskless bond ( B )
• SDEs:
  • ( dS_t = \mu S_t dt + \sigma S_t dW_t )
  • ( dB_t = r B_t dt )
• Market Assumptions: As outlined in Hull’s book.
• Itô's Lemma: Applied to obtain derivative pricing dynamics.

Lognormal Distribution

• PDF and CDF: Defined for lognormal random variables derived from normal variables.
• Conditional Expected Value: Calculation using transformation and integration.

Solving Stochastic Differential Equations (SDEs)

• Stock Price: Solved using Itô’s Lemma for ( \ln S_t ).
  • Follows a lognormal distribution.
• Bond Price: Solved under constant interest rates.
• Discounted Stock Price as a Martingale: Establishing a new measure ( Q ) for martingale pricing.

Black-Scholes Call Price

• Expression for European call option pricing using integration and measure transformations.
• Numeraires: Discussion on different numeraires for pricing.

Feynman-Kac Theorem Application

• Application to solve Black-Scholes partial differential equation (PDE).
• Boundary conditions: ( V(S_T, T) = (S_T - K)^+ ).

Heat Equation Transformation

• Conversion of Black-Scholes PDE to heat equation through transformations.
• Dirac Delta Function: Utilized in defining initial conditions.
• Derivation results in the Black-Scholes call price.

CAPM Approach

• Relationship between expected returns and asset betas.
• Derivation of Black-Scholes PDE from CAPM assumptions.

Incorporating Dividends

• Continuous Dividends: Adjustments for a constant dividend yield ( q ).
• Lumpy Dividends: Future consideration for discrete dividend payments.

References

• Black and Scholes (1973): Original paper on option pricing.
• Other key texts: Hogg and Klugman (1984), Hull (2008), Wilmott et al. (1995).