Skip to content

Black Scholes Base

Background

Black Scholes Base

Educational script demonstrating black scholes base concepts.

Code

```python """ Black Scholes Base

Educational script demonstrating black scholes base concepts. """

============================================================================

black_scholes/black_scholes_base.py

============================================================================

class BlackScholesBase: def init(self, S0, K, T, r, sigma, q=0): self.S0 = S0 # Initial stock price self.K = K # Strike price self.T = T # Time to maturity (in years) self.r = r # Risk-free interest rate self.sigma = sigma # Volatility of the underlying asset self.q = q # Continuous dividend yield

if name == "main": pass ```

Exercises

Exercise 1. List the six parameters stored in BlackScholesBase. For each, give a typical range of values and explain its financial meaning.

Solution to Exercise 1
Parameter Symbol Typical Range Financial Meaning
S0 \(S_0\) $50--$500 Current market price of the underlying asset
K \(K\) $50--$500 Strike (exercise) price of the option
T \(T\) 0.01--5 years Time remaining until the option expires
r \(r\) 0--0.10 Annualized risk-free interest rate
sigma \(\sigma\) 0.05--1.0 Annualized volatility of the asset returns
q \(q\) 0--0.05 Continuous dividend yield

Exercise 2. Write the Black-Scholes PDE that governs the option price \(V(S, t)\). What are the boundary conditions for a European call?

Solution to Exercise 2

The PDE is

\[ \frac{\partial V}{\partial t} + (r - q)S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} = rV \]

Boundary conditions for a European call:

  • Terminal: \(V(S, T) = \max(S - K, 0)\)
  • As \(S \to 0\): \(V(0, t) = 0\) (worthless)
  • As \(S \to \infty\): \(V(S, t) \sim S e^{-q(T-t)} - K e^{-r(T-t)}\) (deep in-the-money)

Exercise 3. Explain why the base class stores the continuous dividend yield \(q\) as a parameter. How does \(q > 0\) modify the Black-Scholes formula?

Solution to Exercise 3

A continuous dividend yield \(q\) reduces the effective growth rate of the stock under the risk-neutral measure from \(r\) to \(r - q\). In the BS formula, \(S_0\) is replaced by \(S_0 e^{-qT}\) (the present value of the stock excluding dividends):

\[ d_1 = \frac{\ln(S_0/K) + (r - q + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \]
\[ C = S_0 e^{-qT}\mathcal{N}(d_1) - K e^{-rT}\mathcal{N}(d_2) \]

When \(q = 0\), the standard BS formula is recovered.

Exercise 4. The base class uses inheritance: subclasses like BlackScholesFormula extend BlackScholesBase. What is the design advantage of this pattern?

Solution to Exercise 4

The inheritance pattern ensures that all Black-Scholes modules share the same parameter set and initialization logic. Advantages:

  1. DRY (Don't Repeat Yourself): Parameters are defined once in the base class.
  2. Consistency: All subclasses automatically have S0, K, T, r, sigma, q.
  3. Polymorphism: Code can accept any BlackScholesBase subclass and access parameters uniformly.
  4. Extensibility: New pricing methods (e.g., tree-based, PDE-based) just extend the base class.

The alternative (composition or standalone functions with parameter dictionaries) would require passing parameters to every function call.