Convergence (General)¶

Background¶

Black Scholes Convergence

Educational script demonstrating black scholes convergence concepts.

Code¶

```python """ Black Scholes Convergence

Educational script demonstrating black scholes convergence concepts. """

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black_scholes_CONVERGENCE_ANALYSIS.py¶

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import black_scholes as bs

Model parameters¶

if name == "main": S0 = 100 # Current stock price K = 105 # Strike price T = 0.25 # 3 months to expiration r = 0.05 # 5% risk-free rate sigma = 0.2 # 20% volatility q = 0.02 # 2% dividend yield mu = 0.1

bs.BlackScholes(S0=S0, K=K, T=T, r=r, sigma=sigma, q=q).plot_convergence(
    option_type='call',
    methods=['explicit', 'implicit', 'cn'],
    grid_points=[50, 100, 200, 400, 800, 1600]
    )

```

Exercises¶

Exercise 1. Describe a comprehensive convergence analysis for the BS PDE solver. What should be varied independently?

Solution to Exercise 1

Test spatial and temporal convergence separately: (1) fix \(N_t\) large, vary \(N_S\), measure error vs \(\Delta S\), (2) fix \(N_S\) large, vary \(N_t\), measure error vs \(\Delta t\). Then test joint refinement with \(\Delta t \propto \Delta S^2\) (maintaining stability for explicit schemes) or \(\Delta t \propto \Delta S\) (for implicit/CN schemes to balance errors).

Exercise 2. The BS formula provides the exact reference solution. What if no analytical solution exists (e.g., for stochastic volatility)?

Solution to Exercise 2

Use: (1) a very fine-grid numerical solution as a reference, (2) Richardson extrapolation to estimate the exact value, (3) comparison with Monte Carlo (independent method), (4) convergence of the error ratio \(e(h)/e(h/2)\) to determine the order without knowing the exact solution.

Exercise 3. The convergence analysis compares multiple schemes (explicit, implicit, CN). What convergence orders do you expect?

Solution to Exercise 3

Scheme	Time order	Space order
Explicit	1	2
Implicit	1	2
CN	2	2
CN-log	2	2

All schemes are second-order in space (centered differences). CN achieves second-order in time via the trapezoidal rule. The log-space transformation does not change the convergence order but may improve the constant.

Exercise 4. Explain Richardson extrapolation and how it can improve the effective convergence order.

Solution to Exercise 4

For a method with error \(V_h = V_{\text{exact}} + ch^p + O(h^{p+1})\), the Richardson extrapolation \(V_R = (2^p V_{h/2} - V_h)/(2^p - 1)\) eliminates the leading error term: \(V_R = V_{\text{exact}} + O(h^{p+1})\).

For CN (\(p = 2\)): \(V_R = (4V_{h/2} - V_h)/3\) gives \(O(h^4)\) effective accuracy. This doubles the convergence order at the cost of two solves instead of one. Richardson extrapolation is especially useful when grid refinement is expensive.