Brownian Bridge™

1. Why is the log-spot transformation $x = \ln S$ applied before discretising the Black-Scholes PDE?

It converts the PDE from parabolic to elliptic, which is easier to solveIt eliminates the variable coefficient

\sigma^2 S^2

in the second-order term, giving constant coefficientsIt maps the semi-infinite domain

(0, \infty)

to a bounded interval

[0, 1]

It converts the terminal condition from nonlinear to linear

2. The explicit (FTCS) scheme for the heat equation requires $\lambda = \sigma^2 \Delta\tau / \Delta x^2 \leq 1/2$ for stability. If $\Delta x$ is halved (grid refined by 2×), by what factor must $\Delta\tau$ be reduced to maintain stability?

Factor of 2Factor of 4Factor of 8No change needed

3. The Crank-Nicolson scheme achieves $O(\Delta\tau^2 + \Delta x^2)$ accuracy but can produce oscillations near payoff discontinuities. What is the recommended fix?

Use a finer spatial grid only near the strikeApply two fully implicit (BTCS) time steps first, then switch to Crank-Nicolson (Rannacher smoothing)Replace Crank-Nicolson with the explicit scheme to eliminate the oscillationsIncrease the time step to damp the oscillations

4. The Lax equivalence theorem states that, for a consistent finite difference approximation of a well-posed linear PDE, stability is equivalent to convergence.

truefalse

5. Von Neumann stability analysis of the BTCS scheme gives amplification factor $\xi = 1/(1 + 2\lambda(1 - \cos\alpha\Delta x))$ . What is the maximum value of $|\xi|$ over all frequencies $\alpha$ ?

Unbounded for large

\lambda

Exactly 1, achieved at

\alpha = 0

Exactly

1/(1 + 4\lambda)

, achieved at

\alpha = \pi/\Delta x

Depends on the boundary conditions

6. For a two-factor model (e.g., Heston), the two-dimensional Black-Scholes-Heston PDE cannot be solved efficiently by direct Crank-Nicolson because:

The PDE is hyperbolic in two dimensions and has no finite difference approximationThe implicit system at each time step involves an

O(M^2) \times O(M^2)

matrix, not tridiagonalCrank-Nicolson is only conditionally stable in two dimensionsThe cross-derivative term makes the PDE nonlinear

Quiz: Finite Difference Schemes and Convergence Analysis

Quick Quiz