Black–Scholes

rl.SwitchingBlackScholesProcess(chain, S0, r, q, sigma) is QuantLib’s BlackScholesMertonProcess with the volatility sigma allowed to switch.

The model

\[\frac{dS_t}{S_t} = (r - q)\,dt + \sigma_{y_t}\,dW_t .\]

Write \(S_T = F\,e^{X_T}\) with \(F = S_0e^{(r - q)T}\) the forward, so that \(e^{X}\) is a martingale.

Reduction

Over an interval \(dt\) spent in regime \(i\), \(X\) moves by a normal amount with mean \(-\tfrac12\sigma_i^2\,dt\) and variance \(\sigma_i^2\,dt\), which multiplies \(e^{iuX}\) on average by \(e^{g_i\,dt}\). Meanwhile the regime jumps at the rates in \(Q\). There is no state to factor out:

\[\phi_i(u) = \mathbb{E}\big[e^{iuX_T} \mid y_0 = i\big] = a_i(T), \qquad g_i = -\tfrac12\sigma_i^2\,(u^2 + iu).\]

Closed form

The forcing is constant, so the two-regime characteristic function is exact:

\[\phi_{1,2}(u) = e^{(\bar g - \lambda)T}\Big(\cosh sT + \frac{\lambda \pm \tilde g}{s}\,\sinh sT\Big), \qquad s = \sqrt{\lambda^2 + \tilde g^2},\]

\[\bar g = -\tfrac12\bar s\,(u^2 + iu), \qquad \tilde g = -\tfrac12\tilde s\,(u^2 + iu), \qquad \bar s, \tilde s = \tfrac12(\sigma_1^2 \pm \sigma_2^2).\]

Expanding in \(\varepsilon = 1/\lambda\) gives Black–Scholes at the averaged variance, times the Green–Kubo factor \(e^{\varepsilon\tilde g^2T/2}\) and the memory of the starting regime. A call is Lewis’s formula,

\[C = e^{-rT}\Big[F - \frac{\sqrt{FK}}{\pi}\int_0^\infty\operatorname{Re}\big(e^{iu\log(F/K)}\,\phi(u - \tfrac i2)\big)\frac{du}{u^2 + 1/4}\Big].\]

Python

S0, r, q, sigma, lam, T = 100.0, 0.03, 0.0, [0.30, 0.15], 25.0, 1.0

def phi(u):
    g = [-0.5 * s * s * (u * u + 1j * u) for s in sigma]
    return exact_two_state(g[0], g[1], lam, T)

model = rl.SwitchingBlackScholesProcess(rl.RegimeChain.twoState(lam, lam), S0, r, q, sigma)
for K in (90.0, 110.0):
    print(f"K = {K:5.0f}   closed form, exact  {lewis_call(phi, S0, K, r, q, T):.6f}"
          f"   library  {library_call(model, K, T):.6f}")

K =    90   closed form, exact  16.600838   library  16.600838
K =   110   closed form, exact  6.790125   library  6.790125

regimelib.symbolic.TwoStateConstantForcing holds the same characteristic function as a sympy expression, for unequal switching rates as well (Symbolic).

Instruments

Vanilla and digital options by the characteristic function; the continuous geometric Asian option, whose forcing in time to maturity \(\tau\) is \(iu(r - q - \tfrac12\sigma_i^2)\tau/T - \tfrac12u^2\sigma_i^2\tau^2/T^2\); barrier and American options on the coupled grid.