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 Incoming links: Stock-inversion reflection for the Black-Scholes equation

Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / B / 8 / a / Solution 2026-10-31

x = B^{2} / S

and

q = (S / B)^{α}

, so

W = q V (t, x)

. Direct differentiation gives

W_{t} = q V_{t}, S W_{S} = q (α V - x V_{x}),

S^{2} W_{SS} = q [α (α - 1) V + 2 (1 - α) x V_{x} + x^{2} V_{xx}] .

Substitute into the Black-Scholes equation and divide by

q

:

\frac{L W}{q} = V_{t} + \frac{1}{2} σ^{2} x^{2} V_{xx} + [σ^{2} (1 - α) - r] x V_{x} + [\frac{1}{2} σ^{2} α (α - 1) + r α - r] V .

To recover the same operator on

V

, its first-derivative coefficient must be

r

. This gives

α = 1 - \frac{2 r}{σ ^{2}} .

For this choice the last coefficient is

- r

, since

\frac{1}{2} σ^{2} (α - 1) + r = 0

. Hence

L W = q L V = 0

, proving the required invariance. The transformation is the stock-inversion reflection for the Black-Scholes equation.

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