Recall: \underline{Definition 1.3}: European floating strike lookback put option

P ayoff = max(M − S_{T} , 0)  (1)

where S_{T} is the final stock price and M = max_{0≤τ≤ T} S_{τ}. Let P = P (S, M, t) stand for the option price, which satisfies

\{ \begin{matrix}\frac{∂P}{∂t} + \frac{σ^{2}}{2} S^{2} \frac{∂^{2}P}{∂S^{2}} + (r_{d} − r_{f})S \frac{∂P}{∂S} − r_{d}P = 0, & 0 ≤ S ≤ M and 0 ≤ t < T , \\ P (S, M, T) = M − S \\ \frac{∂P}{∂M} (M, M, t) = 0 \end{matrix}  (2)

where r_{d} > 0 and r_{f} > 0 represent the risk-free domestic rate and foreign rate, respectively. With the transformation:

x = ln \frac{M}{S}  V (x, t) = \frac{P (S, M, t)}{S},  (3)

Then (2), can be reduced to

\{ \begin{matrix}\frac{∂V}{∂t} + \frac{σ^{2}}{2} \frac{∂^{2}V}{∂x^{2}} + (r_{d} − r_{f} − \frac{σ^{2}}{2} ) \frac{∂V}{∂x} − r_{f}V = 0, & x > 0 and 0 ≤ t < T , \\ V (x, T) = e^{x} − 1 \\ \frac{∂V}{∂x} (0, t) = 0 \end{matrix}  (4)

u = e^{σ^{\sqrt{Δt}}} d = \frac{1}{u} K_{0} = \frac{M}{S} = 1  (5)

u^{j} = \frac{M_{n}}{S_{n}} V_{j}^{N} = u^{j} − 1  (6)

p = \frac{ρe^{−r_{f} Δt} − d}{u − d}  (7)

K_{2} = 1 = u^{0} K_{2} = u^{1} K_{2} = \frac{S}{d^{2}S} = u^{2} u^{3} u^{4}  (8)

p 1 − p

p = \frac{ρe^{−r_{f} Δt} − d}{u − d}

ρ = e^{r_{d}Δt} V_{0}^{n} = \frac{1}{ρ} (puV_{0}^{n+1} + (1 − p)dV_{1}^{n+1}) for j = 0 V_{j}^{n} = \frac{1}{ρ} (puV_{j−1}^{n+1} + (1 − p)dV_{j+1}^{n+1}) for j > 0

ρ = e^{r_{d}Δt}  (9)

V_{i+1}^{−1} = V_{i+1}^{1}  (10)

V_{i+1}^{j−1} = V_{i+1}^{j} = V_{i+1}^{j+1}  (11)

V = e^{x} − 1  (12)