The Markov Property: How Much of the Past Do You Need?

What to remember

• The risk-neutral up-probability $q = (1 + r - d)/(u - d)$ makes the discounted stock $S_n/(1 + r)^n$ a martingale; the raw price still grows at $r$.
• The discounted wealth of any self-financing portfolio is then a martingale, which rules out arbitrage and gives $V_0 = \tilde E[V_N/(1 + r)^N]$.
• A Markov process forecasts from its current state alone, so prices become functions $v_n(s)$ computed backward on $n + 1$ nodes.
• A lookback depends on the path; the pair $(S_n, M_n)$ is Markov, so price it with $v_n(s, m)$.
• The augmented state stays small: 2,601 states at step 100 against $2^{100}$ paths.
• Markov is a property of the chosen state: the P&L $I_n = \tfrac12 Z_n^2 - \tfrac n2$ needs only itself, $(S_n, M_n)$ folds 1,024 paths into 36 states, but the Asian's $(S_n, Y_n)$ barely merges (79,680 states at 20 steps against 121 for the lookback).