What is the difference between a submartingale and a supermartingale?

The risk-neutral pricing formula for a derivative with terminal payoff $V_N$ at time $N$ is $V_n = \mathbb{E}_n\left[\frac{V_N}{(1+r)^{N-n}}\right]$. This means the price at time $n$ is the conditional expectation of the discounted terminal payoff.

Conditions: Used in the risk-neutral pricing setting.; Discounting uses the factor $(1+r)^{N-n}$.; $V_N$ is the payoff at time $N$, and $V_n$ is the price at time $n$.

The constant expectation property states that the unconditional expectation of a martingale does not change over time and remains equal to its initial value. Formally, $\mathbb{E}[M_n] = M_0$ for all $n$.

Conditions: Applies for all times $n$ in the discrete-time setting.; This is the unconditional expectation, not the conditional one.

The risk-neutral probability $p$ of the up move is computed using the formula $p = \frac{(1+r)-d}{u-d}$. Given $r = 0.1$, $u = 1.2$, and $d = 0.9$, the calculation is $p = \frac{1.1 - 0.9}{1.2 - 0.9} = \frac{0.2}{0.3} = \frac{2}{3}$.

Conditions: One-period binomial model with up factor $u$, down factor $d$, and interest rate $r$.; $u > 1+r > d$ to ensure $0 < p < 1$.

A sequence of random variables $M_0, M_1, \ldots, M_N$ is a martingale if it is adapted and satisfies the condition $M_n = \mathbb{E}_n[M_{n+1}]$ for all $n = 0, 1, \ldots, N - 1$. This means the expected value of the next variable, given all prior information, is exactly equal to the current variable's value.

Conditions: The sequence must be adapted, meaning $M_n$ depends only on the first $n$ events.; The equality must hold for all time steps from $0$ to $N-1$.

Time consistency means that the expected future value of the martingale, conditioned on the information available today, is always equal to today's value. Formally, $M_n = \mathbb{E}_n[M_m]$ for all $n \le m$.

Conditions: Applies for all pairs of times with $n \le m$.; The expectation is conditional on information at time $n$.