In discrete time, an integrable stochastic process MtM_t that is adapted to the information Ft\mathcal{F}_t available up to each time is a martingale if E[Mt+1∣Ft]=MtE[M_{t+1} \mid \mathcal{F}_t] = M_t. This condition implies E[Mt+1]=E[Mt]E[M_{t+1}] = E[M_t] as well1. In continuous time with Brownian motion, the Ito integral of an adapted integrand is a martingale under conditions such as square integrability2.

For example, consider a process that is now 3 and moves next to either 4 or 2. If each has probability 0.5, the expected next value is (4+2)/2=3(4 + 2)/2 = 3, so it is a martingale. If the probability of 4 is 0.6, the expected value is 0.6×4+0.4×2=3.20.6 \times 4 + 0.4 \times 2 = 3.2. The process then has a drift of 0.2 per step on average and is not a martingale.

In pricing theory, one chooses, for example, a probability (measure) under which the discounted stock price (discounted using a cash account as the numeraire) is a martingale, and expresses the derivative price as an expected value under it. The sources present this as the statement that an option price is given by an average under an equivalent martingale measure34.

Notes

  1. Ichiro Shigekawa, "数理ファイナンス入門" (Introduction to Mathematical Finance), Kyoto University, intensive course at Ehime University, 2009 (in Japanese), pages 21 to 22, equation (2.20) and the equivalent conditions E[ΔMt+1∣Ft]=0E[\Delta M_{t+1} \mid \mathcal{F}_t] = 0 and E[Mt+1]=E[Mt]E[M_{t+1}] = E[M_t] that follow it. ↩

  2. Massachusetts Institute of Technology, "Lecture 18: Itō Calculus", pages 4 to 5, Theorem 2.3 (the Ito integral of a square-integrable adapted process is a martingale) and Example 2.4. ↩

  3. Shigekawa, "数理ファイナンス入門" (Introduction to Mathematical Finance), Kyoto University, in Japanese, page 4 (the goal of the course: option prices are given by averages under an equivalent martingale measure). ↩

  4. Martin Haugh, "Market Models" (Columbia University, Term Structure Models: IEOR E4710, 2010), page 6, "Numeraire-EMM Pairs", and page 8 (zero-coupon bond prices divided by the reference asset are martingales under the corresponding measure). ↩