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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...
Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options , futures , interest rate derivatives , credit derivatives , etc.
Each is non-negative and their sum is 1. This is the risk-neutral measure! Now it remains to show that it works as advertised, i.e. taking expected values with respect to this probability measure will give the right price at time 0. Suppose you have a security C whose price at time 0 is C(0). In the future, in a state i, its payoff will be C i.
In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, [1] also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random ...
A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the blackboard font letter "". The relationship must hold for all times t: therefore the processes used for derivatives pricing are naturally set in continuous time.
When stock price returns follow a single Brownian motion, there is a unique risk neutral measure.When the stock price process is assumed to follow a more general sigma-martingale or semimartingale, then the concept of arbitrage is too narrow, and a stronger concept such as no free lunch with vanishing risk (NFLVR) must be used to describe these opportunities in an infinite dimensional setting.
Compare with futures prices, which are martingales under the risk neutral measure. Note that when interest rates are deterministic, this implies that forward prices and futures prices are the same. For example, the discounted stock price is a martingale under the risk-neutral measure:
For instance, let () be the price at time t of $1 that was invested in the money market at time 0. The fundamental theorem of asset pricing says that all assets S ( t ) {\displaystyle S(t)} priced in terms of the numéraire (in this case, M ), are martingales with respect to a risk-neutral measure , say Q {\displaystyle Q} .