Theorems & Results

There are a few basic Fin-Math results which is great to be able to point to or cite, so I’m going to gather here the basic results which are recurrently used in the field.

Theorem (The First Fundamental Theorem)
The following two statements about a market are equivalent:

  1. The market is arbitrage-free, i.e. no arbitrage exists.
  2. The market admits a risk-neutral measure, i.e. there exists an equivalent risk-neutral measure.

Theorem (Arbitrage Pricing)
In an arbitrage-free market the price process of any contingent-claim (i.e. derivative) is given by:

\displaystyle \Pi_t = B_t \Bigg\langle \frac{H_T}{B_T} \Bigg| \mathcal{F}_t \Bigg\rangle_Q

Theorem (The Second Fundamental Theorem)
The following two statements about an arbitrage-free market are equivalent:

  1. The market is complete, i.e. every contingent-claim (i.e. derivative) is hedgable.
  2. The market admits a unique risk-neutral measure, i.e. there exists a unique, equivalent risk-neutral measure.

In a complete fundamental market, a contingent-claim has a unique price determined by the unique risk-neutral measure and is equal to the value-process of the replicating hedging strategy:

\displaystyle \Pi_t = B_t \Bigg\langle \frac{H_T}{B_T} \Bigg| \mathcal{F}_t \Bigg\rangle_Q = V_t(\boldsymbol{\phi})

The above theorems are called the fundamental theorems of financial mathematics. The a priori given probability space contains what is called “the real-world measure”, which measures the probability of events happening in the real world. It depends on numerous factors, among others the risk preferences of investors etc. A risk-neutral measure on the other hand, is one under which all the traded securities have mean rate of return equal to zero (i.e. they are all martingales). Such a measure has nothing to do with the probabilistic interpretation of the real market under investigation, instead, it corresponds to a world where all investors have completely neutral risk preferences. One could say that such a world corresponds to all investors having an infinite investment horizon, they can wait for all means to be realized.

Theorem (Generalized Black-Scholes Market)
The generalized Black-Scholes market, dynamics given by

\displaystyle \textrm{d} S^{(k)}_t = \big( \alpha(t,\boldsymbol{S}_t) \big)^{(k)} S^{(k)}_t \, \textrm{d} t + S^{(k)}_t \, \sum_l \big( \sigma(t,\boldsymbol{S}_t) \big)^{(kl)} \, \textrm{d} W^{(k)}_t

is an arbitrage-free and complete market.