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Introduction to Implied, Local and Stochastic Volatility

Ornstein-Uhlenbeck Process

By definition, a stochastic process {YT : t ≥ 0} is:

stationary if, for all t1 < t2 < ... < tn and h > 0, the random n−vectors (Yt1 , Yt2 , ..., Ytn) and ¡Yt1+h , Yt2+h , ..., Ytn+h¢ are identically distributed. That is, changes in time do not modify the probability or distribution.

Gaussian if, for all t1 < t2 < ... < tn, the n vector (Yt1 , Yt2 , ..., Ytn) is multivariate normally distributed.

Markovian if, for all t1 < t2 < ... < tn, the P (Ytn ≤ y|Yt1 , Yt2 , ..., Ytn−1) = P (Ytn ≤ y|Ytn−1). That is, the future is determined only by the present and not the past.

Also, a process {YT : t ≥ 0} is said to have independent increments if, for all t0 < t1 < t2 < ... < tn, the random variables Yt1 −Yt0 , Yt2 −Yt1 , ..., Ytn −Ytn−1 are independent. This condition implies that {YT : t ≥ 0} is Markovian, but not conversely. Furthermore, the increments are said to be stationary if, for any t > s and h > 0, the distribution of (Yt+h − Ys+h) is the same as the distribution of (Yt − Ys). This additional provision is needed for the following definition. A stochastic process {WT : t ≥ 0} is a Wiener-Lévy process or Brownian motion if it has stationary independent increments, if WT is normally distributed, the E(Wt) = 0 for each t > 0, and if W0 = 0. It then follows that {WT : t ≥ 0} is Gaussian and that Cov(Wt, Ws) = σ2 min {t, s}, where the variance parameter σ2 is a positive constant. Almost all paths of Brownian motion are always continuous but nowhere differentiable. One technical stipulation is required for the following. A stochastic process {YT : t ≥ 0} is continuous in probability if, for all u ε R+ and ε > 0,

P (|Yv − Yu| ≥ ε) --> 0 as v --> u

This holds true if Cov(Yt, Ys) is continuous over R+×R+. Note that this is a statement about distributions, not simple paths. Using these definitions, we can now define our intended topic. A stochastic process {XT : t ≥ 0} is an Ornstein-Uhlenbeck Process or a Gauss- Markov process if it is stationary, Gaussian, Markovian, and continuous in probability.

Arbitrage Possibility

An arbitrage possibility on a financial market is a self-financed portfolio "h" such that its value "V ” has the following behavior during a period of time:

V h(0) = C, C > 0

V h(T ) > C, − a.s.

We say that the market is arbitrage free if there are no arbitrage possibilities. An arbitrage possibility is thus equivalent to the possibility of making a positive amount of money out of nothing with probability 1 or a.s. (almost sure). It is thus a riskless money making machine or, if you will, a free lunch on the financial market, and our main assumption is that the market is efficient in the sense that no arbitrage is possible. This definition is given by [2].

Prof. Klaus Schmitz

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