I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 A. Single step binary tree argument. Risk neutral probability. Delta hedging.
 B. Why Ito process?
 C. Existence of risk neutral measure via Girsanov's theorem.
 D. Self-financing strategy.
 E. Existence of risk neutral measure via backward Kolmogorov's equation. Delta hedging.
 F. Optimal utility function based interpretation of delta hedging.
 2 Change of numeraire.
 3 Basics of derivative pricing II.
 4 Market model.
 5 Currency Exchange.
 6 Credit risk.
 7 Incomplete markets.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Why Ito process?

e first assume that the price of some asset is Markovian and has continuous path. We further assume that we can divide the whole time interval into sufficiently small subintervals such that the properties of the price process are uniform within each . We further divide such subintervals into smaller intervals of length . Let us denote the price changes over and denote the price change over . According to the central limit theorem (see the section ( Central limit theorem )) applied to the sum we have or where the and are the volatility and mean of the smaller price changes . The above expression must be invariant with respect to any sufficiently large number of subdivisions . Therefore, we must have that and for large enough

Consider now two time intervals, put together: . We presume that is still sufficiently small to maintain the identical distribution assumption for all . For the price change over the union we have where the variables and are independent (Markovian property). The and have the same dependence on as noted above. Therefore we find that the price change at time t over a small time interval has two components with the structure originated from the CLT. The first component is deterministic conditionally on and depends linearly on the length of the time interval. The second component is a normal random variable (also conditionally on ) with standard deviation proportional to the square root of the time interval. Hence we use the Ito process to model the Markovian price process with continuous path.

If the price does not have continuous path then we use a sum of an Ito process and a Poisson process with stochastic intensity (see the section on Credit risk). If the price is not Markovian and the total state variable is observable then we increase the dimensionality of the model to include the full information and consider a multidimensional Ito process. If some part of the filtration is not observable then we base our modelling on the Kalman filter technique, see section ( Kalman filter II ). If we are not sure about precise manner of interaction between the missing information and the price process then we may perform Bayesian modification of the model possibly combined with the hierarchical techniques, see the section ( Baysian statistics ).

It is common in financial analysis to use some limited model because hedging strategy restricts most sources of uncertainty or because the contract under consideration is not sensitive to some degrees of freedom. The section on incomplete markets contains a technique for combining several models into a consistent pricing policy.

More remarks on the subject of modelling are placed at the beginning of the part ( Data Analysis part ) and the part ( Implementation tools II ).

Principal insight into composition of stationary Markov process is given by the proposition ( Construction of generic Levy process ).

 Notation. Index. Contents.