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With the distribution of the loss function, we are ready to introduce so-called risk measures. Their main purpose is stated by Föllmer and Schied in [7] as: . a risk measure is viewed as a capital requirement: We are looking for the minimal amount of capital which, if added to the position and invested in a risk-free manner, makes the position acceptable. For completeness, we state: A risk measure ρ is a real-valued mapping defined on the space of random variables (risks). g. [7, 14, 17]). 1 10 Computational Challenges in Finance 17 As this discussion is beyond the scope of this survey, we restrict ourselves to the introduction of two popular examples of risk measures: The one which is mainly used in banks and has become an industry standard is the value-at-risk.

Then, for a step size of Δ = T /n > 0, the discretized process S(Δ ) (t) generated by the EMS is defined by S(Δ ) (0) s0 , S(Δ ) (kΔ ) S(Δ ) ((k − 1) Δ ) + μ S(Δ ) ((k − 1) Δ ) Δ + σ S(Δ ) ((k − 1) Δ ) Δ Wk , k = 1, . . , n. Here, Δ Wk , k = 1, . . , n, is a sequence of independent, N (0, Δ )-distributed random variables. Between two consecutive discretization points, we obtain the values of S(Δ ) (t) by linear interpolation. The EMS can easily be generalized to a multidimensional setting. s.

In most cases, the model will depend on a set of model parameters that are not directly observable from the market. We denote this set by M. e. the market parameters, such as spot prices or interest rates, is denoted by O. Finally, there is a third parameter set P, entering a pricing formula, which contains parameters of the financial product, also referred to as product parameters. For instance, for a European call on a stock we have P = {K, T }, where K is the strike and T the maturity of the call.

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