Abstract The volatility ambiguity is a major mode

l of risk in Black-Scholes pricing formula.

We will study, from the point of view of a supervising angent, how to provide a coherent and dynamic monetory risk measure for margin requirements of a composition of different types of risky positions.

Theoretically, we have devoloped a new notion sublinear, or coherent, expectation --G-Expenctation derived by a nonlinear heat equation. Random variables with G-normal distributions are then defined and, accordingly, we introduce the so-called G-normal Brownian.

We then establish the related stochastic calculus, especially stochastic integrals of Itô's type with respect to a G-Brownian motion and derive the related Itô's formula. We have also proved the existence and uniqueness of stochastic differential equation under this new stochastic calculus.

The whole framework is out of Wiener probability measure based on the foundation of probability theory established by Kolmogorov 1933.

It is in fact a new theory of nonlinear probability. Moreover, the above Wiener probability space as well as many other types of spaces of risk measures, or sublinear expectations, can be treated universally in our new framework.

Practically, we have developed an algorithm for the numerical calculation of G-expectation generalied the well-known risk measure SPAN (System of Portfolio Analysis) to a dynamically consistent risk measure.

(g –Expectation)

Model without Ambiguity

To use the language of the nonlinear expectation, instead of probability 1933 Foundation of Prob.

Kolmogorov1918-1921 1925

Daniell Wiener Sublinear expextation Interagal Wiener measure

Coherent Risk Measure

Workshop3 talks: 2006

2006

Random Variable

Space of Lipschitz functions

[Lyons1995AMF] Uncertain volatility and the risk free synthesis of derivatives.

Coherent Risk Measure

Coherent Risk Measure

(Bounded)