Abstract
In general, the term risk signifies the possibility that some future event might have some negative consequences. Since the future is uncertain, the term risk is tightly connected with the probability or likelihood that an uncertain future event actually becomes real.
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- 1.
Naked short selling, i.e. selling of shares without owning or borrowing (or at least ensuring that they can be borrowed) has been largely restricted in many countries, though it is still possible to sell shares that have been borrowed beforehand, e.g. by a security lending trade.
- 2.
Here, the term liquidity risk means market liquidity risk rather than the risk not to be able to pay.
- 3.
The difference to a in Eq. 21.11 lies in the sign of the VaR.
- 4.
This can be seen quite easily: Let c = N(x). By definition, x is the percentile associated with c. Or equivalently, the inverse of the cumulative distribution function gives the percentile \(Q_{c}=x=\text{N}^{-1}\left [ \text{N}(x)\right ] =\text{N}^{-1}(c)\). Applying N−1 to the symmetry equation N(−x) = 1 −N(x) gives
$$\displaystyle \begin{aligned} \text{N}^{-1}\left[ \text{N}(-x)\right] & =\text{N}^{-1}\left[ 1-\text{N}(x)\right] \\ \Leftrightarrow \qquad -x & =Q_{1-\text{N}(x)}\;. \end{aligned} $$Substituting c for N(x) and Q c for x immediately yields − Q c = Q 1−c.
- 5.
\(\exp (x)\approx 1+x+\cdots \)
- 6.
Note that \(Q_{1-c}^{{ }_{\text{N}(0,1)}}<0\) for all reasonable confidence levels c, see for instance Eq. 21.13. Since for all time spans δt usually considered and for all reasonable values of μ the drift term is smaller than the volatility term, the value at risk is a positive number.
- 7.
Here, we make use of the so called Kronecker delta often appearing in the science. It is defined as
$$\displaystyle \begin{aligned} \delta_{ij}=\left\{ \begin{array}{cc} 1 & \mbox{for }i=j\\ 0 & \mbox{for }i\neq j \end{array} \right. \end{aligned}$$ - 8.
A short and simple overview of matrix algebra can be found in [78], for example.
- 9.
Since A is invertible we have: \((\mathbf {AA}^{-1})^{T}=\mathbf {1}\Rightarrow ({\mathbf {A}}^{-1} )^{T}\,{\mathbf {A}}^{T}=\mathbf {1}\Rightarrow ({\mathbf {A}}^{-1})^{T} \,\underset {\mathbf {1}}{\underbrace {{\mathbf {A}}^{T}({\mathbf {A}}^{T})^{-1}} }=\mathbf {1}({\mathbf {A}}^{T})^{-1}\).
References
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1972)
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Deutsch, HP., Beinker, M.W. (2019). Fundamentals. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_21
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DOI: https://doi.org/10.1007/978-3-030-22899-6_21
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Publisher Name: Palgrave Macmillan, Cham
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