Motivation of Forward Entropic Risk Mea- sures Sample Clauses

Motivation of Forward Entropic Risk Mea- sures. Maturity-independent Risk Measure- ment‌ In the first section of this chapter, we motivate the study of the forward entropic risk measures, which were first introduced in Zariphopoulou and Zˇitkovi´c (2010). Let W = {Wt}t≥0 be a d-dimensional Brownian motion on a probability space (Ω, F, P). Denote by F = {Ft}t≥0 the augmented filtration generated by W . Assume that we have a financial market environment building upon this filtered probability space. We first recall how dynamic risk measures on the risk positions in the financial market are defined. At the initial time t = 0, fix any terminal date T > 0, and thus, the investment horizon [0, T ] is fixed. Denote L∞(FT ) as the space of all FT -measurable and uni- formly bounded random variable defined on the filtered probability space. The conditional risk measure ρt,T , at any intermediary time t ∈ [0, T ], is defined as a mapping from L∞(FT ) to L∞(Ft), the space of all Ft-measurable and uniformly bounded random variable, which satisfies certain reasonable axioms, namely, con- ditional translation invariance and monotonicity. The dynamic risk measure ρ·,T is then defined as the family of conditional risk measures ρ·,T = {ρt,T }t∈[0,T]. From the definition of dynamic risk measures, we observe an implicit but crucial assumption being made at the initial time t = 0. It is assumed that all risk positions in consideration will be introduced and matured at any time in the fixed investment horizon [0, T ]. However, this is not a realistic assumption, and hence, the dynamic risk measure, solely defined in the fixed investment horizon [0, T ], fails to manage any new risk positions with larger maturities. Mathematically, although we may have complete knowledge of all risk positions maturing in the fixed investment horizon [0, T ] at the initial time t = 0, a new risk position, with the maturity Tr > T , may be introduced at a post-initial time s ∈ (0, T ). The dynamic risk measure ρ·,T = {ρt,T }t∈[0,T], defined at the initial time t = 0, cannot evaluate the new risk position since the conditional risk measures are not defined for times t ∈ (T, Tr]. At time s ∈ (0, T ), the simplest way to remedy this situation is to extend the dynamic risk measure ρ·,T = {ρt,T }t∈[0,T] to a new dynamic risk measure ρ˜·,T′ = {ρ˜t,T′}t∈[s,T′]. If the intrinsic time consistency is essential, ρ·,T and ρ˜·,T′ must be related by ρ˜t,T′ = ρt,T ∀t ∈ [s, T ]. On the other hand, if the intrinsic time consistency needs to be preserved, we mu...