Dynamic risk management with Markov decision processes by André Philipp Mundt PDF

Risk Management

By André Philipp Mundt

A huge device in probability administration is the implementation of danger measures. We research dynamic types the place probability measures and dynamic danger measures might be utilized. particularly, we clear up a number of portfolio optimization difficulties and introduce a category of dynamic hazard measures through the proposal of Markov choice tactics. utilizing Bayesian keep watch over thought we additionally derive an extension of the latter surroundings once we face version uncertainty.

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A simple choice would be to set R(v) := R, v ∈ S, for fixed R. However, the idea of the model (UM’) is that based on the current state, in every trading period from t − 1 to t it is ensured that the risk does not exceed a certain maximal level. But after some periods the current wealth could already be very high. Thus, the risk constraint R could become redundant if its level is not adapted over time by using the current wealth. Indeed, we will model the function R such that we can ensure that the risk of the relative gain or loss of wealth from one point of time to the next one is not too high.

We have seen in the previous chapter that this is a economically founded choice. The Markov decision model defined below is similar to the one implicitly used in Runggaldier et al. (2002) and Favero and Vargiolu (2006). The time horizon is finite and discrete. Hence, the trading takes place at times t ∈ {0, 1, . . , T } for some T ∈ N. Assuming that the bond is constantly equal to 1 we can model the price of the asset as follows. ,T with distribution P(Y1 = u) = p = 1 − P(Y1 = d) for some probability p ∈ (0, 1) and some 0 < d < 1 < u, which implies no–arbitrage.

Then D(v) = ∅ and as above J0 (v) = −∞. 3 (ii). Remark. It can be observed from the proof that a further generalization is possible. Indeed, the parameters R and γ can be chosen time–dependent if we assume ∗ Rt ≥ 0, t = 1, . . , T . The result would change for γt ≥ p−p , p > p∗ , where now 1−p∗ T J0 (v) = v · 1 + E [Yt − 1] · min{1, δu (γ, R)} . 4). Interpretation of the optimal policy First, let p > p∗ , i. e. we have an upward tendency in the asset, and let the safety level γ be sufficiently large.

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