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Citation: Qiuli Liu, Xiaolong Zou. Dassios A, Jang J-W. The book presents Markov decision processes in action and includes various state-of-the-art applications with a particular view towards finance. See all 1 customer reviews. For small c the risk loading is the same as in the case of no derivative held, since the probability that the derivative has a positive payoff is small.
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nanicamo.cf MR  J. McNeil, R. Frey, and P. In the statement of [ 2 , Theorem 8. For proving Theorem 3. We need to check the conditions given in Definition 3. Now we need to show that the backward equation 11 has a solution and hence also 9 has a solution. Let F be the distribution function of Y 1,1. Expanding the above expression yields the first and the second claim of the lemma.
The right-hand side is finite by Assumption 2. The function h is a bounded linear operator. Thus h is bounded by Lemma 3. Since q is also continuous, the first statement follows. The claim that g is locally Lipschitz follows from Lemma 3. Denote the corresponding Lipschitz constant by L R. The first claim follows from Lemma 3. The latter follows from Lemmas 3. Now we prove that 11 has a unique maximal local solution. Then the backward equation 11 has a unique maximal local solution.
By Lemma 3. By Lemmas 3. Along the lines of the proof of [ 2 , Theorem 8. Given the value of the index c and the amount of wealth x at time t , the maximum expected utility of terminal wealth can be written as. Therefore 7 simplifies to. In particular, p does not depend on x and we omit that parameter from p henceforth.
The function w 0 does not depend on c. So w 0 is the solution of an ordinary differential equation.
Since w 0 does not depend on c , the backward equation 11 becomes. From 15 we can derive a backward equation for p. Since w 0 does not depend on the second variable, we have. In this section, we present a convenient numerical method for computing the expected value in 17 for the case where the distribution of Y 1,1 has a smooth density.
We have. The first claim now follows by linearity of the convolution. Our aim is to price a CAT spread option, i. We assume that there are M clients in the market who potentially contribute to the claims process. Let a be the fair annual premium for one client, i.
The annual premium for one contract therefore has to be greater or equal than a , since otherwise the insurance company will make an almost sure loss in the long run. Furthermore, the company faces an exogenously given demand curve d for insurance. For our numerical example we choose. We see that the price increases in c.
Partially Observable Markov Decision Processes and Piecewise Determinsitic. Markov Decision .. to Finance, Universitext, DOI / 1. zeimurorenou.cf: Markov Decision Processes with Applications to Finance ( Universitext) (): Nicole Bäuerle, Ulrich Rieder: Books.
Further, we observe that in the SC-model the price is always lower than in the CC-model, since the latter more accurately accounts for a clustering of claims. Utility indifference prices p in the SC-model left and in the CC-model right units on the axes are 10 6 units of currency. For small c the risk loading is the same as in the case of no derivative held, since the probability that the derivative has a positive payoff is small.
As time increases the probability that the payoff of the derivative grows in c and hence compensates losses during the remaining time decreases, but also the expected number of claims before T decreases. An interesting observation is that for small c the first effect dominates and hence the risk loading increases, whereas for large c the latter effect dominates and hence the risk loading decreases.