By Kraft
This thesis summarizes so much of my contemporary study within the box of portfolio optimization. the most issues which i've got addressed are portfolio issues of stochastic rates of interest and portfolio issues of defaultable resources. the place to begin for my learn was once the paper "A stochastic keep an eye on ap proach to portfolio issues of stochastic rates of interest" (jointly with Ralf Korn), within which we solved portfolio difficulties given a Vasicek time period constitution of the quick cost. Having thought of the Vasicek version, it used to be noticeable that I may still learn portfolio difficulties the place the rate of interest dynamics are gov erned by means of different universal brief price versions. The correct effects are provided in bankruptcy 2. the second one trouble issues portfolio issues of default capable resources modeled in an organization price framework. because the resources of an organization then correspond to contingent claims on company price, I sought for how to simply take care of such claims in portfolio difficulties. accordingly, I constructed the pliancy method of portfolio optimization that's offered in bankruptcy three. despite the fact that, this fashion of tackling portfolio difficulties isn't really constrained to portfolio issues of default capable resources merely, however it offers a basic framework taking into consideration a compact formula of portfolio difficulties whether rates of interest are stochastic.
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Additional info for Optimal Portfolios with Stochastic Interest Rates and Defaultable Assets
Sample text
Obviously, the percentage invested in the savings account is given by 1 - 71"'1 where 1:= (1, ... ,1)' E md. 5 The wealth equation can be interpreted as a controlled SDE with the control being the portfolio process 71"(')' In this setting the investor chooses a portfolio process to maximize utility. We assume that his preferences can be represented by the utility function U (x) = x', x ::::: 0, 0 < I < l. 1 and leads to a positive wealth process X7r. We are now in a position to formulate his optimization problem: 6 4 5 6 See for example Musiela/Rutkowski (1997), pp.
1) lead to the explicit solution Hence, applying the Holder inequality, we arrive at E( exp (,I t 1'(s)ds)) ~ E( exp (, lt {e-O. 50" t {Y(O) l u l u O5 e . l
29) in the Ho-Lee model. Vasicek Model Our candidate for the value function is ,T] 00. 2 Ho-Lee and Vasicek Model 33 where H2 is a continuously differentiable and deterministic function. With suitable constants K i , i = 1, ... ,6, we find that G(t, X(t), r(t)) = X(t)' . exp (1~'Y (H2(t) - H2(T)) + ;(1 - exp(a(t - T)))r(t)) ::::: Kl . X(t)' . exp l,t +,l,t l,t ::::: K2 . 5(n(s)a(s))2 ds n(s)a(s) dW(S)) . exp (;(1- exp(a(t - T)))· r(t)) ::::: K3 . exp (, r(s) ds +, l,t n(s)a(s) dW(S)) . exp (;r(t)) .