×

zbMATH — the first resource for mathematics

Bounds on process of contingent claims in an intertemporal economy with proportional transaction costs and general preferences. (English) Zbl 0935.91014
The authors consider an economy with two securities, a bond with price \(B(t)\) and stock with price \(S(t)\) at time \(t\geq 0\). The bond pays no coupons, is default free and has price dynamics \(B_{t}=e^{rt}B_0\), \(t\geq 0\), where \(r\) is the constant rate of interest. The stock price is the diffusion process \[ S_{t}=S_0\exp\left\{\left( \mu-{\sigma^2\over 2}\right)t+ \sigma W_{t}\right\}, \] where the mean rate of return \(\mu\) and the volatility \(\sigma\) are constants such that \(\mu>r\), \(\sigma\neq 0\), \(W_{t}\) is a one-dimensional standard Brownian motion. The investor holds \(x_{t}\) dollars of the bond and \(y_{t}\) dollars of the stock at date \(t\), and consumes at the rate \(c_{t}\) dollars out of the bond account. The stock account process, starting with \(y_0=y\), is \[ y_{t}=y+\int_0^{t}\mu y_{\tau} d\tau+ \int_0^{t}\sigma y_{\tau} dW_{\tau} +L_{t}-M_{t}, \] where \(L_{t}\) represents the cumulative dollar amount transferred into the stock account and \(M_{t}\) the cumulative dollar amount transferred out of the stock account, \(L_0=M_0=0\). The bond account process, starting with \(x_0=x\), is \[ x_{t}=x+\int_0^{t}\{r x_{\tau} -c_{\tau}\} d\tau-\beta L_{t}+\alpha M_{t},\quad 0<\alpha<1<\beta. \] Analytic bounds on the reservation write price of European-type contingent claims are derived. The option prices are obtained via a utility maximization approach by comparing the maximized utilities with and without the contingent claim.

MSC:
91B28 Finance etc. (MSC2000)
93E20 Optimal stochastic control
60G40 Stopping times; optimal stopping problems; gambling theory
PDF BibTeX XML Cite
Full Text: DOI