On pricing contingent claims in a two interest rates jump-diffusion model via market completions. (English) Zbl 1195.91159

Kane, S.; Melnikov, A.

Teor. Jmovirn. Mat. Stat. 77, 52-63 (2007) and Theory Probab. Math. Stat. 77, 57-69 (2008).

Well-known financial market models deal with unique interest rate for both deposit and credit (see, for example, the books by R. Elliott and P. E. Kopp [Mathematics of Financial Markets, Springer-Verlag, Berlin, 1998], I. Karatzas and S. E. Shreve [Applications of Mathematics, Berlin: Springer (1998; Zbl 0941.91032)] ). In reality, the credit rate is always higher than the deposit rate. This market constraint brings new difficulties into the problem of hedging contingent claims. In contrast with complete markets, there is no symmetry between seller and buyer positions in the case of a market with constraints. The fair price of the derivative security (option) is split into the upper and lower prices. Hence, the problem of hedging a given contingent claim is to find these prices.
The authors consider the problem of hedging contingent claims in the framework of a two-factor jump-diffusion model with different credit and deposit rates and derive the formulas for the above prices in terms of parameters of the initial model. The upper and lower hedging prices are derived for European options by means of auxiliary completions of the initial market. An extension of the methodology of completions of the initial market in a two interest rates jump-diffusion financial market is described. It is shown how the obtained results are applied in the Black-Scholes model [R. Korn, ZOR, Math. Methods Oper. Res. 42, No. 3, 255–274 (1995; Zbl 0836.90010)]) and in the Merton model [R. C. Merton, Continuous-time finance, Cambridge, MA: Blackwell (1999; Zbl 1019.91502)]).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

91G20	Derivative securities (option pricing, hedging, etc.)
60H30	Applications of stochastic analysis (to PDEs, etc.)
60J75	Jump processes (MSC2010)
60G44	Martingales with continuous parameter

Keywords:

European options; market; hedging; contingent claims; jump-diffusion model; credit rate; deposit rate

Citations:

Zbl 0941.91032; Zbl 0836.90010; Zbl 1019.91502