The benchmark model introduced by Platen (2002) has not been well studied in literature for the case of incomplete information whereas the incompleteness of the information in financial modeling is usual in practice. Some of the factors that characterize the evolution of the market may be hidden. It is well-known that a financial market with constant parameters can only model for a relatively short period of time. Thus, there is a need to use stochastically varying parameters. One possibility is to introduce a continuous-time Markov chain, as a random source to modulate the coefficients of the stochastic differential equation (SDE) representing the general market direction. In the present work, we consider a jump diffusion model with Markov-modulated coefficients. First we derive the growth optimal portfolio (GOP) for the case of complete information by using the Hamilton-Jacobi-Bellman equation to solve the optimization problem.
[...] Along these lines, in Platen (2002, 2004), pricing and hedging are performed for complete markets with jump diffusions without the measure transformation. Platen and Runggaldier (2004) have generalized the model by using the GOP for pricing and hedging in incomplete markets when there are unobserved factors to be ﬁltered. The paper is structured in the following way: Section 2 introduces the ﬁnancial model under which we derive option price for the European kind contingent claim with benchmark approach. GOP has been derived in section 3 and the fair price of a European option is obtained. [...]
[...] for t T Deﬁnition 3.3 : We call a price process Λ = t T fair if the corresponding benchmarked process Λ(t) , t T V form an P )-martingale, i.e., it satisﬁes the conditions Λ = = Λ(T ) ) < and Λ(t) = E[Λ(T ) Ft Proposition The fair price1 ΛHT at time t for a given contingent claim HT is given by the fair pricing formula ΛHT = V In Follmer and Schweizer (1991), it is shown that minimizing the intrinsic risk rising due to the incompleteness of the market is similar to ﬁnding a mean self-ﬁnancing strategy where the cost process C is orthogonal to the martingale part of S which is called optimal strategy. [...]
[...] Without loss of generality, we can assume It can be written as dVtπ = Vtπ ((πt σ1 (Xt ) + πt σ2 (Xt (dWt + θ1 (Xt + (πt ρ1 (Xt ) + πt ρ2 (Xt + θ2 (Xt where θ1 (Xt ) and θ2 (Xt ) are market price of risk characterized by the equations σ1 (Xt )θ1 + ρ1 (Xt )θ2 = µ1 (Xt ) σ2 (Xt )θ1 + ρ2 (Xt )θ2 = µ2 (Xt We assume that V0π = x0 is the given initial wealth G ROWTH OPTIMAL PORTFOLIO In this section we deﬁne the GOP and derive the fair price for European option. [...]
[...] In particular, π is a growth optimal portfolio strategy. We now take the V π , the wealth process corresponding to a GOP, as benchmark or reference unit and we call prices when expressed in unit of V π as benchmarked prices. Furthermore, we call a model with the above prescribed form of prices a benchmark model. That is, we have the benchmarked primary security account process = V , = and the benchmarked value of the wealth process corresponding to the portfolio π is Vtπ t 4 t ˆ Vtπ = V for t T By Ito's formula t the benchmarked value process will satisfy the SDE Vπ dV π = V π (t)σ1 (Xt ) + π2 (t)σ2 (Xt ) (Xt ))dWt + (π1 (t)ρ1 (Xt )+π2 (t)ρ2 (Xt )+1)θ2 λ(t) 1 dMt the above deﬁned fair price coincides with the corresponding risk-neutral price in case of complete (incomplete) market. [...]
[...] H can be written as Using the fair price concept introduced by Platen (2002) the European option is priced and T T πt dS2 + LT by the help of martingale representation theoπt dS1 + HT = H0 + rem the option has been hedged in complete information and later in incomplete information where LT is martingale orthogonal to the mar- by taking the optimal projection of the strategy tingale part of S. Which is Follmer-Schweizer towards F S . [...]
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