Optimal Portfolio Strategy with Discounted Stochastic Cash Inflows When the Stock Price Is a Semimartingale

This paper discusses optimal portfolio with discounted stochastic cash inflows (SCI). The cash inflows are invested into a market that is characterized by a stock and a cash account. It is assumed that the stock and the cash inflows are stochastic and the stock is modeled by a semi-martingale. The Inflation linked bond and the cash inflows are Geometric. The cash account is deterministic. We do some scientific analyses to see how the discounted stochastic cash inflow is affected by some of the parameters. Under this setting, we develop an optimal portfolio formula and later give some numerical results.


Introduction
For example in financial mathematics, the classical model for a stock price is that of a geometric Brownian motion.However, it is argued that this model fails to capture properly the jumps in price changes.A more realistic model should take jumps into account.In the Jump diffusion model, the underlying asset price has jumps superimposed upon a geometric Brownian motion.The model therefore consists of a noise component generated by the Wiener process, and a jump component.It involves modelling option prices and finding the replicating portfolio.Researchers have increasingly been studying models from economics and from the natural sciences where the underlying randomness contains jumps.According to Nkeki [1], the wars, decisions of the Federal Reserve, other central banks, and other news can cause the stock price to make a sudden shift.To model this, one would like to represent the stock price by a process that has jumps (Bass [2]).Liu et al. (2003) [3] solved for the optimal portfolio in a model with stochastic volatility and jumps when the investor can trade the stock and a risk-free asset only.They also found that Liu and Pan (2003) [4] extended this paper to the case of a complete market.Arai [5] considered an incomplete financial market composed of d risky assets and one riskless asset.Branger and Larsen [6] solved the portfolio planning problem of an ambiguity averse investor.They considered both an incomplete market where the investor can trade the stock and the bond only, and a complete market, where he also has access to derivatives.In Guo and Xu (2004) [7], researchers applied the mean-variance analysis approach to model the portfolio selection problem.They considered a financial market containing 1 d + assets: d risky stocks and one bond.The security returns are assumed to follow a jump-diffusion process.Uncertainty is introduced by Brown motion processes and Poisson processes The general method to solve mean-variance model is the dynamic programming.Dynamic programming technique was firstly introduced by Richard Bellman in the 1950s to deal with calculus of variations and optimal control prob-lems (Weber et al. [8]).Further developments have been obtained since then by a number of scholars including Florentin (1961,1962) and Kushner (2006), among others.In Jin and Zhang [9], researchers solved the optimal dynamic portfolio choice problem in a jump-diffusion model with some realistic constraints on portfolio weights, such as the no-short-selling constraint and the no-borrowing constraint.Beginning with work of Nkeki [1] which involves optimization of the portfolio strategy using discounted stochastic cash inflows, this work explores optimal portfolio strategy using jump diffussion model.
In Nkeki [1], the stock price is modelled by continuous process which is geometric and but in this work we assume that the stock price process is driven by a semimartingale; defined in Shiryaev et al. [10].The jump diffusion model combines the usual geometric Brownian motion for the diffusion and the general jump process such that the jump amplitudes are normally distributed.
Semimartingales as a tool of modelling stock prices processes has a number of advantages.For example this class contains discrete-time processes, diffusion processes, diffusion processes with jumps, point processes with independent increments and many other processes (Shiryaev [11]).The class of semimartingales is stable with respect to many transformations: absolutely continuous changes of measure, time changes, localization, changes of filtration and so on as stated in (Sharyaev [11]).Stochastic integration with respect to semimartingales describes the growth of capital in self-financing strategies.In this research, a sufficient maximum principle for the optimal control of jump diffusions is used showing dynamic programming and going applications to financial optimization problem in a market described by such process.
For jump diffusions with jumps, a necessary maximum principle was given by Tang and Li, see also Kabanov and Kohlmann (∅ksendal and Sulem [12]).If stochastic control satisfies the maximum principle conditions, then the control is indeed optimal for the stochastic control problem.It is believed that such results involves a useful complicated integro-differential equation (the Hamilton-Jacobi-Bellmann equation) in the jump diffusion case.The investor's stochastic Cash inflows (CSI) into the cash account, on inflation-linked bond and stock were considered.Most calculations and methods used were influenced by the works of Nkeki [1], Nkeki [13] ∅ksendal [14], ∅ksendal and Sulem [12], Klebaner [15] and Cont and Tankov [16].

Model Formulation
Let ( ) be a probability space where denotes the "flow of information" as discussed in the definition.Mathematically the latter means that  consists of σ-algebras, i.e. for all is a 2-dimensional process on a given filtered probability space , where  is the real world probability measure, t is the time period, T is the terminal time period, ( ) I W t is the Brownian motion with respect to the "noise" arising from the inflation and ( ) I W t is the Brownian motion with respect to the "noise" arising from the stock market.
The dynamics of the cash account with the price

( )
Q t is given by: ( ) where r is the short term interest as defined in Nkeki [1].
The price of the inflation-linked bond is given by the dynamics: ( ) ( ) is the volatility of inflation-linked bond, I φ is the market price of inflation risk, t I is the inflation index at time t and has the dynamics: where q is the expected rate of inflation, which is the difference between nominal interest rate, r real interest r and I σ is the volatility of inflation index.Suppose the financial process ( stock return) is given on a filtered probability space.Assume that ( ) where is a semi-martingale with respect to  and  .Using Itô formula for semimartingales (see Appendix) and then differentiating the process we have where ( ) Using random measure if jumps (see [11]) ( ) Substituting on Equation (7) into Equation (50) we have ) ) We know that differential of our stock price can written as where ( ) ( ) W t defined as before.
Now comparing Equation (8) with Equation ( 9), we can now see that when we equate the predictable parts we have ( ) Equating the continuous parts we get and the jump parts give ( ) ( ) ( ) ( ) and hence we let ( ) ( ) ( ) From (11) it follows that and hence it follows that ( ) Substituting Equation (12) into Equation (9) we have and further simply it to where ( ) ( ) )( ) Hence we define the following , , , 0 The market price of the market risk is given by where, s φ is the market price of stock market risk.We assume the process ( ) which is geometric and with the no arbitrage conditions applied to it obtain the following stochastic differential equation, ( ( ) P t is a martingale that is always positive and satisfies ( ) . Now we have the price density given by ( ) where

The Dynamics of Stochastic Cash Inflows
The dynamics of the stochastic cash inflows with process, ( ) where , is the volatility of the cash inflows and k is the expected growth rate of the cash inflows. 1 D σ is the volatility arising from inflation and 2 D σ is the volatility arising from the stock market.
Solving for ( ) D t we use Itò's formula for continuous processes.Let ( )

The Dynamics of the Wealth Process
If ( ) X t is the wealth process and ( ) ( ) 0 , , is the admissible portfolio where 0 θ is number of units in the cash account, I θ is the number of units in the inflation bond and S θ is the number of units in the stock.In an incomplete market with no arbitrage we have 0 1 I S θ θ θ = − − .The dynamics of the wealth process is given by ( ) (see Appendix).For we have the dynamics of the wealth process as For the Poisson jump measure we have the dynamics of the wealth process as where N is the Poisson measure and ν is the compensator on the Poisson measure N .

The Discounted Value of SCI
In this Section, we introduce Definition 1.The discounted value of the expected future SCI is defined as where ( ) is the conditional expectation with respect to the Brownian Filtration { } 0 F t t≥ and ( ) ( ) ( ) is the stochastic discount factor which adjust for nominal interest rate and market price of risks for stock and inflation-linked bond (Nkeki [1]).
is the discounted value of the expected future SCI, then Proof.By definition 1, we have that Applying change of variable on 30, we have { } We further take note that for 0 ν = we have the discounted value of the SCI as The differential form of ( ) is given by Equation ( 32) is obtained by differentiating as shown in the proof below The current discounted cash inflows can be obtained by putting ( ) we can change the horizon by allowing our T to go up to ∞ i.e. ( In case of deterministic case, we have and for r k > , and T → ∞ we have  ( ) Ψ with respect to r, we have Differentiating 0 Ψ with respect to D σ , we have The following calculations shows how we differentiated ( ) Ψ with respect to 0 D , we have Differentiating 0 Ψ with respect to r, we have Differentiating 0 Ψ with respect to D σ , we have Differentiating 0 Ψ with respect to k, we have Table 1 shows the sensitivity of variables.Sensitivity analysis can be incorporated into discounted cash inflows analysis by examining how the discounted cash inflows of each project changes with changes in the inputs used.These could include changes in revenue assumptions, cost assumptions, tax rate assumptions, and discount rates.It also enables management to have contingency plans in place if assumptions are not met.It also shows that the return on the project is sensitive to changes in the projected revenues and costs.Looking at Table 1, one can see that changing a variable can make Table 1.Simulation of the sensitivity analysis.an impact on the SCI.An investor must do the sensitivity analysis in order to know changes can be made on the market to improve the results of an investment.

The Dynamics of the Value Process
Proposition 2. If ( ) V t is the value process and ( ) ( ) ( ) where Ψ is the discounted value of the expected future SCI then the differential form of ( ) Proof.Differentiating ( ) V t and substituting Equations ( 32) and ( 26) on the differential obtained we have ) For 0 µ ν = = , the jump part becomes zero and we obtain

Finding Optimal Portfolio
Theorem 3. Let ( ) X t be the worth process whose dynamics is defined by Equation ( 23), the discounted value of expected future stochastic cash inflow as defined in proportion (1), ( ) V t the value process as defined in proportion (2) and ( ) the utility function and if we assume that 0 ν = , the optimal portfolio is given by ( ) ( ) The proof is given in Appendix.
From Equation (71),   70).This figure shows that when 0 t = , the portfolio value is 0.151 which is equivalent to 15.1% when the value of the wealth is 40,000 and the portfolio value is 0.159 which is equivalent to 15.9% when the value of the wealth is 1,000,000.When 10 t = , the portfolio value is 0.16 which is equivalent to 16% when the value of the wealth is 40,000 and the portfolio value is 0.1604 which is equivalent to 16.04% when the value of the wealth is 1,000,000.This shows that there is a huge increase on the portfolio value from

Some Numerical Values
when the value of the wealth is small and there in less change when the value of the wealth is large.71).This figure shows that when 0 t = , the portfolio value is 0.907 which is equivalent to 90.7% when the value of the wealth is 40,000 and the portfolio value is 0.9019 which is equivalent to 90.19% when the value of the wealth is 1,000,000.When 10 t = , the portfolio value is 0.9017 which is equivalent to 90.17% when the value of the wealth is 40,000 and the portfolio value is 0.9017 which is equivalent to 90.17% when the value of the wealth is 1,000,000.This shows that there is a huge decrease on the portfolio value from when the value of the wealth is small and there in less change when the value of the wealth is large.72).This figure shows that when 0 t = , the portfolio value is −0.057 which is equivalent to −5.7% when the value of the wealth is 40,000 and the portfolio value is −0.0613 which is equivalent to −6.13% when the value of the wealth is 1,000,000.When 10 t = , the portfolio value is −0.0615 which is equivalent to −6.15% when the value of the wealth is 40,000 and the portfolio value is −0.0613 which is equivalent to 6.13% when the value of the wealth is 1,000,000.This shows that there is a huge decrease on the portfolio value from

Conclusion
Semimartingales seems to model financial processes better since the cater for the jumps that occur in the system.The continuous processes may be convenient because one can easily produce results.For example, in our situation we managed to find the portfolio for continuous processes but we couldn't for the ones with jumps.This work can be extended designing a MATLAB program that will solve the equation for portfolio θ .
to find our SDE, assume that ( ) x f x e = and substitute on Equation (47).Simplifying will give the following results

( ) ( ) ( )
Differentiating will give; ( ) ( ) Now the differential of the stock process is given by where ( ) then, using Ito's formula for semimartingales (Protter [?]), we have and in differential form, this can be expressed as ( )

Appendix B
Assuming ln and substituting it on the formula we get )( ) ) and substituting on 58 to have Taking the expectations on both sides we have ( For simplicity we have By Equation (57), we have the integral on the right hand side being equals to zero.That is substituting , Σ ∆ and D σ as defined , we obtain the following ( ) Using Itó's formula for jump diffusion ( )

Ψ
we are interested to see how it behaves with respect to 0 , , , and D T D k r σ we need to take ( ) 0 Ψ as a function of 0 , , , and D T D k r σ .Then we can look at the sensitivity analysis of with respect to T, we have ( ) We repeated the following procedure for all other variables.When we have a deterministic case, differentiating 0 Ψ partially we have the fol- that offset shock from the SCI at time t.

Figure 3
the value of the wealth is small and

Figure 2 .
Figure 2. Portfolio value in stock.

Figure 3 .
Figure 3. Portfolio value in cash account.


Differentiating both sides we obtain following partial differential equation with jumps.as in Equation (57) Under technical conditions, the value function V satisfies We consider the function of θ which is can solve for θ because we have a linear equation below next manuscript to SCIRP and we will provide best service for you:Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.A wide selection of journals (inclusive of 9 subjects, more than 200 journals) Providing 24-hour high-quality service User-friendly online submission system Fair and swift peer-review system Efficient typesetting and proofreading procedure Display of the result of downloads and visits, as well as the number of cited articles Maximum dissemination of your research work Submit your manuscript at: http://papersubmission.scirp.org/Or contact jmf@scirp.org