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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.

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 super- imposed 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 [

In Nkeki [

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 [

Let

The dynamics of the cash account with the price

where r is the short term interest as defined in Nkeki [

The price of the inflation-linked bond

where

where

Suppose the financial process ( stock return)

where

Using Itô formula for semimartingales (see Appendix) and then differentiating the process we have

where

Using random measure if jumps (see [

hence

Substituting on Equation (7) into Equation (50) we have

We know that differential of our stock price can written as

where

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

Substituting Equation (12) into Equation (9) we have

and further simply it to

where

Using Itó’s formula for jump diffusion

(see Appendix). Hence we define the following

The market price of the market risk is given by

where,

Using Itó’s formula for jump diffusion equation on 17 we have

where

Now we have the price density given by

where

The dynamics of the stochastic cash inflows with process,

where

rate of the cash inflows.

Solving for

If

where

(see Appendix). For

For the Poisson jump measure we have the dynamics of the wealth process as

where

In this Section, we introduce

Definition 1. The discounted value of the expected future SCI is defined as

where

Proposition 1. If

Proof. By definition 1, we have that

Applying change of variable on 30, we have

starting with

we have

and lastly

We further take note that for

The differential form of

Equation (32) is obtained by differentiating

differentiating both sides,

The current discounted cash inflows can be obtained by putting

If

our T to go up to

In case of deterministic case, we have

and for

Since

Finding partial derivatives of

Differentiating

Differentiating

Differentiating

Differentiating

Differentiating

where

The following calculations shows how we differentiated

differentiating with respect to T

We repeated the following procedure for all other variables.

When we have a deterministic case, differentiating

Differentiating

Differentiating

Differentiating

Differentiating

Differentiating

1 | 1.00 | −4.01 | −4.83 | 100.4359 | −50.15 | −12.54 | −18.05 | 50.15 |

2 | 2.01 | −16.09 | −19.36 | 100.87 | −201.16 | −50.29 | −72.42 | 201.16 |

3 | 3.02 | −36.31 | −43.69 | 101.31 | −453.93 | −113.42 | −163.42 | 453.93 |

4 | 4.04 | −64.753 | −77.90 | 101.76 | −809.34 | −202.34 | −291.36 | 809.34 |

5 | 5.05 | −101.46 | −122.07 | 102.20 | −1268.27 | −317.07 | −456.58 | 1268.27 |

6 | 6.08 | −146.53 | −176.29 | 102.64 | −1831.63 | −457.91 | −659.39 | 1831.63 |

7 | 7.11 | −200.02 | −240.65 | 103.09 | −2500.31 | −625.08 | −900.11 | 2500.31 |

8 | 8.14 | −262.02 | −315.24 | 103.54 | −3275.22 | −818.80 | −1179.08 | 3275.22 |

9 | 9.18 | −332.58 | −400.14 | 103.99 | −4157.27 | −1039.32 | −1496.62 | 4157.27 |

10 | 10.22 | −411.79 | −495.44 | 104.45 | −5147.39 | −1286.85 | −1853.06 | 5147.39 |

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.

Proposition 2. If

Proof. Differentiating

For

Theorem 3. Let

given by

and

The proof is given in Appendix.

From Equation (71),

classical portfolio strategy at time t and

temporal hedging term that offset shock from the SCI at time t.

Some Numerical Valuesand 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

there in less change when the value of the wealth is large.

For

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

We thank the editor and the referee for their comments. We also thank Professor E. Lungu for the guidance he gave us on achieving this. Lastly, we thank the University of Botswana for the resources we used to come up with this paper. Not forgetting the almighty God, the creator.

Baraedi, O. and Offen, E. (2016) Optimal Portfolio Strategy with Discounted Stochastic Cash Inflows When the Stock Price Is a Semimartingale. Journal of Mathematical Finance, 6, 660- 684. http://dx.doi.org/10.4236/jmf.2016.64047

Assume that

to find our SDE, assume that

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

Assuming

Let

then

where

with

Let

take

Choosing

Substituting

Integrating both sides we get

Taking the expectations on both sides we have

For simplicity we have

Where

Which gives us

By Equation (57), we have the integral on the right hand side being equals to zero. That is

Differentiating both sides we obtain following partial differential equation with jumps.

Consider the value function

where J is as in Equation (57) Under technical conditions, the value function V satisfies

This takes us to the HJB equation;

where

Taking our utility function as

We consider the function of

Differentiating

Since

For

and

substituting

where

and

We can now see that

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