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

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1. 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 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 [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] ex- tended 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 assets: 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 sem- imartingale; 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] ). Sto- chastic 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 (Æksendal 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] Æksendal [14] , Æksendal and Sulem [12] , Klebaner [15] and Cont and Tankov [16] .

2. Model Formulation

Let be a probability space where denotes the “flow of infor- mation” as discussed in the definition. Mathematically the latter means that consists of σ-algebras, i.e. for all. The Brownian motions 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, is the Brownian motion with respect to the “noise” arising from the inflation and is the Brownian motion with respect to the “noise” arising from the stock market.

The dynamics of the cash account with the price is given by:

(1)

where r is the short term interest as defined in Nkeki [1] .

The price of the inflation-linked bond is given by the dynamics:

(2)

where is the volatility of inflation-linked bond, is the market price of inflation risk, is the inflation index at time t and has the dynamics:

where is the expected rate of inflation, which is the difference between nominal interest rate, real interest and is the volatility of inflation index.

Suppose the financial process ( stock return) is given on a filtered probability space. Assume that is of “exponential form”.

(3)

where is a semi-martingale with respect to and.

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

(4)

where

(5)

Using random measure if jumps (see [11] )

(6)

hence

(7)

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

(8)

We know that differential of our stock price can written as

(9)

where and defined as before.

Now comparing Equation (8) with Equation (9), we can now see that when we equate the predictable parts we have

(10)

Equating the continuous parts we get

(11)

and the jump parts give

and hence we let

(12)

From (11) it follows that and hence it follows that

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

(13)

and further simply it to

(14)

where

Using Itó’s formula for jump diffusion

(see Appendix). Hence we define the following

(15)

The market price of the market risk is given by

(16)

where, 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,

(17)

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

(18)

where (see Appendix).

is a martingale that is always positive and satisfies.

Now we have the price density given by

(19)

where

(20)

3. The Dynamics of Stochastic Cash Inflows

The dynamics of the stochastic cash inflows with process, is given by

(21)

where is the volatility of the cash inflows and k is the expected growth

rate of the cash inflows. is the volatility arising from inflation and is the volatility arising from the stock market.

Solving for we use Itò’s formula for continuous processes. Let and

(22)

4. The Dynamics of the Wealth Process

If is the wealth process and is the admissible portfolio where is number of units in the cash account, is the number of units in the inflation bond and is the number of units in the stock. In an incomplete market with no arbitrage we have. The dynamics of the wealth process is given by

(23)

where

(24)

(see Appendix). For we have the dynamics of the wealth process as

(25)

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

(26)

where is the Poisson measure and is the compensator on the Poisson measure.

5. The Discounted Value of SCI

In this Section, we introduce

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

(27)

where is the conditional expectation with respect to the Brownian Filtration 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] ).

Proposition 1. If is the discounted value of the expected future SCI, then

(28)

Proof. By definition 1, we have that

(29)

Applying change of variable on 30, we have

(30)

starting with

we have

and lastly

We further take note that for we have the discounted value of the SCI as

(31)

The differential form of is given by

(32)

Equation (32) is obtained by differentiating as shown in the proof below

differentiating both sides,

The current discounted cash inflows can be obtained by putting into Equation (28),

(33)

If and we can change the horizon by allowing

our T to go up to i.e.

In case of deterministic case, we have and, so

(34)

and for, and we have

Since is a constant, if we are interested to see how it behaves with respect to we need to take as a function of. Then we can look at the sensitivity analysis of,

Finding partial derivatives of we obtain the followng

Differentiating with respect to T, we have

(35)

Differentiating with respect to, we have

(36)

Differentiating with respect to k, we have

(37)

Differentiating with respect to r, we have

(38)

Differentiating with respect to, we have

(39)

where and and

The following calculations shows how we differentiated with respect to T

differentiating with respect to T

We repeated the following procedure for all other variables.

When we have a deterministic case, differentiating partially we have the fol- lowing

Differentiating with respect to T, we have

(40)

Differentiating with respect to, we have

(41)

Differentiating with respect to r, we have

(42)

Differentiating with respect to, we have

(43)

Differentiating with respect to k, we have

(44)

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.

6. The Dynamics of the Value Process

Proposition 2. If is the value process and where is the discounted value of the expected future SCI then the differential form of is given by

(45)

Proof. Differentiating and substituting Equations (32) and (26) on the dif- ferential obtained we have

For, the jump part becomes zero and we obtain

7. Finding Optimal Portfolio

Theorem 3. Let 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), the value process as defined in proportion (2) and

the utility function and if we assume that, the optimal portfolio is

given by where

and

(46)

The proof is given in Appendix.

From Equation (71), represents the

classical portfolio strategy at time t and represents the inter-

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

Some Numerical Values

Figure 1 was obtained by setting, , , , , , , , , and in Equation (70). This figure shows that when, 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, the portfolio value is 0.16 which is equivalent to 16% when the value of the wealth is 40,000

Figure 1. Portfolio value in inflation-linked bond.

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

Figure 2 was obtained by setting, , , , , , , , , and in Equation (71). This figure shows that when, 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, 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 to when the value of the wealth is small and there in less change when the value of the wealth is large.

Figure 3 was obtained by setting, , , , , , , , , and in Equation (72). This figure shows that when, 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, 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 to when the value of the wealth is small and

Figure 2. Portfolio value in stock.

Figure 3. Portfolio value in cash account.

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

For, we have a problem because we cannot solve the equation explicitly. we need to come up with a computer program.

8. 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.

Acknowledgements

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.

Appendix

Appendix A

Assume that and. Using Itô formula for sememartingales (see Jacod [?], Protter [?], Shiryaev [11] , Shiryaev [10] ), one obtains

(47)

to find our SDE, assume that and substitute on Equation (47). Simplifying will give the following results

(48)

Differentiating will give;

(49)

Now the differential of the stock process is given by

(50)

where

(51)

then, using Ito’s formula for semimartingales (Protter [?]), we have

(52)

and in differential form, this can be expressed as

(53)

Appendix B

Assuming and substituting it on the formula we get

(54)

Appendix C

Let and

(55)

Appendix D

then

(56)

where

with and

was found by simply dividing by i.e.

Appendix E

Let and define. Then is a sto- chastic process with jumps and

take and substituting on 58 to have

Choosing such that for a given portfolio strategy (not necessarily optimum, we introduce the associated utility

(57)

Substituting, and we now have

Integrating both sides we get

Taking the expectations on both sides we have

For simplicity we have

Where and. Since we know that , we now have

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

(58)

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

(59)

This takes us to the HJB equation;

(60)

where is the second linear operator for jump diffusion. Hence

(61)

Taking our utility function as

(62)

We consider the function of which is

(63)

Differentiating and substitute on (63), we get

(64)

Since is a concave function of, to find its maximum we differentiate (64) with respect to to obtain

(65)

For we can solve for because we have a linear equation below

(66)

and will be given by

(67)

substituting and as defined , we obtain the following

(68)

(69)

where

(70)

and

(71)

We can now see that

(72)

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