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In this paper, we compare equilibrium equity premium under discrete distributions of jump amplitudes. In particular, we consider the binomial and gamma distributions because of their applicability in finance. For the binomial, we assume that the price movement is allowed to either increase or decrease with probability <i>p</i> or 1 − <i>p</i> respectively. <i>n</i> is the trading period thereby forming a vector <i>x</i> of jump sizes (shifts) whose distribution is a binomial over time. For the gamma, the jumps are taken to be rare events following a Poisson distribution whose waiting times between them follows a gamma. In both distributions, the optimal consumption of the investor is affected by the deterministic time preference function but it has no effect on the diffusive and rare-events premia thereby not affecting the equilibrium equity premium. Also, for , the volatility effect on the equity premium is the same in both the power and square root utility functions although the equity premium is not affected by the wealth process . However, the wealth process affects the equity premium of the quadratic utility fuction. We observe no significant differences in equity premium for the two discrete distributions.

The equity risk premium or simply equity premium, the rate by which risky stocks are expected to outperform safe fixed-income investments, such as government bonds and bills, is perhaps the most important index in finance. This is the investor’s compensation for taking on the relatively higher risk of the equity market. The equity risk premium is found by subtracting the estimated bond return from the estimated stock return. In our early work, we had considered the impact of utility functions in the production economy with jumps under an arbitrary jump size and derived analytical formulae for an equity premium for the power, exponential, square root and quadratic utility functions. However, we were unable to simulate graphs because of the jump size being arbitrary. In this paper, we derive numerical formulae for an equity premium and simulate graphs by imposing a Binomial distribution on the jump sizes. We then compare the results with those obtained by simulating the Gamma distribution of Jump Amplitudes. Jump diffusion has been widely explored in the area of option pricing but little work has been done to ascertain the behaviour of equity premium under jump diffusion models.

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Jump Diffusion Option Valuation in Discrete Time was proposed by [

This paper is related to a number of papers including [

This paper is based on theoretical model of [

The gamma distribution arises naturally when we consider waiting times between Poisson distributed events as relevant. It can be thought of as a waiting time between Poisson distributed events.

The probability density function is the waiting time until the

Now, for

where x is a vector of jump amplitudes, k is the number of occurrences of an event and

the gamma function. The value

between jumps.

We still subtract the expected value from the drift so that the process becomes more volatile and hence a martingale because its future is unexpected. If we apply Itô Lemma with Jumps we have,

By integration we have

where

Suppose also that, at the risk-free rate

whose total supply is assumed to be zero. Consider here that

We study comparatively the general equilibriums of one investor who wishes to maximize his expected reward function

subject to

in an economy with jumps when jump amplitudes follow the binomial and gamma distributions for some time preference function

Theorem 1. If X is a vector of binomially distributed jump sizes, an investor’s equilibrium equity premium with

CRRA power utility function

where

Proof. If X is a random variable with a binomial distribution, then

In particular, if

and so

Let

rare-event premium

Now

Therefore, our rare-event premium

The optimal consumption of the investor is affected by the deterministic time preference function

As can be seen in

Theorem 2. For a gamma distribution of jump sizes, an investor’s equilibrium equity premium with CRRA

power utility function

where

Proof. If x follows a gamma distribution, that is

for some constant u. This is just the moment generating function of x evaluated at u.

For the power utility function, the equilibrium equity premium

where our rare-event premium

[

Now since

Therefore our rare-event premium

which implies that our equilibrium equity premium is

The optimal consumption of the investor is affected by the deterministic time preference function

price of the jump risk.

We realize in

Theorem 3. In the production economy with jump diffusion under a vector of binomially distributed jump sizes, the investor’s equilibrium equity premium with square root utility function

where

Proof. For the square root utility function, the rare-event premium is given by

Since

and

Also

Thus our rare-event premium is

and therefore our equity premium is

The equity premium is neither affected by the wealth value nor the time preference function and the diffusive risk premium is always positive.

Just as for the power utility function and normally distributed jump size,

Theorem 4. In the production economy with jump diffusion under a vector x of jump sizes whose distribution follows a gamma, the investor’s equilibrium equity premium with square root utility function

where

Proof. For the square root utility function, the rare-event premium is given by

Now, since

therefore

and thus our equilibrium equity premium is

The equity premium is neither affected by the wealth value nor the time preference function and the diffusive risk premium is always positive. For

Theorem 5. For the binomially distributed jump sizes, the investor’s equilibrium equity premium with quadratic utility function

where

rare-event premium.

Proof. For the HARA Quadratic utility function,

where

Now since

and

thus our rare-event premium is

which implies that our equity premium is

It is not affected by the time preference function

Theorem 6. For the gamma distribution of jump sizes, the investor’s equilibrium equity premium with quadratic utility function

where

premium.

Proof. For the HARA Quadratic utility function,

where

Now since

thus

which is just

So that our equilibrium equity premium is now

It is not affected by the time preference function

In conclusion, the optimal consumption of the investor is affected by the deterministic time preference function

We thank the Editor and the referee for their comments.

George M.Mukupa,Elias R.Offen,DouglasKunda,Edward M.Lungu, (2016) A Comparative Study of Equilibrium Equity Premium under Discrete Distributions of Jump Amplitudes. Journal of Mathematical Finance,06,232-246. doi: 10.4236/jmf.2016.61020