The Risk Averse Investor ’ s Equilibrium Equity Premium in a Semi Martingale Market with Arbitrary Jumps

In this paper, we study the risk averse investor’s equilibrium equity premium in a semi martingale market with arbitrary jumps. We realize that, if we normalize the market, the equilibrium equity premium is consistent to taking the risk free rate ρ = 0 in martingale markets. We also observe that the value process affects both the diffusive and rare-event premia except for the CARA negative exponential utility function. The bond price always affect the diffusive risk premium for this risk averse investor.


Introduction
Much work in finance has been based on martingale markets whose future is deemed fair and unpredictable by normalizing prices.This gives investors a fair chance to either gain or lose out on their investments.In our case, we make the market partly predicable in order to give certainty of some degree on a fair compensation an investor receives for having taken up some risk.In this case, we allow X t to be a semimartingale with a decomposition X t = X 0 + M + A, such that M = (M t ) 0≤t≤T is a square-integrable martingale with M 0 = 0 and A = (A t ) 0≤t≤T is a predictable process of finite variation |A| with A 0 = 0.In this paper, we use the semi martingale approach to determine equilibrium equity premium in a production economy with jumps as opposed to option pricing.The problem of deriving ordering results for option prices has been adressed in several papers [ (Karoui & Shreve, 1998), (Hobson, 1998), (Bellamy, 2000), (Henderson, 2002), (Hendersonn & Hobson, 2003), (Hendersonnn & Kluge, 2003), (Moller, 2003), (Eberlein & Jacod, 1997), (Frey & Sin, 1999), (Jakubenas, 2002), (Gushchin & Mordecki, 2002)].The results for models with nontrivial pricing intervals and the corresponding comparison results are less complete.Comparison results for diffusion processes are discussed in (Karoui & Shreve, 1998) and nontrivial bounds for stochastic volatility models are given in (Frey & Sin, 1999).(Bellamy, 2000) (see also (Henderson & Hobson, 2003)) prove that the price of a European call for a diffusion with jumps is bounded below by the corresponding Black-Scholes price and above by the trivial upper price [see also (Bergman & Wiener, 1996) and (Hobson, 1998) for alternative proofs].An important generalization of the technique introduced in (Karoui & Shreve, 1998) and (Bellamy, 2000) has been established by (Gushchin & Mordecki, 2002) who derive a general comparison result for one-dimensional semimartingales to some Markov process w.r.t convex ordering of terminal values.This paper is comparable to (Zhang & Chang, 2012) and also further elaboration by (Mukupa & Offen, 2015) and (George & Offen, 2016) who considered the martingale case of equilibrium equity premium.

Method
Our price process evolves according to the stochastic differential equation; which is a semi martingale with discontinuities because of the presence of jumps.
We take µ, δ and λ as constants and x as a vector of arbitrary distributed jump sizes.The processes B t and N t are independent since Brownian motion is a continuous process while the Poisson process is discrete.The parameter λ denotes the frequency of the Poisson process.In this model, we have set the coefficient (e x − 1) in the jump process such that e x − 1 = 0 if there is no jump, that is, for x = 0. E denotes the expectation which makes the process E(e x − 1) deterministic.dN t models the sudden changes as a result of rare events happening and dB t models small continuous changes generated by the noise whose volatility is a constant δ.
Note that the compensated compound Poisson process (e x − 1)dN t − λE(e x − 1)dt has the mean of zero since To solve we do not need to apply Itô Lemma with Jumps because the diffusion part is a continuous semi martingale whose procedure for solution does not require the integrating factor.We solve for the price process at the terminal time T as follows; Integrating (1), we obtain where τ = T − t is the investment period.
Suppose that an investor holds two assets, the risk-free asset, X 0 (t), and the risky asset, X 1 (t) = X t given by equation ( 1).The risk-free asset is assumed to evolve according to the equation where ρ is a constant risk-free rate.Denote Y t = (X 0 (t), X 1 (t)) and the corresponding portfolio by ϕ = (1 − ω, ω) consisting of 1 − ω non risky assets and ω risky assets.
We have, by the self financing strategy, that dV t = ϕ • dY t so that the total wealth at any time t is where V 0 (t) is the value of the money market account and V 1 (t) is the value of the investment in the stock market at time t.Now Since the equity premium φ = µ − ρ, we have that µ = φ + ρ, hence The investor's optimal control problem then is to maximize his expected utility function subject to The wealth ratio ω and consumption rate r t are control variable.The general equilibrium occur when ω = 1.

Results
Proposition 1 Equilibrium Equity Premium For CRRA Power Utility Function.
In a semi martingale market, an investor's equilibrium equity premium with CRRA power utility function U(r t ) = r β t β , 0 < β < 1, in the production economy with jump diffusion is given by where ] is the rare-event premium.
Proof.We optimize the investor's utility based on the Hamilton-Jacobi-Bellman (HJB) equation (2) Now, Equation ( 2) can be written as where then Ito's formula gives The generalized process of equation ( 3) gives Equation ( 2) can now be written as Dividing through by dt, we obtain To find the optimal values, we solve max (r t ,ω) by taking partial derivatives with respect to r t and ω to obtain the first order conditions Solving for φ from the second equation and taking the equilibrium condition ω = 1 yields the general equilibrium equity premium Substituting this φ into the Bellman equation Consider now the power utility function we solve for J(V t , t) based on the indirect utility .
The optimal consumption will be solved from the first order condition (1) as: is the optimal consumption we require.
Substituting the functions differentiating with respect to V t and dividing through by V β−1 t gives the terminal conditions It is easy to see that the RRA is 1 2 by virtue of square root since the CRRA family is of form U(c) = c β for some RRA = β > 0.
Substituting J t , J V t , J V t V t and r t into the integro-partial differential equation This gives Differentiating with respect to V t and dividing through by and thus are terminal conditions.
Substituting into the general equilibrium equity premium formula gives the equilibrium equity premium for the square root utility function as For this utility function, the value process affects both the diffusive and rare event premia.Again, if the wealth process V t = 0, the equilibrium equity premium is undefined.If the wealth process increases, the diffusive risk reduces and vice-versa.The rare event premium also reduces with the increase in the wealth value.
Proposition 4 Equilibrium Equity Premium For Quadratic Utility Function.
Proof.Suppose now that this investor consumed quadratically from the investment, that is U(r t ) = r t − ar 2 t , a > 0,