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This paper studies equilibrium equity premium in a semi martingale market when jump amplitudes follow a binomial distribution. We take n to be the number of times. An investor is trading in this market with p being the probability that there is a shift in the price at the trading time t . We find significant variations in the equilibrium equity premium for the martingale and semi martingale markets in terms of wealth value, volatility and other parameters under study. In this market, the equilibrium equity premium remains constant regardless of volatility and wealth value.

A semi martingale market is a partially predictable market 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 being a predictable process of finite variation | A | with A 0 = 0 . This market is so attractive to investors as it is deemed fair and enables uncertainity risks to be compensated fairly. This fair compensation is usually termed as Risk Premium which of late has attracted a lot of attention from researchers. [

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More recent studies [

A discrete random variable (RV) is a Binomial if it arises from Bernoulli trials. There is a fixed number, n, of independent trials which by independence, means the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances, we define the binomial random variable X as the number of successes in n trials. The notation X ~ B ( n , p ) is usually used to imply that X is a random variable following a binomial distribution. The mean is μ = n p and the standard deviation is σ = n p q for some probability of failure q. The probability of exactly x successes in n trials is p. In this market, everytime we observe a jump (shift in price), we record it as a success with probability of occurence, p. Therefore, p = E ( d N t ) = λ d t which studies by [

In literature, Jump Diffusion has been widely used in Option Valuation as opposed to modelling equity premium. In discrete time, the theory was proposed by [

Our contribution in this paper is comparable to [

To formulate the model, we consider that the price process is arising from a Binomial distribution with one parameter p, the probability of observing a jump in a given time interval [ 0, t ) and since the market is partially volatile, our process will evolve as a compensated compound poisson process similar to that of [

Let’s consider a Jump Diffusion process;

d X t = μ d t + δ d B t + ( e x − 1 ) d N t − λ E ( e x − 1 ) d t .

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 jump sizes following a binomial distribution. The processes B t and N t are independent. This follows directly from the definition of Brownian motion as being a continuous process and the Poisson being discrete which we obviously know that continuous processes and discrete are independent. λ is the frequency of the Poisson process. We set ( e x − 1 ) in the jump process so that e x − 1 = 0 if there is no jump as x is then a zero vector. E is the expectation which makes the process e x − 1 deterministic. d N t models the sudden changes as a result of rare events happening and d B t models small continuous changes generated by the noise whose volatility is a constant δ .

The compensated compound Poisson process ( e x − 1 ) d N t − λ E ( e x − 1 ) d t has the mean of zero because

E [ ( e x − 1 ) d N t − λ E ( e x − 1 ) d t ] = E ( e x − 1 ) E ( d N t ) − E ( e x − 1 ) E ( λ d t ) = 0

and E ( d N t ) = λ d t .

To solve

d X t = μ d t + δ d B t + ( e x − 1 ) d N t − λ E ( e x − 1 ) d t ,

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:

d X t = [ μ − λ E ( e x − 1 ) ] d t + δ d B t + ( e x − 1 ) d N t

By integration we have

X T = X t + [ μ − λ E ( e x − 1 ) ] τ + δ B τ + ∑ i = 1 N τ ( e x i − 1 ) , for τ = T − t

as the investment period.

Suppose also that, at the risk-free rate ρ , the money market account X 0 ( t ) is such that

d X 0 ( t ) = ρ ( t ) X 0 ( t ) d t

whose total supply is assumed to be zero. Consider here that ρ is risk-free because it is the rate for the non risky asset (money account).

Since the value of someone’s investment in this production economy at any time t is given by V t = ϕ X t , for some portfolio ϕ = ( 1 − ω , ω ) consisting of 1 − ω non risky assets and ω risky assets, we have that by the self financing strategy,

d V t = ϕ d X t

so that the total wealth at any time t is

V t = V 0 ( t ) + V 1 (t)

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

d V t = d V 0 ( t ) + d V 1 ( t ) = ( 1 − ω ) d X 0 ( t ) + ω d X ( t ) = ( 1 − ω ) ( ρ X 0 ( t ) d t ) + ω [ μ − λ E ( e x − 1 ) ] d t + ω δ d B t + ω ( e x − 1 ) d N t .

Since the equity premium ϕ = μ − ρ , we have that μ = ϕ + ρ , hence

d V t = [ ρ X 0 ( t ) − ω ρ X 0 ( t ) + ω ϕ + ω ρ − λ ω E ( e x − 1 ) ] d t + ω δ d B t + ω ( e x − 1 ) d N t .

The investor’s optimal control problem then is to maximize his expected utility function

m a x E t ∫ t T y ( t ) U ( r t ) d t ,

subject to

d V t = [ ρ X 0 ( t ) − ω ρ X 0 ( t ) + ω ϕ + ω ρ − λ ω E ( e x − 1 ) − r t ] d t + ω δ d B t + ω ( e x − 1 ) d N t

The wealth ratio ω and consumption rate r t are control variable. The general equilibrium occur when ω = 1 .

Theorem 1. In a semi martingale market with binomial jumps, an investor’s equilibrium equity premium with CRRA power utility function

U ( r t ) = r t β β , 0 < β < 1 , in the production economy with jump diffusion is given by

ϕ = ρ X 0 ( t ) − ρ − ( β − 1 ) V t − 1 δ 2 + λ ( 1 + ( e − 1 ) p ) n − λ q ( 1 − p + p e β ) n − λ + λ q ( 1 − p + p e β − 1 ) n

where ϕ δ = ρ X 0 ( t ) − ρ − ( β − 1 ) V t − 1 δ 2 is the diffusive risk premium and ϕ N = λ ( 1 + ( e − 1 ) p ) n − λ q ( 1 − p + p e β ) n − λ + λ q ( 1 − p + p e β − 1 ) n is the rare-event premium.

Proof. If X is a random variable with a binomial distribution, then Y = e X is a logbinomial random variable.

In particular, if X ~ B ( n , p ) and Y = e X then Y k = e k X . Also

E [ e k X ] = m X (k)

where m X ( k ) is the moment-generating function of X evaluated at k. Hence

E [ e k X ] = ( 1 − p + p e k ) n

and so

E [ e X ] = ( 1 − p + p e ) n = ( 1 + ( e − 1 ) p ) n = m X ( 1 ) .

Let X = x be a vector of binomially distributed jump sizes then for the power utility function of [

ϕ N = λ E [ ( e x − 1 ) ( 1 − V t ( e x ) β − 1 ) ] which becomes ϕ N = λ E [ e x − V t e x + x ( β − 1 ) − 1 + V t e x ( β − 1 ) ] .

Now, taking E [ V t ] = q , we realise that:

E [ e x + x ( β − 1 ) ] = E [ e x ( 1 + β − 1 ) ] = E [ e β x ] = ( 1 − p + p e β ) n = m X ( β ) .

E [ e x ( β − 1 ) ] = ( 1 − p + p e β − 1 ) n = m X ( β − 1 ) .

Therefore, our rare-event premium

ϕ N = λ [ E ( e x ) − q E ( e x + x ( β − 1 ) ) − 1 + q E ( e x ( β − 1 ) ) ]

now becomes

ϕ N = λ [ ( 1 + ( e − 1 ) p ) n − q ( 1 − p + p e β ) n − 1 + q ( 1 − p + p e β − 1 ) n ]

which implies that our equity premium is now

ϕ = ρ X 0 ( t ) − ρ − ( β − 1 ) V t − 1 δ 2 + λ ( 1 + ( e − 1 ) p ) n − λ q ( 1 − p + p e β ) n − λ + λ q ( 1 − p + p e β − 1 ) n .

are assured of some compensation regardless of the fall or rise in the trading prices. Infact, when there is no jump expected, the premium is symmetrical about zero volatility and increases on either side (see

Theorem 2. In the semi martingale market with binomial jumps, the investo’s equilibrium equity premium with square root utility function U ( r t ) = r t , r t > 0 is given by

ϕ = ρ X 0 ( t ) − ρ + δ 2 2 V t + λ ( 1 + ( e − 1 ) p ) n − λ q ( 1 − p + p e 1 2 ) n − λ + λ q ( 1 − p + p e − 1 2 ) n

where ϕ δ = ρ X 0 ( t ) − ρ + δ 2 2 V t is the diffusive risk premium and ϕ N = λ ( 1 + ( e − 1 ) p ) n − λ q ( 1 − p + p e 1 2 ) n − λ + λ q ( 1 − p + p e − 1 2 ) n is the rare-event premium.

Proof. For the square root utility function, the rare-event premium is given by ϕ N = λ E [ ( e x − 1 ) ( 1 − V t e − 1 2 x ) ] (see [

ϕ N = λ E [ ( e x − 1 ) ( 1 − V t e − 1 2 x ) ] = λ E [ e x − V t e 1 2 x − 1 + V t e − 1 2 x ] = λ [ E ( e x ) − q E ( e 1 2 x ) − E ( 1 ) + q E ( e − 1 2 x ) ]

Since x ~ B ( n , p ) , we have that

E [ e X ] = ( 1 − p + p e ) n = ( 1 + ( e − 1 ) p ) n = m X (1)

and

E [ e 1 2 X ] = ( 1 − p + p e 1 2 ) n = m X ( 1 2 ) .

Also

E [ e − 1 2 X ] = ( 1 − p + p e − 1 2 ) n = m X ( − 1 2 ) .

Thus our rare-event premium is

λ [ ( 1 + ( e − 1 ) p ) n − q ( 1 − p + p e 1 2 ) n − 1 + q ( 1 − p + p e − 1 2 ) n ]

and therefore our equity premium is

ϕ = ρ X 0 ( t ) − ρ + δ 2 2 V t + λ ( 1 + ( e − 1 ) p ) n − λ q ( 1 − p + p e 1 2 ) n − λ + λ q ( 1 − p + p e − 1 2 ) n

We observe a constant premium of 1.722 in

martingale market. Investors should therefore take advantage and consider investing in a market like this one.

In practice, the square root utility has many advantages in finance and economics including it’s ability to minimize shocks in the stock market. We were also able to see that the results for this utility function in terms of equilibrium equity premium were significantly reasonable compared to other utility functions in the martingale market.

Theorem 3. An investor’s equilibrium equity premium with quadratic utility function U ( r t ) = r t − a r t 2 , a > 0 in the semi martingale market with normal jumps is given by

ϕ = ρ X 0 ( t ) − ρ + 2 a δ 2 1 − 2 a V t + λ [ ( 1 + ( e − 1 ) p ) n − q ( 1 + ( e − 1 ) p ) n 1 − 2 a q + 2 a q 2 ( 1 − p + p e 2 ) n 1 − 2 a q − 1 + q 1 − 2 a q − 2 a q 2 ( 1 + ( e − 1 ) p ) n 1 − 2 a q ]

where ϕ δ = ρ X 0 ( t ) − ρ + 2 a δ 2 1 − 2 a V t is the diffusive risk premium and ϕ N = λ [ ( 1 + ( e − 1 ) p ) n − q ( 1 + ( e − 1 ) p ) n 1 − 2 a q + 2 a q 2 ( 1 − p + p e 2 ) n 1 − 2 a q − 1 + q 1 − 2 a q − 2 a q 2 ( 1 + ( e − 1 ) p ) n 1 − 2 a q ] is the rare-event premium.

Proof. For the HARA Quadratic utility function, ϕ N = λ E [ ( e x − 1 ) ( 1 − V t ( 1 − 2 a V t e x ) 1 − 2 a V t ) ] so that

ϕ N = λ E [ ( e x − 1 ) ( 1 − V t ( 1 − 2 a V t e x ) 1 − 2 a V t ) ] = λ E [ e x − V t e x − 2 a V t 2 e 2 x 1 − 2 a V t − 1 + V t 1 − 2 a V t − 2 a V t 2 e x 1 − 2 a V t ] = λ E [ e x − V t e x 1 − 2 a V t + 2 a V t 2 e 2 x 1 − 2 a V t − 1 + V t 1 − 2 a V t − 2 a V t 2 e x 1 − 2 a V t ] = λ [ E ( e x ) − q E ( e x ) 1 − 2 a q + 2 a q 2 E ( e 2 x ) 1 − 2 a q − 1 + q 1 − 2 a q − 2 a q 2 E ( e x ) 1 − 2 a q ]

Now since x ~ B ( n , p ) , we have that

E [ e X ] = ( 1 − p + p e ) n = ( 1 + ( e − 1 ) p ) n = m X (1)

and

E [ e 2 X ] = ( 1 − p + p e 2 ) n = m X ( 2 ) .

thus our rare-event premium is

λ [ ( 1 + ( e − 1 ) p ) n − q ( 1 + ( e − 1 ) p ) n 1 − 2 a q + 2 a q 2 ( 1 − p + p e 2 ) n 1 − 2 a q − 1 + q 1 − 2 a q − 2 a q 2 ( 1 + ( e − 1 ) p ) n 1 − 2 a q ]

which implies that our equity premium is

ϕ = ρ X 0 ( t ) − ρ + 2 a δ 2 1 − 2 a V t + λ [ ( 1 + ( e − 1 ) p ) n − q ( 1 + ( e − 1 ) p ) n 1 − 2 a q + 2 a q 2 ( 1 − p + p e 2 ) n 1 − 2 a q − 1 + q 1 − 2 a q − 2 a q 2 ( 1 + ( e − 1 ) p ) n 1 − 2 a q ]

The results in

All in all, it is important to note that the real life processes are not martingales but prices are normalized so that the processes can then be martingales. This means that jumps must be expected in any normalized market although most scholars have generally assumed the processes without jumps. In comparison, the results in the martingale market are similar to those of the semi martingale market but the premium is not as attractive as the one realised from the semi martingale market. Infact, in this market, the equity premium is always positive regarless of the utility function the investor is following. We therefore urge investors to consider investing in this market.

The martingale and semi martingale markets differ significantly in terms of how much compensation an investor recieves for having taken some risk in the investment. This is the case whenever jump amplitudes follow a binomial distribution in a semi martingale market. We observe consistent results in the equity premium of the power, square root and quadratic utility functions in terms of volatility effect, but the quadratic utility is affected also by the wealth process V t . We therefore advise investors consuming quadratically to consider investing in the semi martingale market with jumps as long as the amplitudes follow a binomial distribution. This is to avoid external shocks and variance in the premium when jumps are not expected.

We thank the editor and the referee for their comments.

The authors declare no conflicts of interest regarding the publication of this paper.

Mukupa, G.M. and Offen, E.R. (2018) Equilibrium Equity Premium in a Semi Martingale Market When Jump Amplitudes Follow a Binomial Distribution. Journal of Mathematical Finance, 8, 599-612. https://doi.org/10.4236/jmf.2018.83038