# Firm Stock market

Let's say that I live in my own world, with 20 inhabitants.

I own a firm called mc Donald's. ( my people purchase food there of course).

Each income to mc Donald's gets into the firm bank.

McDonald's is also tracking profit each hour , day, week and month.

As McDonald's owner , I will be able to take out any of the profit whenever I want to.

Here comes the question

Let's say I want to make mc Donald's a stock market , where my inhabitants can buy and sell stocks on mc Donald's firm.

How can I accomplish this ? How and what algorithms can I use to do stock price changes to make it appear to be working like how real stock firms works ?

How many stocks should I allow the firm to have ?

Any suggestions or spoon feed on how I can make this mc Donald's function like a stock market in itself ?

What will be needed for a price drop / increase on the stock's price ?

I would approach this by building a general equilibrium model of the economy. Here is the high-level idea:

• There is a single perishable consumption good and all prices are measured in terms of it.

• There is a fully equity financed firm with one unit of share outstanding. The firm pays a continuous dividend at a stochastic rate $\delta$. You could for example start with a geometric Brownian motion $$\mathrm{d}\delta_t = \mu \delta_t \mathrm{d}t + \sigma \delta_t \mathrm{d}W_t$$

• There is a continuum of identical risk-averse individuals that maximize their expected lifetime utility $$\mathbb{E}_{\mathbb{P}} \left[ \int_0^\infty e^{-\rho v} u \left( C_v \right) \mathrm{d}v \right]$$ by, at each point in time, choosing their optimal consumption and portfolio compositions. Here, $u \left( C_t \right)$ is their utility of consumption and $\rho$ is their rate of time preference. You can treat these as one "representative agent".

• You can introduce additional derivatives into your economy that are in zero net supply and whose payoffs are measurable w.r.t. the filtration generated by $W$. A trivial such example are zero-coupon bonds.

• You find the competitive equilibrium that solves the agents' expected utility maximization problem. In this equilibrium, the representative agent holds the one unit of stock, no derivatives and his consumption is equal to the dividend $C_t = \delta_t$.

• In equilibrium you obtain the dynamics for the stock price as well as valuation functions for all derivatives (and thus a risk-free rate). In my above example, you will find that the stock price also follows a geometric Brownian motion.

Note that there are many ways to setup such an economy and often your assumptions will be a bit reversed-engineered from the desired result. You could certainly add other things like production to make your setup more realistic. Here are some references to get you started:

• The above setup is the pure-diffusion version of the Naik and Lee (1990) exchange economy. They consider a Merton jump-diffusion model. Many variations of this model have been used to find economically motivated measure changes for Levy processes, see e.g. Milne and Madan (1991) for the variance gamma model or Kou (2002) for the double exponential jump-diffusion model.

• A good reference for the general topic of continuous time consumption and portfolio choice are the books by Back (2010) and Pennacchi (2008).

• One of the ground braking papers is Cox et al. (1985). They construct a very general multi-factor production economy. This paper is a tough read though.

References

Back, Kerry E. (2010) "Asset Pricing and Portfolio Choice Theory", Financial Management Association Survey and Synthesis Series: Oxford University Press

Cox, John C., Jonathan E. Ingersoll and Stephen and Stephen A. Ross (1985) "An Intertemporal General Equilibrium Model of Asset Prices", Econometrica, Vol. 53, No. 2, pp. 363-384

Kou, Steven G. (2002) "A Jump-Diffusion Model for Option Pricing", Management Science, Vol. 48, No. 8, pp. 1086-1101

Milne, Frank and Dilip B. Madan (1991) "Option Pricing with V.G. Martingale Components", Mathematical Finance, Vol. 1, No. 4, pp. 39-55

Pennachi, George G. (2008) "Theory of Asset Pricing", Pearson

• Do you have time for a chat ?;) Oct 29 '16 at 14:08
• Sorry - I don't. I gave you an answer to your question with literature references to start from. If you have a specific question where sth. in my answer is not clear then please write it here. Oct 29 '16 at 14:10
• Please use the forum itself to get additional clarification, there the information is visible and useful for a wider audience. Jun 9 '17 at 19:29