# Is complete market or not if appreciation rate is random?

Consider the stock price process satisfies the following SDE:

$dS_t=\mu_t S_tdt + \sigma S_t dW_t , S_0=s$

and the appreciation rate process $\mu_t$ satisfies the following SDE:

$d\mu_t=(a-\mu_t)dt +dB_t, \mu_0=\mu$

where $W_t, B_t$ are two independent Brownian motions.

Hence, there are two sources of uncertainty in the model, but only one stock available for investment.

My question is: Is this market complete?

And, is it similar to the stock consist of two independent Brownian motions?

@Neeraj I think he meant: is the market complete considering that i have two sources of risk and only one asset? This market is incomplete. Stochastic drift isn't really used in derivative pricing because under risk neutral proability the drift is given by the rf rate. Depends on what you need to do.

• Thanks for your answer. Yes, it's just what I had in mind. So, under derivative pricing. Although there are not unique risk neutral probability measure, in each risk neutral probability measure the drift is meaningless. Am I right? If I consider an European call option, the price will be same as Black-Scholes case? – N.chan Feb 16 '16 at 14:53
• @LorenzQF The OP has not asked about the application of model for specific purpose. There is sufficient empirical evidence that suggest drift is not constant. Drift on the stock gets influenced by risk free rate which in itself is stochastic. If the OP has considered drift as stochastic then what is wrong with it? No doubt, model is subject to empirical test, only then we can determine how good model is in explaining real world behavior exhibit by stock prices. – Neeraj Feb 16 '16 at 15:03
• @N.chan, Black and Scholes case has several assumptions and simplified assumptions (For example deterministic risk free rate). But in this case if you want to price the option with the underlying following the process you described, you simply change the drift (change probability) and compute evrything under the risk neutral probability. Remember that if you have only ONE risk neutral probability then the market is complete, while the opposite is not true. – LorenzQF Feb 16 '16 at 15:10
• @Neeraj of course it is possible to assume a stochastic drift for an asset, but i guess the aim of the question was another. Not if the model fits good to the real world, but if the model represents a complete or incomplete market. Which is different. – LorenzQF Feb 16 '16 at 15:14
• @LorenzQF The OP has asked about the "is model complete", not about the "is market complete". These two terms are very different. – Neeraj Feb 16 '16 at 15:17

In your model, you assumed drift ($\mu$) of the process is stochastic. You can better write your model like this: $$dS_t=\mu_t S_t dt + \sigma S_t dW_t$$ $$d\mu_t=\theta(\gamma-\mu_t)dt + \eta dB_t$$ where, we used Ornstein–Uhlenbeck process to model drift. Statistically your model is complete but behind every model there must be some theoretical base or some observed behavior from the reality. In finance, there are similar model that assume volatility is stochastic (stochastic volatility model) and such model is used to price various derivative products.

Further, There is sufficient empirical evidence that suggest drift is not constant. Drift of the stock price gets influenced by risk free rate which in itself is stochastic. So, You may reconsider your model which incorporates risk free rate into your model.

There are numerous studies that assume interest rate as stochastic and price derivative contract. The assumption of constant interest rate in Black-Scholes was first relaxed by Merton(1973) [highly technical paper]. After that various author tested Black-Scholes model efficiency under stochastic interest rate model framework. For example Haowen(2012) used Vasicek Model for interest rate. You may check other studies too.

• Thanks for your answer. But how can we know that this model is complete? I saw this model in some theses which about partial information. And I can't open your link of (stochastic volatility model). – N.chan Feb 16 '16 at 13:57
• @N.chan I donot know what you want to convey from the word complete model. I have changed the link of the reference paper. You may also read comment I posted on LorenzQF answer. – Neeraj Feb 16 '16 at 15:09
• @N.chan In derivative world, market is said to complete if derivative payoff can be replicated with underlying assets and cash only. But the word complete market has entirely different meaning in economics context. Because your question is silent on purpose and objective of your model, can you please tell which meaning of complete market has in your mind? – Neeraj Feb 16 '16 at 15:32
• I consider it in derivative world. Further, if we consider Black-Scholes model with above drift process rather than constant. The European option price will be same with constant drift case? – N.chan Feb 16 '16 at 15:50
• @N.chan I found various studies that assume interest rate as stochastic and then price derivative. Just check my edit answer. – Neeraj Feb 16 '16 at 18:16