Market Completeness with a Stochastic Stock Drift
Summary
The discussion asks whether a market is complete when a stock price is exposed to two independent Brownian motions: one drives returns directly, while another drives a mean-reverting stochastic drift. With only the stock available for investment, one response identifies the market as incomplete because there are two sources of risk but only one traded risky asset to span them.
A second response disputes that conclusion, calling the model statistically complete and suggesting stochastic volatility as a related modeling example. It also points to stochastic interest rates as a more conventional extension of Black–Scholes. These answers conflict and do not provide a rigorous derivation. In particular, market completeness is a pricing and replication property that depends on the model’s traded assets and filtration; a stochastic drift alone is not automatically a separately priced risk under a standard risk-neutral setup. The exchange is useful as a prompt to distinguish physical dynamics from pricing assumptions, but its conclusions require careful qualification.
Key ideas
- The model includes independent return and drift shocks but only one traded stock.
- If both shocks create independent, unhedgeable risks, one risky asset cannot span every contingent claim.
- Market completeness depends on the traded assets and assumptions governing risk-neutral pricing.
- The answers disagree, and the exchange does not supply a full completeness proof.
- Stochastic interest rates and stochastic volatility are mentioned as related extensions of Black–Scholes modeling.
Tags
Full text
# Is complete market or not if appreciation rate is random? # 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? ## Answer by LorenzQF (score 1) https://quant.stackexchange.com/a/24326 @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. ## Answer by Neeraj (score 0) https://quant.stackexchange.com/a/24325 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.
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