# Stochastic (volatility) models with the elements of fundamental analysis - are there such models and why not?

I read about stochastic volatility models (e.g. https://en.wikipedia.org/wiki/Stochastic_volatility) and those models are quite simple, but the most important feature is that parameters are quite empirical and I can not clearly see how the paramaters of stochastic differential equations are connected with the fundamental analysis. E.g. - how it is reflected in the parameters that one firm can heavily invest in knowledge, R&D, patent building, human resources and expect big increase in the share price (e.g. ModernaTX and mRNA technologies) and the second firm invest just in manufacturing capabilities. E.e. how it is reflected in the model parameters that one firm is in the field that is business cycle proof and the other firm is in the field that is highly cyclical and how - in turn - the phase of the current business cycle is included in the parameters of boths firms?

So - my question is - is there connection between the parameters of the stochastic financial engineering models with the fundamental analysis at the firm-level and macroeconomic (business cycle, technology shocks) level? Does such interpretation of the parameters and possible further explication explication of those parameters in some analytical form already exists in financial literature? What are the most sophisticated stochastic models that include the elements of the fundamental analysis? I have seen none so far, why there is so little developments in this direction? All these are only the subquestions in my efforts to understand are there stochastic (volatility) models with fundamental analysis and why so few?

• your question extends more generally to the difference between time-series models and econometric models. The latter sometimes use other variables ( interest rates, gdp, the market return ) besides error terms and unobserved components. Time-series models are usually restricted to the use of error terms and-or unobserved components. For example, an arch model is a time-series model. There are only error terms and the unobserved volatility in the arch model. If someone could build an sv model that used variables like you describe and had predictive power, they wouldn't talk about it anyway. Sep 25 '21 at 3:12
• There are models of the firm where the exogenous state variables (e.g. output price, demand or productivity of the firm’s homogenous good) is driven by a diffusion with stochastic volatility (and jumps and mean regression and any other feature studied in financial engineering). In asset pricing, you’d then calibrate or estimate these fundamental parameters to match, for example, returns of value strategies and see whether your model can explain this “anomaly”. Is that something you’re looking for? Sep 25 '21 at 5:55
• Kevin: I think the OP saying that the response is stochastic volatility and the predictors are exogenous variables. That type of model I've never seen. Sep 25 '21 at 11:13
• @markleeds perhaps I’m misinterpreting the question but to me it sounds as if we consider a stochastic volatility model and seek economic interpretation of the parameters governing the distribution of the state variables. An easy such model would link stochastic volatility to firm decisions via the exercise of real options (to invest, disinvest, produce, etc.) but I may very well be wrong and OP may want to clarify. Sep 25 '21 at 12:39
• I’ll add an answer tonight going down this direction Tom!:) stochastic volatility can very well be an exogenous variable! That’s no problem at all. Indeed, it’s totally sufficient to consider a partial equilibrium model to see how a firm’s fundamentals relate to stochastic volatility in the state variables. Sep 25 '21 at 12:46

## A simple model of the firm

Consider a firm with the following properties

• The firm is a monopolist.
• The firm is fully equity-financed.
• The firm owns production assets which the firm can switch on or off, depending on the level of an exogenous process $$X$$ (that may be the productivity of the assets, demand for the output good, etc.).
• The firm can choose the number of production assets it owns (i.e., the firm can invest and disinvest).

I made the first two assumptions to make our life much easier (there's no competition about when to make firm decisions and there are no debt and leverage considerations (option to default)). Of course, these assumptions can be relaxed. There are many further simplifying assumptions: The firm only produces a homogenous good, there are no labour decisions, no capital depreciation, no taxes, no time-to-build or time-to-produce, no inventory or working capital, etc. etc. etc. The above model set up is a simple "real options model of the firm".

But what does this (simple) model set up buy us?

• We can interpret the firm as a collection (portfolio) of (real) options: the option to produce, the option to invest and the option to disinvest. The total market value of the firm is then the sum of these three components.

• Suppose $$X$$ is productivity of the firm's installed capacity units. Then, the production asset are essentially call options, the investment options compound call options and the disinvestment options are compound put options.

• The production and investment options (call options) depend positively on $$X$$, the disinvestment options (put options) negatively on $$X$$. The value of an economically distressed firm drives from its deep ITM disinvestment options. The value of profitable firms drives from its deep ITM investment options.

• Hackbarth and Johnson (2015, RES) and Aretz and Pope (2018, JF) show how such a model with investment and disinvestment options can explain the positive returns of profitable firms and momentum stocks, see also this answer about asset pricing with real options.

To give more details on the firm value, we would need to make assumption about the functional form of the firm's technology (variable costs, fixed costs, demand function, capital adjustment costs, etc.).

## How can stochastic volatility models be linked to firm fundamentals?

Note that I made no distributional assumptions about $$X$$ thus far. Suppose now our state variable $$X$$ follows a Heston (1993) stochastic volatility process \begin{align} \text{d}X_t&=\alpha X_t\text{d}t+\sigma_t X_t\text{d}B^X_t,\\ \text{d}\sigma^2_t&=\kappa(\theta-\sigma^2_t)\text{d}t+\xi\sigma_t \text{d}B^\sigma_t, \end{align} where $$\text{d}B_t^X\text{d}B_t^\sigma=\rho\text{d}t$$.

From the Heston model, we know the role the different parameters play

• $$\kappa$$ controls the persistence of the variance process
• $$\theta$$ controls the width of the distribution of $$X$$
• $$\xi$$ controls the tails of the distribution of $$X$$
• $$\rho$$ controls the skewness of the distribution of $$X$$

Choosing a different stochastic volatility process allows us to study the impact of different parameters. If we now additionally take a stance on the firm's technology, we can solve the model (perhaps numerically), simulate panels of firms and see how each parameter impact gross profitability, investment rates, etc. of firms.

• If firms produce and sell immediately, stochastic volatility (about the future) does not impact immediate production decisions. However, the value of the production options (portfolio of call options) does depend on stochastic volatility.
• Investment decisions also depend on volatility parameters because investing and disinvesting means to give up or to gain the "option to wait and see" and such option values are very sensitive to volatility. Intuitively, if $$\sigma_t^2$$ is high, option values are large and firms don't like to adjust their capacity (you see the effect of these uncertainty shocks following the 2016 Brexit vote after which British firms invested less due to the higher uncertainty).
• You can also think about how the (negative) volatility risk premium impacts the firm's expected return, see the working papers from McQuade (2018) and Barinov and Chabakauri (2021). You can use this mechanism to explain the value premium (firms with high book/market ratios tend to have high returns).

Note that it is difficult to solve these models analytically (most real options models rely on geometric Brownian motions). Also, note that the productivity of a production asset (or the demand for its output good) are unobservable. Its time-varying second moment is even more unobservable. So it may be hard to quantify all the parameters (you can't easily calibrate the model as with financial options).