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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?

TomR
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    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. – mark leeds Sep 25 '21 at 03:12
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    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? – Kevin Sep 25 '21 at 05:55
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    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. – mark leeds Sep 25 '21 at 11:13
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    @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. – Kevin Sep 25 '21 at 12:39
  • @Kevin Yes, I would be glad and thankful to see some references (keywords, terms, articles) about such models, please, drop them, if possible. Thanks! And why there can not be stochastic volatility models based on exogenous variables? That would be great answer to my question. – TomR Sep 25 '21 at 12:42
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    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. – Kevin Sep 25 '21 at 12:46
  • I can imagine that the firm's decision and/or external environment can be defined as the Markov process/state space (or the product of state spaces - one for firm, another for competitors, still another - for macroeconomic environment) and at each state of this state space can contain the certain vector of numerical values and those numerical values can describe the stochastic volatility (parameters of SDE) for certain state (product of states). Maybe this is the way to account for changes? The estimation can be quite hard - to identify states and numbers in for each of them. – TomR Sep 25 '21 at 12:46
  • @Kevin Thanks! I value your time, no hurry, it can wait and pointers can be sufficient, I am self-learning person. Google not always understand what I am searching for. – TomR Sep 25 '21 at 12:47
  • @TomR you’re ideas about the Markov process are spot on! I’ll elaborated on that tonight. Estimating these models is indeed difficult. Thus many papers use calibration instead. [Note: My doctoral research is in the area of estimating these models.] May I ask what your background is and why you’re interested in the topic/what application you have in mind? – Kevin Sep 25 '21 at 14:17
  • Kevin: It sounds like, based on TomR's response, that you were correct and I was wrong. I'll step away and listen. Thanks for correction. – mark leeds Sep 25 '21 at 16:34
  • @TomR I apologise for the late answer. I was quite busy over the last few days. Please let me know if you have any further questions or would like me to expand a bit on the topic!:) Sorry again!! – Kevin Sep 27 '21 at 14:48
  • @Kevin Thanks for your comprehensive answer, especially about detailing the assumptions, reasoning and points of extension. About my background and interests. I am software developer by my side project (which may become MS thesis or startup) is development of cognitive architecture for Artificial General Intelligence (http://agi-conference.org/) and I have narrowed down my efforts to the automated program synthesis - e.g. one can formulate the constraints on the unknown function, one can search the symbolic space for this unknown function and at the end one can search the proof that the... – TomR Sep 28 '21 at 10:19
  • @Kevin ...discovered function satisfies all the constraints and hence it the the solution of the synthesis problem. This is very good exercise because the theorem provers (like https://isabelle.in.tum.de/) can give the yes/no answer to the proposed solution and the algorithm can to the exploration itself. No need for big data. This can be implemented as reinforcement learning algorithm in which the theorem prover provides the reward signal (utmost reward when the function and its correctness proof is found and zero/negative, if not found, reinforcement learning in sparse reward setting) – TomR Sep 28 '21 at 10:22
  • @Kevin Such efforts (proof synthesis, theorem proving, program synthesis, automation of math) is very topical now, e.g. https://mathai-iclr.github.io/papers/, especially paper about TacticZero. So, I just had idea to grab some financial engineering problem and to try see what can be automated here. This is more like symbolic regression for stochastic differential equations and automated solution determination. This is my side project that is why the progress is very slow and interrupted by explorations in the other directions and generalizations. E.g. this summer was hot time... – TomR Sep 28 '21 at 10:25
  • @Kevin ...for the theory of neural networks and machine learning. At last (after all those years of empirical work) there are articles about the theory (https://arxiv.org/abs/2106.14587 and https://arxiv.org/abs/2106.07032). Automation efforts are my passion and this seemed to be hot topic that can be considered for automation. Your answer provided be excellent guidance about the practical steps for my efforts. Lot of thanks! – TomR Sep 28 '21 at 10:28
  • @Kevin https://www.isa-afp.org/entries/DiscretePricing.html is the formalization of some financial math in the Isabelle/HOL. Isabelle have automated proof search components, maybe they can be used in financial math as well if only this branch of math if formalized enough. Automation of the math is not only about the theorem proving but about the stating the "interesting" theories and therems as well, e.g. https://sensilab.monash.edu/publications/?author=Simon%20Colton;%20Colton,%20Simon and... – TomR Sep 28 '21 at 10:36
  • @Kevin especially book https://www.springer.com/gp/book/9781852336097 – TomR Sep 28 '21 at 10:37
  • Sorry for lot of grammatical mistakes. Comments close so fast that I can not correct them. – TomR Sep 28 '21 at 10:52

1 Answers1

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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).

Kevin
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