Application of Control Theory in Quantitative Finance

I have recently completed an MSc in Control Systems from a top university. It seems to me that control theory must have an application within quantitative finance. I would like to apply my degree within finance, but I want to be sure that it is relevant to the role of a quantitative analyst.

The topics which I have particular interest and experience in are State Space Control, Systems Identification, Model Predictive Control and Optimal Control. I imagine that effective management of finances must involve modelling of financial systems in terms of transfer functions/state space models (based on large sets of historical data). These models could then be used to predict the evolution of a market over time, and therefore optimise a given cost function such as profit, risk etc.

If this kind of role exists within quantitative finance/ other areas, can you please give me more information/ ideas of job roles/ industries to research.

Of course, optimal control is at the core of math finance. Take few applications:

• Option Pricing: you have an exposure to a time dependent combination of market factors; you have some knowledge of their dynamics. They are partly deterministic, partly stochastic (i.e. random). At each "time step" you can adjust your portfolio at a given cost. Your goal is to lower your risk. Of course it is a control program, see Continuous-time Stochastic Control and Optimization with Financial Applications, by Huyen Pham, for examples (and here for a summary by the same author).

• Portfolio Construction: you have a given amount of money to invest, you will build a portfolio with it. You have some expectations in terms of the dynamics of returns of the available investment instruments (stocks, bonds, etc) and estimated for the associated risk. Changing your allocation has a cost at each time step. Again it is a control program, see for instance Dynamic Portfolio Choice with Frictions, Garleanu and Pedersen.

• Trading: you have a large amount of stocks to buy or sell today. You have the whole day (around 8h) to do it, and you can take a decision every millisecond. The faster you go the more you move the market at your detriment, the slower you go the more your are exposed to a bad price change. It is nothing else than control again, see Market Microstructure in Practice, by L and Laruelle. Market Microstructure Knowledge Needed for Controlling an Intra-Day Trading Process for an overview (there is a preprint available here).

• Thanks a lot for the reply. You say that these concepts are at the core of mathematical finance. However, is this purely just of interest to academics and software packages which have already been designed? For instance would someone involved in trading actually be employed to come up with trading algorithms, or is this already included on the software that a trader would use? basically, I would like advice on what job title to seek out if I was to apply this theory in the realm of (for instance) trading. Thanks again. – mark roche May 17 '15 at 18:28
• Most firms make the difference between the "operator" (trader, portfolio manager), the "engineer" (the "quant" who designs the models) and the "IT" (who takes care of the implementation). You have to look after "quant" positions, or "quantitative analyst", or "financial engineer" (sometimes they are called "structurers" in some derivatives small teams) @markroche – lehalle May 18 '15 at 5:28

The problem of when to exercise an option with Bermudan features is an optimal stopping problem. I have a done a lot of work on how to do these things when the state space is high dimensional. There are various more complicated problems where the contract is more difficult eg swing options.

Actually, a lot of finance and economics are centered around optimal control problems. Traditionally, most economies are modeled as dynamic systems. In finance, portfolio optimizations, advanced option pricing etc are all optimal control problems.

You could look at the book Non Linear Option Pricing, it has a lot of optimal control problems.