# What are the main differences between discrete and continuous time models when modeling asset price dynamics?

My intuition says that both approaches, discrete time models and continuous time models will be models (i.e. approximations) of reality. Therefore it should be possible to develop useful models in both domains.

Continuous time models have more mathematical elegance and can therefore probably bring more mathematical machinery to bear on the problem which presumably helps with deriving analytical solutions and asymptotic limits.

Discrete models more easily correspond to observed data and measurements and are easier to simulate on computers.

I have been told that it is possible to discretise continuous time models and vice-versa but care has to be taken when performing this transformation. Could you please highlight what the common pitfalls are, in particular when modelling asset prices? If there are other differences in the dynamics between the two approaches (for example possibly something like non-linearities, chaos, ... in one and not the other) then I would like to know about that as well.

I mainly speak as market practitioner when I say that I believe in the end all models that are applied to data and real life pricing issues are discretized. Think about it, even the BS hedge argument is in the end just a "theoretical continuous time overlay" of actual discrete time steps and re-hedges. Thus some of the limiting assumptions re BS. You do not have continuous prices, even if ticks come in millisecond frequency they are still discretely timed. Thus you cannot hedge continuously but only re-hedge when you receive new price discovery.

Continuous pricing models are elegant to work with from a mathematical standpoint. However, in the end whatever derivative or mortgage security you attempt to price you must resort to discretized version of pricing algorithms. Thats my take of it. I would not waste too much time on trying to figure out how to move from one version to the other. I rather recommend you think about the problem at hand and what you actually want to solve for. From my experience 90% of pricing complexities boil down to finding a replication of the to-be-price asset in question. Monte Carlo simulators have become my best friend because their applicability is so versatile.

• The fact that in practice all models become discrete doesn't imply the question is not important, even from the application point of view. For example, a question that steams from the OP's question is whether a pricing from a multi-period discrete model converges to that of a continuous one. – Anna Taurogenireva Jul 9 '14 at 16:54
• One thing that I am not sure if we are both taking into account is that it is not the same thing (1) a discretization of a continuous model and (2) a discrete model (as in a multi-period discrete model). There is a claim in Hunt & Kennedy's book that I don't have enough knowledge to justify that says "Though of mathematical interest, the [discrete] multi-period case is not important in practice and when we again take up the story of derivative pricing in Chapter 7 we will work entirely in the continuous time setting." – Anna Taurogenireva Jul 10 '14 at 14:11
• In other words, they are saying that, in practice, what is more useful is continuous models. Of course, for actually computing discretizations of continuous models. A question that I have myself is: If one applies no-arbitrage pricing to multi-period discrete models and take limit as the number of periods tends to infinity, do we get the same price as with the no-arbitrage price for a continuous model? – Anna Taurogenireva Jul 10 '14 at 14:15
• I think you might be confusing a few things. What do you mean by ... numerical methods do not exist? I think you really meant ... explicit formulas for pricing do not exist. Numerical methods is all the plethora of computational techniques used to make a computer give you a number that approximate the value you are looking for. About models not agreeing. Different models may have different assumptions. They don't have to agree on a price. Not even theoretically. Under the assumptions of completeness of a single market and no arbitrage you have a unique price but ... – Anna Taurogenireva Jul 13 '14 at 23:41
• ... but different models may make different assumptions about the market itself. This means that essentially they are working in different (theoretical) markets. Each of these theoretical markets get, in practice, calibrated to look as close as possible to the data of the real market. But, even under the assumptions of completeness and no arbitrage the theoretical price for a derivative in each of them need not be the same. – Anna Taurogenireva Jul 13 '14 at 23:44

Continuous time has a so-called elegance, but it is rarely correct. Most Q-measure people rarely care about correctness anyway, since they usually don't root their models in statistics. With no goodness of fit measures, continuous time models are elegant theory.

In general, we also see that most ex-ante hedges are rarely good, ex-post. They have large elements of directionality. There are lots of minor alterations, and even kluges (e.g., hedging delta by using the 'smile'). Even simple things like calibrating implied vol is technically wrong (recall the P-measure dynamic in Black-Scholes uses the same vol-i.e., the change of measure doesn't change the vol, so technically it must be the same as given by the historical dynamic, the DGP-in a Black-Scholes world, there is no implied-realized premium!). Of the many standard methods for hedging swaptions I have seen and used and backtested, it's pretty clear none are great.

In some ways you could say that continuous time finance with its beautiful formulas and elegant equations is merely a method for splining (calibrate to 4 points on the smile, and infer all others). But as I say rarely do Q-measure types care about the reality of P-measure. (exception: the failed attempt of the Macro-Affine community). Model Validation people try to do this correctly, but as far as I can tell, their methods are not altogether satisfactory.

Continuous time finance gives us some nice formulas and rules of thumb. But the world around us can be modelled more effectively for the most part in discrete time. Moreover, the dynamics are far richer in discrete time. Autocorrelation, seasonality, long-lag lengths--all of these phenomena are impossible to fit into SDEs.

One must know continuous time mathematics to understand and get an intuition about optionality and nonlinear payoffs. But until academics and practitioners start using delay-differential stochastic differential equations (used in areas such as signal processing/electrical engineering, and other physical sciences) in finance, we can model many more interesting phenomena in discrete time than we can in continuous time.