2. From Model to Application: Calibration to Market Data
2.2 Model Calibration: A General View
When valuing financial derivatives, like options for example, certain assumptions on the development
of the underlying financial instruments, like for instance the stock price or some interest rates, have to be made. Usually, these assumptions give rise to a mathematical model, which in our case typically is a stochastic differential equation, describing the future random movement of the underlying. Then, the model can be used to derive pricing formulas for financial derivatives, or to simulate possible future evolutions of the financial market.
In most cases, the model will depend on a set of model parameters that are not directly
observable from the market. We denote this set by . The set of parameters directly observable from the market, i.e. the market parameters, such as spot prices or interest rates, is denoted by . Finally, there is a third parameter set , entering a pricing formula, which contains parameters of the financial product, also referred to as product parameters. For instance, for a European call on a stock we have
, where is the strike and the maturity of the call. For t = 0, we have with today’s price of the underlying and interest rate .
While the meaning of market and product parameters generally is quite clear, the model
parameters might not even have a financial meaning at all. Still, in order to use the model, sensible parameter values have to be found such that the model mirrors market reality. We will call this process model calibration.
Let f denote the type of a financial instrument, for example f = call for calls. Combined with a particular choice of market and product parameters, this completely defines the specific product of type f. Throughout this contribution, the corresponding market price of this product is denoted by
This price can generally be obtained as market quote from recent trades on an exchange. In fact, we assume to observe N such prices, that result for different product parameters , and we will refer to them as
Accordingly, let
be the model price of a particular product of type f. Hence, for given model and market parameters,
refers to the model price, calculated for the product parameter set .
Since can be observed and is stipulated in the contract of the product, the purpose of calibration is to determine such that the resulting model prices are as close as possible to the observable market prices, hence
simultaneously for all i. If the model reflects reality, model and market prices coincide. However, since a model always is only an approximation of reality, equality cannot be expected in general and the fit might not be perfect.
Therefore, we can only expect the distance
Mean squared error:
Relative errors:
Logarithmic errors:
(2.1) to be small, where dist(⋅ , ⋅ ) is a suitable distance or objective function on , which will be discussed in more detail later on. Finally, we are left with the optimization problem to find a
particular set such that the given distance is small.
The remainder of this section deals with possible choices of an objective function, focuses on financial instruments that can be used for calibration and briefly discusses types of optimization algorithms.
2.2.1 Objective Function
Obviously, the primary objective of calibration is to find an optimal fit between market and model prices. However, optimality cannot be guaranteed and the choice of a particular distance function is quite arbitrary.
In the following, we state the most common objective functions detailing Eq. (2.1).
A first obvious choice to specify the distance function is to use the
“smoothest” norm on which is the L 2-norm. The minimization problem then corresponds to the common least-squares problem and reads as
Instead of the L 2-norm any other L p -norm with 1 ≤ p < ∞ can also be used. Larger values of p put more emphasis on larger deviations.
This type of objective function emphasizes out-of-the-money products. It is given by
Similar to relative errors, logarithmic errors sometimes are advantageous, compare for example [14]. Here, the objective function is given as
It is important to note, that depending on the particular objective function, the resulting model parameters may vary, see for example [14]. So, in applications, one can incorporate further desirable properties. Mostly these features deal with the stability of the calibration, as detailed in the following list.
If market conditions change, market prices change. Ideally, the model is designed in such a way that it can reflect these changes. However at certain times a re-calibration of the model
parameters has to be performed in order to capture new market beliefs. If the change in the market prices is rather small, we expect the change in the parameters to be relatively small as
well, i.e. we want the calibration to be continuous and smooth in the market prices.
If market conditions remain constant, still, as time goes by, prices of financial derivatives change due to the decreasing time to maturity. Calibrating on the progressed prices, however, should then not distort the resulting model parameters. This is what we call time homogeneity.
The model as well as the calibration is required to price derivatives consistently with respect to time.
Prices of illiquid products with a wide bid-ask spread are less reliable and often might change even randomly. These prices imply different calibration results. This opposes the stability conditions above.
The consequential disparity of information is often incorporated into the calibration by assigning different weights and to the influence of the market prices depending on their importance, i.e. the influence of the prices might be stressed or damped. In applications, the weights are often set in such a way that they correspond to the liquidity of the product. Since
frequently traded instruments are more liquid, hence contain more reliable information, they might be of higher importance.
For example, in the case of equity options, setting the weights to the Vega of the product, i.e. to the derivative of the option price with respect to volatility, emphasizes at-the-money products.
Even using weights, calibration problems typically are ill-posed in the sense of Hadamard, see for instance [11]. Usually, the surface of the objective function is very rugged. It is not convex and does not feature any regular shapes. Often there exist several local minima, separated by relatively high ridges, what makes the calibration process even more difficult. Further, small changes in market prices typically lead to quite different model parameters.
In general, a stabilizing functional that incorporates prior knowledge will smooth the
minimization problem to some extent. In our case, the prior knowledge, or penalty term, can be seen as the expectation of the market participants, who usually assume that the model parameters remain relatively constant over time. In terms of the common least-squares formulation, for , the minimization problem changes to
(2.2) where are previously calibrated model parameters, if such parameters are available.
In (2.2), the multiplier α has to be chosen with care. If too small, the problem is still unstable, if too large, the dependence on prior data is too intense and we are in a situation of a self-fulfilling prophecy. For a broader view on ill-posed problems and possible regularization methods, we refer to [11] or [21].
Although model prices rarely match observed market prices exactly, they should at least lie within the corresponding bid-ask spread. In some cases, this relaxation is very reasonable because, not only due to noisy data, accuracy in the context of model calibration quickly might be spurious. In [17] Kalman filters are used to incorporate bid and ask prices into the calibration.
A further possibility to smooth calibration is to consider functionals of the product prices. These can for example be the time value of an equity option or the implied volatility, which we will discuss in Sect. 2.3.1.1.
Local or deterministic algorithms
Global or stochastic algorithms
2.2.2 Relevant Instruments
The selection of financial instruments for calibration is an important task and has to be done carefully.
First of all, the chosen instruments must allow to determine model prices fast and accurately for different model parameter sets. Note that, even though the time needed for calculating a single model price might be almost negligible, calculating the prices for a whole set of instruments and for several sets of model parameters is very time consuming. Moreover, especially when valuing derivatives for trading, calibration is done on demand for each trade rather than only occasionally, compare for instance [14].
For equity and foreign exchange markets this requirement leads to considering plain vanilla European exercise feature options almost exclusively. In interest rate markets the corresponding instruments are options on the forward rate, so called caps and floors. Although slightly more
complicated, options on the swap rate, so called swaptions, are also computable in closed-form, and hence are used for calibration too.
In general, more complicated products can be relevant as well. In particular, for multi-asset models correlation parameters must be calibrated. In this case, basket options can be used.
However, the pricing of more complex products often requires advanced numerical schemes like flexible Monte Carlo methods or finite difference schemes. But due to the slow convergence of Monte Carlo simulations and the strong dependence between grid size and pricing accuracy of finite
difference schemes, products that require advanced numerical methods are generally not usable for calibration.
In a nutshell: “Therefore, closed pricing formulae are the basis of a convenient model calibration.”, see [16, Remark 5.30].
2.2.3 Optimization: Local Versus Global
Obviously, minimizing the objective function is a non-linear problem. Due to the rugged shape of the objective function, both local and global optimization algorithms should be applied.
like the downhill simplex algorithm are simple and fast.
Usually, in each step these algorithms determine the direction in which the objective function has its steepest descend. However, they will terminate in a local minimum and should be restarted with different initial parameters again.
These algorithms are most useful if a strong expectation about the model parameters is available, such as if previously calibrated parameters exist. In this case it makes sense to also use these parameters as starting values for the optimization.
like adaptive simulated annealing do not depend on the initial parameter set, since they randomly sweep the search space.
Obviously, these algorithms are able to continuously improve the value of the objective function, but this comes with a high computational cost.
This type of algorithms should in particular be used for an initial calibration or when the market situation changed drastically.
Further, it is important to note that for many practical applications a perfect fit is not really
necessary. Rather, stable, more reliable values are usually preferred. Inaccuracies in pricing financial derivatives come from many sources and calibration of model parameters is just one of them. Others
Volatility clustering
include the appropriateness of the model and the use of approximation formulas.