Motivation: In most of the examples we dealt with we have assumed that the underlying asset price process is a GBM or is nice enough that we could avoid simulating from the increments or it was easy to simulate them.
Other than the GBM, there are two other important cases where the transition density is known exactly: the Vasicek and the CIR square root rate models.
Vasicek Model
The Vasicek spot rate model (also known as Ornstein-Uhlenbeck model) is
where
- is the long-term mean interest rate
- is the speed of mean reversion
- is the instantaneous volatility
It can be derived that for , the conditional distribution of given is
This process has as the stationary distribution.
Remarks:
- Vasicek’s model was the first one to capture mean reversion (by ), an essential characteristic of interest rate
- The parameter controls how fast or slow does mean reversion happens
- The main disadvantage is that, under Vasicek’s model, it is theoretically possible for the interest rate to become negative. This shortcoming is fixed by the Cox-Ingersoll-Ross (CIR) square root model.
CIR model
The Cox-Ingersoll-Ross (CIR) square root model is given by
The CIR model retains the mean reversion property by . Besides, ensures that the process never becomes negative.
The conditional distribution is not Gaussian but a non-central distribution. For , the conditional distribution of given is given by
where and , .
The non-central chi-square with degrees of freedom and non-centrality parameter is obtained by taking sum of squares of independent random variables where . No matter what sequence of we take, as long as , we get the same distribution.
We can get samples from non-central distribution by
X = np.random.noncentral_chisquare(d, _lambda, n) .Euler Discretization
Basic Idea (Forward Euler Method)
Suppose we have an SDE of the form
We want to simulate values of but we do not know its distribution. We can instead simulate a discretized version of the SDE , where is the number of steps, is the step-size and .
All discretization methods start by integrating the SDE on :
Euler discretization using the approximations: for ,
Since , we have
Backward Euler Method
Backward Euler method instead use
and then
Backward Euler discretization scheme is sometimes used to bring stability that lacks with the Forward Euler method.
Examples
Black-Scholes Model
The BS model is
the exact increment process in this case:
Hence
Euler discretization in this case is
and thus
Note that for small enough .
Heston Model
The Heston model is described by the bivariate stochastic process for the stock price and its volatility as
where ( and are correlated at each ).
Firstly apply the Euler scheme to and then to
where is a standard normal random variable such that is . can be constructed as where is another independent variable.
Lamperti Transform
Basic Idea
Sometimes it can be beneficial to discretize and SDE after transformation, i.e., if we are given an SDE for , find the SDE for for some transformation using Ito’s lemma, and discretize to get , return .
Lamperti transform tries to transform the SDE so that the volatility becomes constant. Suppose the dynamics of are driven by
Define . Then has unit volatility.
Examples
BS model
The BS model is
We have , hence the Lamperti transform is
Taking and applying Ito’s lemma, we have
Applying Euler discretization to , we get
therefore
which is the same as the closed form formula.
Heston Model
For the Heston volatility model
thus . The Lamperti transform is
Taking and applying Ito’s lemma, we have
If we further apply the forward Euler Discretization
which involves and thus is unstable. We can apply the backward Euler Discretization to avoid this as
Milstein Discretization
Basic Idea
The Euler discretization scheme yields
The drift term is order but the volatility term is only of order . The idea of Milstein’s scheme is to expand the volatility term to get a order error by applying the Ito’s lemma.
Consider the specialized SDE:
The drift and volatility are only functions of and not .
Start with
Unlike Euler discretization where we approximated by a constant, we write an expansion for using Ito’s lemma
Applying Euler discretization to this, we get the Milstein scheme as
Milstein’s discretization implies that given
while the Euler discretization only implies
Note that this improvement requires an additional smoothness assumption on the volatility .
Furthermore, Euler discretization is easy to extend to the multivariate case, but Milstein’s discretization is not that easy for the multivariate case because of cross-product terms
that appear for multivariate SDEs.
Evaluation Metrics: Convergence
There are two metrics:
- Strong convergence
- or
- A discretization scheme has a strong convergence of order if for small enough
- Weak convergence
- for smooth functions , e.g. or
- A discretization scheme has a weak convergence of order if for small enough
Euler discretization has a strong convergence of order 1/2 which comes from . It has a weak convergence of order 1.
Milstein discretization has a strong convergence of order 1 and has a weak convergence of order 1.
It is also possible to refine the Euler scheme beyond the Milstein scheme to obtain schemes of order 2. Most of these, similar to Milstein’s discretization, are hard to implement in multivariate cases.
Randomized Midpoint Discretization
The starting point of discretization
The second term has an expectation of zero. Euler discretization replaces the first term with the biased approximation . Shen and Lee (2019, NeurlPS) noticed that for
Hence, an unbiased discretization would start with
But we don’t know , to obtain it we can use Euler discretization.
Starting with and taking , we have
Randomized midpoint discretization uses
The last two terms come from the fact that
Because of unbiasedness, randomized midpoint discretization has a weak convergence of order 2.
Richardson Extrapolation
Above discretization methods aimed at path generation, i.e. looking to get a version of entire path
In practice, it may suffice to construct an estimator of .
The Euler and Milstein can get weak order of convergence 1, i.e.,
for small enough .
Under more regularity conditions, we have
as . This implies that (Richardson extrapolation)
as .
One simple way to implement extrapolation is to generate independently and for and return the estimator
Generating and independently can yield a higher variance and by making them correlated can reduce the variance. More specifically, each Brownian increment in should be the sum of increments in .
This means that if we use as the Brownian increments for , then use as the Brownian increments for .
Brownian Bridge Interpolation
Brownian bridge interpolation is used for reducing the error in estimating the price of options with non-smooth payoff functions.
For estimating , Euler and Milstein discretization schemes can yield a weak convergence of order 1. This only holds if is smooth enough. But some of the payoff functions are not smooth, for example
- Lookback or max option with payoff
- Swing option:
- Barrier options, knock-in or knock-out options
All these payoffs depend on the minimum or the maximum of the process .
However, when we generate the discretized path, we only obtain , and the maximum/minimum of this finite sequence is not a good approximation to the maximum/minimum of the continuous process.
Taking the maximum/minimum of the finite sequence is the same as computing the maximum/minimum of linear interpolation:
for , where .
Assuming constant drift and constant volatility in the interval , we can define
where is a Brownian motion process conditioned to satisfy
Such a Brownian motion with fixed starting and ending values is called a Brownian bridge or a tied-down Brownian motion, which is very well studied.
In particular, we know the distribution of the maximum of a Brownian bridge:
The distribution of the minimum can be obtained using the reflection principle as
So, we can use
to approximate or of process.
For the lookback option having the discounted payoff function
From the discretization techniques, we get (with ). Clearly, no matter what is
This implies that the standard Monte Carlo estimator can be underestimating the price.
The Brownian bridge method would use
where
This method can rectify the underestimation issue.
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