numerical-methods-controlvariate-exercise

In this exercise you should implement a Monte-Carlo control variate to improve the convergence of the Monte-Carlo integration by reducing the variance.

A control variate is (usually) product and model dependent. This is a clear disadvantage of the method. Nevertheless, it can achieve impressive improvements.

In this exercise we consider a Black-Scholes model (as model) and an Asian option (as product).

Implement an Asian Option valuation under the Black-Scholes Model with a Control Variate.

Implement a class with the following properties:

• It implements the interface net.finmath.montecarlo.assetderivativevaluation.products.AssetMonteCarloProduct.
• It has a constructor taking the argument list (final double maturity, final double strike, final TimeDiscretization timesForAveraging, Double callOrPutSign) or (final Double maturity, final Double strike, final TimeDiscretization timesForAveraging, double callOrPutSign).
• The getValue method returns the value of the corresponding Asian option using a control variate for the Black-Scholes model.

The payoff of the Asian call option is max(1/n * sum S(T_i)-K,0) paid in T, where T_i are the times in timesForAveraging, n is the number of times in timesForAveraging, T is maturity and K is strike.

The payoff of the Asian put option is max(K - 1/n * sum S(T_i),0) paid in T, where T_i are the times in timesForAveraging, n is the number of times in timesForAveraging, T is maturity and K is strike.

Your implementation should cover call and put by implementing the payoff max( sign * ( 1/n * sum S(T_i) - K ),0) paid in T, where T_i are the times in timesForAveraging, n is the number of times in timesForAveraging, T is maturity and K is strike and sign is either +1 or -1.

And most importantly

• The Monte-Carlo valuation uses a control variate to improve the accuracy of the valuation (in probability, i.e. for most cases) if a Black-Scholes model is used.

Submission of the Solution

You may just complete the stub implementation provided in the repository in

info.quantlab.numericalmethods.assignments.montecarlo.controlvariate.AsianOptionWithBSControlVariate

Alternatively, if you provide your own implementation of a class implementing AssetMonteCarloProduct to value an Asian option, you may just return an object of your class in the method getAsianOption of AsianOptionWithBSControlVariateSolution. Remark: Our unit test will call this method to test your implementation.

Hints

Getting the parameters of the underlying Black-Scholes model (the ProcessModel)

The valuation method getValue takes as argument a model implementing AssetModelMonteCarloSimulationModel. This interface is comparably parsimonious as it only allows to get the value of the asset process S and the numeraire N (and some information on the simulation time discretization). At this point the model of S may be almost anything (Black-Scholes, Bachelier, Heston, etc.).

In order to construct a control variate it may be necessary to get more information about the ProcessModel used to construct the stochastic process.

When we test your implementation, we will call the getValue with a MonteCarloAssetModel and calling getModel() on this object will return a BlackScholesModel. You can rely on this to obtain the model parameters we used in the test (but you could implement an Exception handling if we don't do it). Hence, you can get the model properties via the following code:

	net.finmath.montecarlo.assetderivativevaluation.models.BlackScholesModel processModel = (BlackScholesModel) ((MonteCarloAssetModel)model).getModel();
	double initialValueOfStock = model.getAssetValue(0, 0).doubleValue();
	double riskFreeRate = processModel.getRiskFreeRate().doubleValue();
	double volatility = processModel.getVolatility().doubleValue();

Note (technical detail): Since the library allows to create objects implementing AssetModelMonteCarloSimulationModel in different ways, a slightly more robust way of getting the underlying model is to use the utility function

info.quantlab.numericalmethods.lecture.montecarlo.models.Utils.getBlackScholesModelFromMonteCarloModel

So you may get the underlying BlackScholesModel via

	net.finmath.montecarlo.assetderivativevaluation.models.BlackScholesModel processModel = info.quantlab.numericalmethods.lecture.montecarlo.models.Utils.getBlackScholesModelFromMonteCarloModel(model);

Test data

You may test you program with the following data.

	// Model properties
	private final double	initialValue   = 1.0;
	private final double	riskFreeRate   = 0.05;
	private final double	volatility     = 0.30;

	// Process discretization properties
	private final int		numberOfPaths		= 200000;
	private final int		numberOfTimeSteps	= 20;
	private final double	deltaT			= 0.5;

	// Product properties
	private final int		assetIndex = 0;
	private final double	maturity = 10.0;
	private final double	strike = 1.05;
	private final TimeDiscretization timesForAveraging = new TimeDiscretizationFromArray(5.0, 6.0, 7.0, 8.0, 9.0, 10.0);
	private final Double	callOrPutSign = 1.0;

For this model and product the value of the product is approximately μ = 0.372. The Monte-Carlo standard deviation is approximately σ = 0.74. Using 200,000 paths, the standard error then is ε = 0.00165.

Using control variates it is possible to bring the standard error below 0.0009 (comparably easy) and even below 0.0001 (a bit more difficult). This would correspond to using 200-times more Monte-Carlo simulation paths (requiring 200-times the computation time).

Unit Tests and GitHub Autograding

The project comes with a unit test that runs four test

• basic: (5 Points) Passes if the valuation of the Asian option appears to be OK (no variance reduction required).
• weak: (5 Points) Passes if at least some variance reduction is performed.
• strong: (5 Points) Passes if good variance reduction is performed.
• stronger: (5 Points) Passes if very good variance reduction is performed.
• strongest: (2 Points) Passes if extremely good variance reduction is performed.

Note: You may consider the exercise solved if you achieve 15 or 20 points, since this is already a good variance reduction. However, the autograding will show a failure unless you reach the full 22 points.

Importing in Eclipse from GitHub

Import this git repository into Eclipse and start working.

• Click on the link to your repository (the link starts with qntlb/numerical-methods… )
• Click on “Clone or download” and copy the URL to your clipboard.
• Go to Eclipse and select File -> Import -> Git -> Projects from Git (with smart import).
• Select “Clone URI” and paste the GitHub URL from step 2.
• Select “master” or "main", then Next -> Next -> Finish.

Note: If you choose "Projects from Git" without the option "(with smart import)" you may expience that the project is not imported into Eclipse, but it was successfully checked out via git, i.e. you find the project files in you local git folder. In that case, you can import the project "as maven project" (see below).

Importing in Eclipse (as Maven Project)

If you checked out the git repository manually (git clone), then import the local git folder as Maven Project;

• File -> Import -> Maven -> Existing Maven Projects
• Select the project folder in you local git folder.

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