Documentation

ts2func

Convert time series arrays to functions of time and state

Description

example

F = ts2func(Array) encapsulates a time series array associated with a vector of real-valued observation times within a MATLAB® function suitable for Monte Carlo simulation of an NVARS-by-1 state vector Xt.

n periods.

example

F = ts2func(___,Name,Value) adds optional name-value pair arguments.

Examples

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Simulate risk-neutral sample paths.

dt      = 1/250;
returns = tick2ret(Dataset.CAC);
sigma   = std(returns)*sqrt(250);
yields  = Dataset.EB3M;
yields  = 360*log(1 + yields);

Simulate paths using a constant, risk-free return

nPeriods = length(yields);  % Simulated observations
rng(5713,'twister')
obj    = gbm(mean(yields),diag(sigma),'StartState',100)
obj =
Class GBM: Generalized Geometric Brownian Motion
------------------------------------------------
Dimensions: State = 1, Brownian = 1
------------------------------------------------
StartTime: 0
StartState: 100
Correlation: 1
Drift: drift rate function F(t,X(t))
Diffusion: diffusion rate function G(t,X(t))
Simulation: simulation method/function simByEuler
Return: 0.0278117
Sigma: 0.231906
[X1,T] = simulate(obj,nPeriods,'DeltaTime',dt);

Simulate paths using a dynamic, deterministic rate of return (get r)

r = ts2func(yields,'Times',(0:nPeriods - 1)');

Simulate paths using a dynamic, deterministic rate of return (r output 1)

r(0,100)
ans = 0.0470

Simulate paths using a dynamic, deterministic rate of return (r output 2)

r(7.5,200)
ans = 0.0472

Simulate paths using a dynamic, deterministic rate of return (r output 3)

r(7.5)
ans = 0.0472

Simulate paths using a dynamic, deterministic rate of return

rng(5713,'twister')
obj = gbm(r, diag(sigma),'StartState',100)
obj =
Class GBM: Generalized Geometric Brownian Motion
------------------------------------------------
Dimensions: State = 1, Brownian = 1
------------------------------------------------
StartTime: 0
StartState: 100
Correlation: 1
Drift: drift rate function F(t,X(t))
Diffusion: diffusion rate function G(t,X(t))
Simulation: simulation method/function simByEuler
Return: function ts2func/vector2Function
Sigma: 0.231906
X2  = simulate(obj,nPeriods,'DeltaTime',dt);

Compare the two simulation trials.

subplot(2,1,1)
plot(dates,100*yields)
datetick('x')
xlabel('Date')
ylabel('Annualized Yield (%)')
title('Risk Free Rate(3-Mo Euribor Continuously-Compounded)')
subplot(2,1,2)
plot(T,X1,'red',T,X2,'blue')
xlabel('Time (Years)')
ylabel('Index Level')
title('Constant vs. Dynamic Rate of Return: CAC 40')
legend({'Constant Interest Rates' 'Dynamic Interest Rates'},...
'Location', 'Best') Input Arguments

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Time series array to encapsulate within a callable function of time and state, specified as a vector, two-dimensional matrix, or three-dimensional array

Data Types: double

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: F = ts2func(yields,'Times',(0:nPeriods - 1)')

Monotonically increasing observation times associated with the time series input array (Array), specified as the comma-separated pair consisting of 'Times' and a vector.

Data Types: double

Specifies which dimension of the input time series array (Array) is associated with time, specified as the comma-separated pair consisting of 'TimeDimension' and a scalar integer.

Data Types: double

Specifies which dimension of the input time series array (Array) is associated with the NVARS state variables, specified as the comma-separated pair consisting of 'StateDimension' and a positive scalar integer.

Data Types: double

Flag to indicate whether the output function is a deterministic function of time alone, specified as the comma-separated pair consisting of 'Determininistic' and a scalar integer flag.

If Deterministic is true, the output function F is a deterministic function of time, F(t), and the only input it accepts is a scalar, real-valued time t. If Deterministic is false, the output function F accepts two inputs, a scalar, real-valued time t followed by an NVARS-by-1 state vectorX(t).

Data Types: logical

Output Arguments

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Callable function F(t) of a real-valued scalar observation time t, returned as a function.

If the optional input argument Deterministic is true, F is a deterministic function of time, F(t), and the only input it accepts is a scalar, real-valued time t. Otherwise, if Deterministic is false (the default), F accepts a scalar, real-valued time t followed by an NVARS-by-1 state vector X(t).

Note

You can invoke F with a second input (such as an NVARS-by-1 state vector X), which is a placeholder that ts2func ignores. For example, while F(t) and F(t,X) produce identical results, the latter directly supports SDE simulation methods.

Algorithms

• When you specify Array as a scalar or a vector (row or column), ts2func assumes that it represents a univariate time series.

• F returns an array with one less dimension than the input time series array Array with which F is associated. Thus, when Array is a vector, a 2-dimensional matrix, or a three-dimensional array, F returns a scalar, vector, or 2-dimensional matrix, respectively.

• When the scalar time t at which ts2func evaluates the function F does not coincide with an observation time in Times, F performs a zero-order-hold interpolation. The only exception is if t precedes the first element of Times, in which case F(t) = F(Times(1)).

• To support Monte Carlo simulation methods, the output function F returns an NVARS-by-1 column vector or a two-dimensional matrix with NVARS rows.

• The output function F is always a deterministic function of time, F(t), and may always be called with a single input regardless of the Deterministic flag. The distinction is that when Deterministic is false, the function F may also be called with a second input, an NVARS-by-1 state vector X(t), which is a placeholder and ignored. While F(t) and F(t,X) produce identical results, the former specifically indicates that the function is a deterministic function of time, and may offer significant performance benefits in some situations.

 Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest Rate.” The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.

 Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, Vol. 54, No. 4, August 1999.

 Glasserman, P. Monte Carlo Methods in Financial Engineering. New York, Springer-Verlag, 2004.

 Hull, J. C. Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.

 Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 2, 2nd ed. New York, John Wiley & Sons, 1995.

 Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer-Verlag, 2004.