Application Center - Maplesoft

Multivariate Distributions In Maple

You can switch back to the summary page by clicking here.

Multivariate Distributions In Maple

with applications in Finance

Igor Hlivka

MUFG Securities International

LONDON

The document demonstrates the extension of Maple's comprehensive Statistical package into multivariate setting. It shows how Maple's symbolic analytics and numerical engines can be seamlessly applied in the field of multivariate statistics. The core concept of multivariate analysis - joint distributions - are discussed in the context of multivariate Normal distribution and particular aspects of "jointness" are presented through marginal and conditional densities. Extension of multinormality into related family of joint distributions is shown on the example of multivariate Student-t distribution.

Application of multivariate analysis is demonstrated in the field of Finance, however many other fields and branches can be naturally explored and tested with the presented set up.

The Multivariate Normal Distribution

Similar to the univariate setting, the core concept of mutivariability resides on the Normal Distribution.

For a given vector of random variables `(X[1], X[2], () .. X[n])" align="center" border="0"> we define the Multivariate Normal Distribution as with mean vector `(mu[1], mu[2] .. mu[n])" align="center" border="0"> and variance-covariance matrix Σ.

The probability density then can be defined as:

where is a vector of real-valued random variables on n-length and Σ is a symmetric and positive definite variance-covariance matrix of size. The function fully resembles the "familiar" univariate Normal PDF, and indeed with , the above formula reduces to the univariate density function.

We build the in several steps:

• Create the relevant vectors of RVs, (X)  means (M)  and standard deviations (S)

• Build the correlation (CorrMat) and variance-covariance matrix (CoVar)

• Combine the components into a multinormal PDF

The Bivariate Normal Distribution

We will first demonstrate the implementation in the bivariate setting:

 >

 >

 (1.1.1)

 >

 (1.1.2)

This is the desired Bivariate Normal PDF with mean vector M and covariance matrix CoVar.

We can obtain the standardized version of the density by setting the mean vector to 0 and variance to 1

 >

 (1.1.3)

The Bivariate CDF can be obtained by integrating the PDF up to a given limit, however the outer integral has be to be computed numerically

 >

 (1.1.4)

Now we can compute various expectations:

• sums

• power sum

• products

 >

 (1.1.5)

Once the join PDF has been defined, we can compute Marginal and Conditional densities

Marginal PDF:             (x)  =

 >

 (1.1.6)

Conditional PDF of

 >

 (1.1.7)

We can define the multivariate Moment Generating Function as the expectation of the vectors product

Multivariate MGF:

 >

 (1.1.8)

We can now compute product moments (the first and the second) using MGF

 >

 (1.1.9)

One can verify that the first moment is the same as the product expectation EV[3] calculated above

Graphics

We will present some graphical representation of densities defined above:

Bivariate Standard PDF will exhibit different shapes depending on the correlation coefficient

 >

 >

This is a Conditional Expectation for x[1]=1..2

 >

 >

The Trivariate Normal Distribution

We can extend the Bivariate PDF for a higher dimension

 >

 >

 (1.2.1)

We will simplify the covariance matrix, knowing that symmetric matrix

At the same time, we will introduce the Trivariate Standard Normal PDF

Maple can handle this complex computational task quite easily...

 >

 (1.2.2)

We can now use the Trivariate Standard density to calculate expectations:

Foe example:

 >

 (1.2.3)

The expectation in this example is a function of single correlation coefficient:

 >

Expectations in trivariate setting are generally more complex and they may exist only for certain set of parameters.  One of the restrictions will apply to covariance matrix which by definition must be positive definite.

The Multivariate Student-t Distribution

We will now show how simple transformation argument leads to an elegant solution for the multivariate Student-t density.

We assume that `(X[1], X[2], () .. X[n])" align="center" border="0">is a vector of random variables drawn from multivariate normal distribution    and let other RV  Y be a Chi-Square-distributed quantity with the degree of freedom  ν:        independent of

Then the joint density of and Y  is:

This relationship then determined the multivariate Student-t distribution with the vector:    `(t[1], t[2], () .. t[n])" align="center" border="0">and the correlation matrix  Σ:

The Bivariate Student-t Distribution

We will demonstrate the above theory in bivariate setting.

Since both and Y and independent, we can obtain the joint density of by simple multiplication of Bivariate Normal and Chi-Square densities

• We will first define the Chi-Square PDF using Maple's Statistics package:

• We will recall the definition of the bivariate standard PDF computed in the previous section:

 >

 >

 (2.1.1)

 >

 (2.1.2)

Then the joint PDF    can be expressed as:

 >

 (2.1.3)

We now want to transform this PDF product into a Student-t quantities:

• We apply the change of variables transformation

• Obtain the determinant of the Jacobian

• Substitute the result into ψ

 >

 (2.1.4)

 >

 (2.1.5)

This is the transformed density expressed in terms of t-parameters:

We will now obtain the bivariate t-density by integrating the above density over y

 >

 (2.1.6)

The crude integration des not work due to ambiguity of the integrand in the exponential term

Noting that the term is negative, we can integrate with this assumption

• Make the term substitution

• Integrate

• Make the reverse substitution

We can verify and examine the negativity of the exponent term visually:

 >

As chart shows, the expterm will always stay negative, hence the integration with the negativity assumption is acceptable

 >

 (2.1.7)

This is the desired Bivariate Student-t density

We can plot it

 >

 >

Application in Finance

We now demonstrate how multivariate analysis can be applied in problems solving arising in Financial Economics.

Two particular cases will be reviewed here:

• Derivation of a hedging scheme

Formal derivation of the hedging concept

Hedging is a transactional process that aims to minimize the risk of the trader's activity. In short, hedging typically refers to the risk offsetting. The risk can be associated with a single transaction (the trade risk) or a number of individual transactions (portfolio risk).

Following trader's activity (buying or selling of financial product), an institution ends up with the position which value can be formally described as:

where:                         A = amount    and                 represents a risky financial product where price changes follow a Normal Distribution:

To eliminate the fluctuation in the value of the position, an institution considers hedging programme, i.e. taking position in different instrument - say - in order to eliminate the risk.

The task is to determine the quantity of hedging instrument - say amount B -  that minimizes the value fluctuation of the original transaction.

We assume that the hedging instrument    is a financial product that similar to follows a Normal Distribution.  Once the quantity is added to the original transaction, the risk is not any longer univariate (two instruments now constitute a mini-portfolio), but now is governed by a bivariate normal distribution

To present this case formally:

• We will create the portfolio of two instruments

• Compute the first two moments - mean and variance

 >

 (3.1.1)

The reader can see that we have computed the portfolio variance by the formula:

Given our objectives, we want to minimize the portfolio's risk, i.e. variance. Although Maple's  Optimization routines can be used here, we will apply step-by-step procedures to derive this result explicitly.

We will:

• Differentiate the portfolio variance w.r.t B

• Set this to zero

• Solve the equation for the quantity B

 >

 (3.1.2)

This is the solution to our problem - we have determined the quantity B = that needs to added to the existing position A to minimize the risk.

The B quantity will be:

• positive        if    (i) correlation is negative   or (ii) amount A is negative    (the so-called short position)

• negative        if    (i) correlation is positive   or (ii) amount A is positive    (the so-called long position)

We still need to verify that this is the minimum (rather than maximum)

The standard condition for minimizations problems stipulates that

 >

 (3.1.3)

The seconded derivative of the portfolio's variance     - MinTest   is always positive, so adding the quantity B shown above is indeed the solution to our minimization exercise.

As seen from above, the efficiency of the hedging is a function of three variables:    volatility of individual assets and the correlation between the two instruments.

How sensitive is the this amount to the functional input?

• The quantity is primarily affected by the volatility of the hedging instrument

• When is low, the correlation will have more pronounced impact on the hedging scheme

• For higher values of correlation effect will diminish

• This is confirmed in the animation chart below:

 >

 >

Spread options naturally arise in Finance when optionality on future value is taken w.r.t  two assets. Spread in finance is usually defined as a difference between the two quantities and can be expressed in terms of difference in (i) prices, (ii) yields, (iii) rates or (iv) spreads themselves (eg. credit spreads).

Spreads options are designed and traded to manage future expectations of assets behavior and allow monetisation of a particular view on their direction and dependency

• If trader expects the spread between two instruments to widen beyond a certain level (the strike),  he(she) will buy a Call Option (benefits if

• If, on the other hand, trader predicts the narrowing of the spread, he(she) will buy a Put Option (benefits if

Mathematically, spread options are class of truncated expectations where the expectation is taken w.r.t joint distribution of two instruments in the defined bivariate space.

To value the spread option, we will define:

• each instrument own stochastic process  (here we assume that the underlyings are "rates" instruments)

• Instrument[1] = S

• Instrument[2] = U

• joint distribution

• option types  Call and Put Spread Option

 >

 (3.2.1)

 >

 (3.2.2)

The formula above is the "adjusted"payoff of the Call Spread Option in the raw format. It differs from the standard notation by having the usual Max(0.....) notation removed, since we will value this option by taking the discounted expectation of the payoff function over the "eligible" positive domain.

The trick to value an option in bivariate setting is to determine the correct limits of integration for each stochastic variable. We will achieve this in three steps:

• Determine the critical threshold for which the stochastic variable takes greater or lower value  -  we compute this threshold w.r.t

• Perform the integration using this critical limit in the inner space

• Keep the outer integration along the whole variable domain  for the second variable  -

 >

 (3.2.3)

This is the required critical threshold

• For Call Option - to ensure the posiitivity of the payoff - we will integrate the probability-weighted payoff  from

• For Put Option - to stay positive- we will do the opposite:

 >

 (3.2.4)

This is the desired Call Spread Option expression - a discounted expectation w.r.t bivariate standard normal density with correlation ρ. The discounting is performed with the DF expression outside the integral.

 >

 (3.2.5)

Maple returns a semi-analytical result that, unfortunately, cannot be evaluated symbolically any further, so we need to resort to numerical routines, which Maples handles with ease.

All we need is to assign numerical values to each parameter and numerically integrate:

We use the particular case:

• Rate[1]                       S = 4.50%

• Rate[2]                      U = 4.60%

• Strike                         K = 0.20%

• σ[1] = 18%

• σ[2] = 15%

• Correlation between S and U:      ρ=0.65

• Option maturity   T  = 1  (year)

• Discount factor       DF = 0.96

Here we are valuating a call option that pays  the unit of currency (i.e. \$1 or ?1 ) if in one year from now the S - U > 0.20%    (irrespective of the fact that today U > S)

 >

 (3.2.6)

This is the Call Option Premium obtained numerically.

Now, what is the sensitivity of the option premium w.r.t functional input?

Although we have computed the value numerically, this does not impact Maple's productivity or efficiency:

• It can first perform the differentiation symbolically

• Evaluate the result numerically

For example, we can compute the option's sensitivity w.r.t. each underling rate  - the deltas  for  S and U

 >

 (3.2.7)

What do these values mean?

• If the value of the rate S changes by +1%, then the call option premium will increase by 0.33%

• If the value of the rate U increases by 1% then the call option premium will drop by 0.28%

Maple can be equally efficient in computing any other desired comparative static - gamma (2nd derivative w.r.t rate), vega (derivative w.r.t sigma) etc.

Reader can verify that in the spread option case, the sensitivity set increases by the factor of two - we will have two deltas, two gammas etc.

At the same time, for example, we can visualize the impact of a certain parameter - such as correlation- on the call option premium computed above:

 >

As seen from the chart, the option premium will increase when the correlation gets lower and will gradually decrease as the correlation increases - as expected. The curvature on the graph also reveals that the relationship between the premium and the correlation is non-linear.

Conclusion

This short presentation shows the ways how Maple's univariate statistical analysis can be easily extended into multivariate platform. The centered focus resides on the Multivariate Normal Distribution that with the help of suitable transformation can be turned into other multivariate distributions.

Two particular examples from Finance demonstrate the advantages of using CAS platforms in applied multivariate analysis. Maple, as the above examples highlight, is well suited and flexible to handle these tasks well.

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.