Heston process simulation values?

[Venkatesh, P.C.]

  Hi, i tried to adapt \test-suite\hestonmodel.cpp to print out path
values of the process.  Specifically, within one of the test sections of
hestonmodel, i inserted a snippet of code from
\test-suite\pathgenerator.cpp to print out path values.  The key
elements then looks as follows:

    boost::shared_ptr<HestonProcess> process(new HestonProcess(
                   riskFreeTS, dividendTS, s0, 0.3, 1.16, 0.2, 0.8,
0.8));
   ....
    typedef PseudoRandom::rsg_type rsg_type;
    typedef MultiPathGenerator<rsg_type>::sample_type sample_type;

    BigNatural seed = 42;  Time length = 10;  Size timeSteps = 12;
    Size assets = process->size();
    rsg_type rsg =
PseudoRandom::make_sequence_generator(timeSteps*assets,
                                                         seed);
    MultiPathGenerator<rsg_type> generator(process,
                                           TimeGrid(length, timeSteps),
                                           rsg, false);
   
    Size i;
    for (i=0; i<100; i++){
      sample_type sample = generator.next();
      wtfil << " i= " << i << " value[0]= "<< sample.value[0].back() <<
          " value[1]= "<< sample.value[1].back()     << std::endl;    
    } // end i loop

This produces [first 11 values]:

 i= 0 value[0]= 8.38959 value[1]= -0.527286
 i= 1 value[0]= 7.17443 value[1]= -0.0744848
 i= 2 value[0]= 31.5997 value[1]= 0.669117
 i= 3 value[0]= 3.29738 value[1]= 0.0869779
 i= 4 value[0]= 7.42287 value[1]= 0.220784
 i= 5 value[0]= 1.56449 value[1]= -0.0911644
 i= 6 value[0]= 4.15224 value[1]= -0.0711199
 i= 7 value[0]= 14.0519 value[1]= -0.16052
 i= 8 value[0]= 7.14489 value[1]= 0.216222
 i= 9 value[0]= 9.32183 value[1]= 0.155854
 i= 10 value[0]= 3.46721 value[1]= -0.333396

I thought value[0] and value[1] would be the the asset value and the
variance, respectively [or their changes] at each time step i.  But
value[0] seems to jump a lot (e.g., from sstep 1 to step 2] and value[1]
is often negative.  Am I doing something wrong, or do these values
represent something else?  How would I recover the stock value and
variance at each time step?  [I tried different parameter sets----some
fixed the negative variance issue, but value[0] still bounces around a
lot.]

 thanks
--------------------------------------------
P.C. Venkatesh
Office of Prudential Supervision & Risk Analysis
Division of Trading and Markets, SEC
Station Place, 100 F Street, N.E., Washington, D.C.  20549

e-mail: [hidden email]
vox: 202 551 5549
fax: [phone] .

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[Klaus Spanderen-2]

Hi

generator.next() generates a complete new path. Therefore please change the
for - loop towards

     for (k=0; k<100; k++){
       sample_type sample = generator.next();
        for (int i=0; i < 12; ++i)
        wtfil << " i= " << i << " value[0]= "<< sample.value[0][i] <<
           " value[1]= "<< sample.value[1][i]     << std::endl;
     } // end i loop

Your suggestion is right, value[0] is the asset path and value[1] is the
variance. By default the Heston process is using an special kind of the Euler
discretization, the so called FullTruncation scheme. Therefore the variance
might become negative but heuristics have shown that these discretization
scheme produces less errors than e.g. the reflexion scheme. If you don't like
it use the ExactVariance discretization scheme.

     boost::shared_ptr<HestonProcess> process(new HestonProcess(
                    riskFreeTS, dividendTS,
                    s0, 0.3, 1.16, 0.2,0.8,0.8,HestonProcess::ExactVariance)));

It's slower but the variance will stay positive (or zero if the Feller
constraint is violated).

Just one additional remark, your vol of vol and your instantaneous variance
are quite high and it seems that you are using only 12 timesteps to simulate
10 years. This will lead to large discretization errors even with the
ExactVariance scheme. Please use much more time steps.

hope that helps

 Klaus


On Tuesday 18 March 2008 21:47:45 Venkatesh, P.C. wrote:

>   Hi, i tried to adapt \test-suite\hestonmodel.cpp to print out path
> values of the process.  Specifically, within one of the test sections of
> hestonmodel, i inserted a snippet of code from
> \test-suite\pathgenerator.cpp to print out path values.  The key
> elements then looks as follows:
>
>     boost::shared_ptr<HestonProcess> process(new HestonProcess(
>                    riskFreeTS, dividendTS, s0, 0.3, 1.16, 0.2, 0.8,
> 0.8));
>    ....
>     typedef PseudoRandom::rsg_type rsg_type;
>     typedef MultiPathGenerator<rsg_type>::sample_type sample_type;
>
>     BigNatural seed = 42;  Time length = 10;  Size timeSteps = 12;
>     Size assets = process->size();
>     rsg_type rsg =
> PseudoRandom::make_sequence_generator(timeSteps*assets,
>                                                          seed);
>     MultiPathGenerator<rsg_type> generator(process,
>                                            TimeGrid(length, timeSteps),
>                                            rsg, false);
>
>     Size i;
>     for (i=0; i<100; i++){
>       sample_type sample = generator.next();
>       wtfil << " i= " << i << " value[0]= "<< sample.value[0].back() <<
>           " value[1]= "<< sample.value[1].back()     << std::endl;
>     } // end i loop
>
> This produces [first 11 values]:
>
>  i= 0 value[0]= 8.38959 value[1]= -0.527286
>  i= 1 value[0]= 7.17443 value[1]= -0.0744848
>  i= 2 value[0]= 31.5997 value[1]= 0.669117
>  i= 3 value[0]= 3.29738 value[1]= 0.0869779
>  i= 4 value[0]= 7.42287 value[1]= 0.220784
>  i= 5 value[0]= 1.56449 value[1]= -0.0911644
>  i= 6 value[0]= 4.15224 value[1]= -0.0711199
>  i= 7 value[0]= 14.0519 value[1]= -0.16052
>  i= 8 value[0]= 7.14489 value[1]= 0.216222
>  i= 9 value[0]= 9.32183 value[1]= 0.155854
>  i= 10 value[0]= 3.46721 value[1]= -0.333396
>
> I thought value[0] and value[1] would be the the asset value and the
> variance, respectively [or their changes] at each time step i.  But
> value[0] seems to jump a lot (e.g., from sstep 1 to step 2] and value[1]
> is often negative.  Am I doing something wrong, or do these values
> represent something else?  How would I recover the stock value and
> variance at each time step?  [I tried different parameter sets----some
> fixed the negative variance issue, but value[0] still bounces around a
> lot.]
>
>  thanks
> --------------------------------------------
> P.C. Venkatesh
> Office of Prudential Supervision & Risk Analysis
> Division of Trading and Markets, SEC
> Station Place, 100 F Street, N.E., Washington, D.C.  20549
>
> e-mail: [hidden email]
> vox: 202 551 5549
> fax: [phone] .
>
> -------------------------------------------------------------------------
> This SF.net email is sponsored by: Microsoft
> Defy all challenges. Microsoft(R) Visual Studio 2008.
> http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/
> _______________________________________________
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> https://lists.sourceforge.net/lists/listinfo/quantlib-users

--
Klaus Spanderen
Ludwig Erhard Str. 12
48734 Reken (Germany)
EMail: [hidden email] (remove NOSPAM from the address)

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