Description

The need for analyzing option portfolios arises because risk/return
tradeoffs grow increasingly more complex as more options are included
in a portfolio. This model evaluates the worst-case scenario for
an option portfolio.

Small Model of Type : NLP

Category : GAMS Model library

Main file : worst.gms

$title Financial Optimization: Risk Management (WORST,SEQ=111)

$onText
The need for analyzing option portfolios arises because risk/return
tradeoffs grow increasingly more complex as more options are included
in a portfolio. This model evaluates the worst-case scenario for
an option portfolio.


Dahl, H, Meeraus, A, and Zenios, S A, Some Financial Optimization
Models: Risk Management. In Zenios, S A, Ed, Financial Optimization.
Cambridge University Press, New York, NY, 1993.

Keywords: nonlinear programming, risk management, portfolio optimization, finance,
          financial optimization
$offText

Set
   t 'expiration date'  / jun, oct, jan    /
   i 'underlying issue' / 9000011, 9020063 /
   j 'options'          / 1*5 /;

Table tdata(t,*) 'time related data'
* Notation:
*  term  time in years to maturing
*  r0    current estimated risk free rate
*  rmax  maximum risk free rate
*  rmin  minimum risk free rate
*  qmax  maximum volatility move
*  qmin  minimum volatility move
            term      r0     rmax     rmin    qmax    qmin
   jun   0.09167  0.0697  0.08570  0.05245  0.0788  0.0388
   oct   0.33889  0.0790  0.09500  0.06175  0.0768  0.0368
   jan   0.58889  0.0779  0.09390  0.06190  0.0768  0.0368;

Table f0(i,t) 'adjusted spot price'
               jun    oct    jan
   9000011   95.54  93.27  91.03
   9020063   95.54  93.27  91.03;

Acronym  future, call, puto;

Table pdata(i,t,j,*) 'portfolio data'
* Notation:
*  type    type of instrument  (futures, call option, put option)
*  strike  strike price for call and put options
*  nom     nominal units of issue (ref. parameters x, y, and z)
*  price   price of futures or options premium
                      type  strike      nom   price
   9000011.jun.1    future           -35000   96.60
   9000011.oct.1    future            15000   96.60
   9020063.jun.1    future            74000   96.15
   9020063.oct.1    future            20000   95.80
   9020063.oct.2    call        95   -30000    3.00
   9020063.oct.3    call        97   -30000    1.50
   9020063.oct.4    puto        95     5000    0.25
   9020063.oct.5    puto        97    15000    1.20
   9020063.jan.1    future          -290000   95.80
   9020063.jan.2    call        95    25000    0.90
   9020063.jan.3    call        97   -50000    0.90
   9020063.jan.4    call        99    25000    0.90
   9020063.jan.5    puto        99    50000    0.90;

Variable
   pval      'portfolio objective function value'
   d1(i,t,j) 'black d1'
   d2(i,t,j) 'black d2';

Positive Variable
    f(i,t)   'futures price'
    c(i,j,t) 'call price'
    p(i,j,t) 'put price'
    r(t)     'risk free rate'
    q(t)     'volatility';

r.lo(t)   = tdata(t,"rmin");
r.up(t)   = tdata(t,"rmax");
q.lo(t)   = tdata(t,"qmin");
q.up(t)   = tdata(t,"qmax");
f.lo(i,t) = .001;
f.l(i,t)  = f0(i,t)*exp(tdata(t,"r0")*tdata(t,"term"));
r.l(t)    = (r.lo(t) + r.up(t))/2;
q.l(t)    = (q.lo(t) + q.up(t))/2;

Equation
   tpv            'total portfolio value'
   futval(i,t,j)  'futures value'
   callval(i,t,j) 'call value'
   putval(i,t,j)  'put value'
   dd1(i,t,j)     'd1 in black model'
   dd2(i,t,j)     'd2 in black model';

tpv..
   pval =e= sum((i,t,j)$pdata(i,t,j,"nom"),
                   (f(i,t)  -pdata(i,t,j,"price")
                            *pdata(i,t,j,"nom"))$(pdata(i,t,j,"type") = future)
                 + (c(i,j,t)*pdata(i,t,j,"nom"))$(pdata(i,t,j,"type") = call)
                 + (p(i,j,t)*pdata(i,t,j,"nom"))$(pdata(i,t,j,"type") = puto));

futval(i,t,j)$(pdata(i,t,j,"type") = future)..
   f(i,t)    =e= f0(i,t)*exp(r(t)*tdata(t,"term"));

callval(i,t,j)$(pdata(i,t,j,"type") = call)..
   c(i,j,t)  =e= exp(-r(t)*tdata(t,"term"))*(f(i,t)*errorf(d1(i,t,j))
              -  pdata(i,t,j,"strike")*errorf(d2(i,t,j)));

putval(i,t,j)$(pdata(i,t,j,"type") eq puto)..
   p(i,j,t)  =e= exp(-r(t)*tdata(t,"term"))*(pdata(i,t,j,"strike")*errorf(-d2(i,t,j))
              -  f(i,t)*errorf(-d1(i,t,j)));

dd1(i,t,j)$pdata(i,t,j,"strike")..
   d1(i,t,j) =e= (log(f(i,t)/pdata(i,t,j,"strike")) + 0.5*sqr(q(t))*tdata(t,"term"))
              /  (q(t)*sqrt(tdata(t,"term")));

dd2(i,t,j)$pdata(i,t,j,"strike")..
   d2(i,t,j) =e= d1(i,t,j) - q(t)*sqrt(tdata(t,"term"));

Model riskmod / all /;

* worst case
solve riskmod using nlp minimizing pval;

* best case
solve riskmod using nlp maximizing pval;