Actual source code: blackscholes.c

  1: /**********************************************************************
  2:     American Put Options Pricing using the Black-Scholes Equation
  3:   
  4:    Background (European Options):
  5:      The standard European option is a contract where the holder has the right
  6:      to either buy (call option) or sell (put option) an underlying asset at 
  7:      a designated future time and price.
  8:   
  9:      The classic Black-Scholes model begins with an assumption that the
 10:      price of the underlying asset behaves as a lognormal random walk.  
 11:      Using this assumption and a no-arbitrage argument, the following
 12:      linear parabolic partial differential equation for the value of the
 13:      option results:
 14:   
 15:        dV/dt + 0.5(sigma**2)(S**alpha)(d2V/dS2) + (r - D)S(dV/dS) - rV = 0.
 16:   
 17:      Here, sigma is the volatility of the underling asset, alpha is a
 18:      measure of elasticity (typically two), D measures the dividend payments
 19:      on the underling asset, and r is the interest rate.
 20:   
 21:      To completely specify the problem, we need to impose some boundary
 22:      conditions.  These are as follows:
 23:   
 24:        V(S, T) = max(E - S, 0)
 25:        V(0, t) = E for all 0 <= t <= T
 26:        V(s, t) = 0 for all 0 <= t <= T and s->infinity
 27:   
 28:      where T is the exercise time time and E the strike price (price paid
 29:      for the contract).
 30:   
 31:      An explicit formula for the value of an American option can be
 32:      found.  See the references for examples.
 33:   
 34:    Background (American Options):
 35:      The American option is similar to its European counterpart.  The
 36:      difference is that the holder of the American option can excercise
 37:      their right to buy or sell the asset at any time prior to the
 38:      expiration.  This additional ability introduce a free boundary into
 39:      the Black-Scholes equation which can be modeled as a linear
 40:      complementarity problem.
 41:   
 42:        0 <= -(dV/dt + 0.5(sigma**2)(S**alpha)(d2V/dS2) + (r - D)S(dV/dS) - rV)
 43:          complements 
 44:        V(S,T) >= max(E-S,0)
 45:   
 46:      where the variables are the same as before and we have the same boundary
 47:      conditions.
 48:   
 49:      There is not explicit formula for calculating the value of an American
 50:      option.  Therefore, we discretize the above problem and solve the
 51:      resulting linear complementarity problem.
 52:   
 53:      We will use backward differences for the time variables and central
 54:      differences for the space variables.  Crank-Nicholson averaging will
 55:      also be used in the discretization.  The algorithm used by the code
 56:      solves for V(S,t) for a fixed t and then uses this value in the
 57:      calculation of V(S,t - dt).  The method stops when V(S,0) has been
 58:      found.
 59:   
 60:    References:
 61:      Huang and Pang, "Options Pricing and Linear Complementarity,"
 62:        Journal of Computational Finance, volume 2, number 3, 1998.
 63:      Wilmott, "Derivatives: The Theory and Practice of Financial Engineering,"
 64:        John Wiley and Sons, New York, 1998.
 65: ***************************************************************************/

 67: /*
 68:   Include "tao.h" so we can use TAO solvers with PETSc support.  
 69:   Include "petscda.h" so that we can use distributed arrays (DAs) for managing
 70:   the parallel mesh.
 71: */

 73: #include "petscda.h"
 74: #include "tao.h"

 76: static char  help[] =
 77: "This example demonstrates use of the TAO package to\n\
 78: solve a linear complementarity problem for pricing American put options.\n\
 79: The code uses backward differences in time and central differences in\n\
 80: space.  The command line options are:\n\
 81:   -rate <r>, where <r> = interest rate\n\
 82:   -sigma <s>, where <s> = volatility of the underlying\n\
 83:   -alpha <a>, where <a> = elasticity of the underlying\n\
 84:   -delta <d>, where <d> = dividend rate\n\
 85:   -strike <e>, where <e> = strike price\n\
 86:   -expiry <t>, where <t> = the expiration date\n\
 87:   -mt <tg>, where <tg> = number of grid points in time\n\
 88:   -ms <sg>, where <sg> = number of grid points in space\n\
 89:   -es <se>, where <se> = ending point of the space discretization\n\n";

 91: /*T
 92:    Concepts: TAO - Solving a complementarity problem
 93:    Routines: TaoInitialize(); TaoFinalize();
 94:    Routines: TaoCreate(); TaoDestroy();
 95:    Routines: TaoApplicationCreate(); TaoAppDestroy();
 96:    Routines: TaoAppSetJacobianRoutine(); TaoAppSetConstraintRoutine();
 97:    Routines: TaoAppSetJacobianMat(); TaoSetOptions();
 98:    Routines: TaoSolveApplication();
 99:    Routines: TaoAppSetVariableBoundsRoutine(); TaoAppSetInitialSolutionVec();
100:    Processors: 1
101: T*/


104: /*
105:   User-defined application context - contains data needed by the
106:   application-provided call-back routines, FormFunction(), and FormJacobian().
107: */

109: typedef struct {
110:   double *Vt1;                /* Value of the option at time T + dt */
111:   double *c;                  /* Constant -- (r - D)S */
112:   double *d;                      /* Constant -- -0.5(sigma**2)(S**alpha) */

114:   PetscReal rate;                /* Interest rate */
115:   PetscReal sigma, alpha, delta; /* Underlying asset properties */
116:   PetscReal strike, expiry;      /* Option contract properties */

118:   PetscReal es;                      /* Finite value used for maximum asset value */
119:   PetscReal ds, dt;              /* Discretization properties */
120:   int ms, mt;                      /* Number of elements */

122:   DA da;
123: } AppCtx;

125: /* -------- User-defined Routines --------- */

127: int FormConstraints(TAO_APPLICATION, Vec, Vec, void *);
128: int FormJacobian(TAO_APPLICATION, Vec, Mat *, Mat*, MatStructure*, void *);
129: int ComputeVariableBounds(TAO_APPLICATION, Vec, Vec, void*);

133: int main(int argc, char **argv)
134: {
135:   int      info;          /* used to check for functions returning nonzeros */
136:   Vec      x;             /* solution vector */
137:   Vec      c;             /* Constraints function vector */
138:   Mat J;                  /* jacobian matrix */
139:   PetscTruth flg;          /* A return variable when checking for user options */
140:   TAO_SOLVER tao;          /* TAO_SOLVER solver context */
141:   TAO_APPLICATION my_app; /* The PETSc application */
142:   AppCtx user;                  /* user-defined work context */
143:   int i, j;               
144:   int    xs,xm,gxs,gxm;
145:   double sval = 0;
146:   PetscScalar *x_array;
147:   Vec    localX;

149:   /* Initialize PETSc, TAO */
150:   PetscInitialize(&argc, &argv, (char *)0, help);
151:   TaoInitialize(&argc, &argv, (char *)0, help);

153:   /* 
154:      Initialize the user-defined application context with reasonable 
155:      values for the American option to price 
156:   */
157:   user.rate = 0.04;
158:   user.sigma = 0.40;
159:   user.alpha = 2.00;
160:   user.delta = 0.01;
161:   user.strike = 10.0;
162:   user.expiry = 1.0;
163:   user.mt = 10;
164:   user.ms = 150;
165:   user.es = 100.0;
166:   
167:   /* Read in alternative values for the American option to price */
168:   info = PetscOptionsGetReal(PETSC_NULL, "-alpha", &user.alpha, &flg); 
169:          CHKERRQ(info);
170:   info = PetscOptionsGetReal(PETSC_NULL, "-delta", &user.delta, &flg); 
171:          CHKERRQ(info);
172:   info = PetscOptionsGetReal(PETSC_NULL, "-es", &user.es, &flg); 
173:          CHKERRQ(info);
174:   info = PetscOptionsGetReal(PETSC_NULL, "-expiry", &user.expiry, &flg); 
175:          CHKERRQ(info);
176:   info = PetscOptionsGetInt(PETSC_NULL, "-ms", &user.ms, &flg); 
177:          CHKERRQ(info);
178:   info = PetscOptionsGetInt(PETSC_NULL, "-mt", &user.mt, &flg); 
179:          CHKERRQ(info);
180:   info = PetscOptionsGetReal(PETSC_NULL, "-rate", &user.rate, &flg); 
181:          CHKERRQ(info);
182:   info = PetscOptionsGetReal(PETSC_NULL, "-sigma", &user.sigma, &flg); 
183:          CHKERRQ(info);
184:   info = PetscOptionsGetReal(PETSC_NULL, "-strike", &user.strike, &flg); 
185:          CHKERRQ(info);

187:   /* Check that the options set are allowable (needs to be done) */

189:   user.ms++;
190:   info = DACreate1d(PETSC_COMM_WORLD,DA_NONPERIODIC,user.ms,1,1,
191:                     PETSC_NULL,&user.da); CHKERRQ(info);
192:   /* Create appropriate vectors and matrices */

194:   info = DAGetCorners(user.da,&xs,PETSC_NULL,PETSC_NULL,&xm,PETSC_NULL,PETSC_NULL); CHKERRQ(info);
195:   info = DAGetGhostCorners(user.da,&gxs,PETSC_NULL,PETSC_NULL,&gxm,PETSC_NULL,PETSC_NULL); CHKERRQ(info);

197:   info = DACreateGlobalVector(user.da,&x);CHKERRQ(info);
198:   /* 
199:      Finish filling in the user-defined context with the values for
200:      dS, dt, and allocating space for the constants 
201:   */
202:   user.ds = user.es / (user.ms-1);
203:   user.dt = user.expiry / user.mt;

205:   info = PetscMalloc((gxm)*sizeof(double),&(user.Vt1)); CHKERRQ(info);
206:   info = PetscMalloc((gxm)*sizeof(double),&(user.c)); CHKERRQ(info);
207:   info = PetscMalloc((gxm)*sizeof(double),&(user.d)); CHKERRQ(info);

209:   /* 
210:      Calculate the values for the constant.  Vt1 begins with the ending 
211:      boundary condition.  
212:   */
213:   for (i=0; i<gxm; i++) {
214:     sval = (gxs+i)*user.ds;
215:     user.Vt1[i] = PetscMax(user.strike - sval, 0);
216:     user.c[i] = (user.delta - user.rate)*sval;
217:     user.d[i] = -0.5*user.sigma*user.sigma*pow(sval, user.alpha);
218:   }
219:   if (gxs+gxm==user.ms){
220:     user.Vt1[gxm-1] = 0;
221:   }

223:   info = VecDuplicate(x, &c); CHKERRQ(info);

225:   /* 
226:      Allocate the matrix used by TAO for the Jacobian.  Each row of
227:      the Jacobian matrix will have at most three elements.
228:   */
229:   info = DAGetMatrix(user.da,MATAIJ,&J);CHKERRQ(info);
230:          
231:   /* The TAO code begins here */

233:   /* Create TAO solver and set desired solution method  */
234:   info = TaoCreate(PETSC_COMM_WORLD, "tao_ssils", &tao); CHKERRQ(info);
235:   info = TaoApplicationCreate(PETSC_COMM_WORLD, &my_app); CHKERRQ(info);

237:   /* Set routines for constraints function and Jacobian evaluation */
238:   info = TaoAppSetConstraintRoutine(my_app, FormConstraints, (void *)&user); 
239:   CHKERRQ(info);
240:   info = TaoAppSetJacobianRoutine(my_app, FormJacobian, (void *)&user); CHKERRQ(info); 
241:   info = TaoAppSetJacobianMat(my_app, J, J); CHKERRQ(info);
242:     
243:   /* Set the variable bounds */
244:   info = TaoAppSetVariableBoundsRoutine(my_app,ComputeVariableBounds,(void*)&user); 
245:   CHKERRQ(info);

247:   /* Set initial solution guess */
248:   info = VecGetArray(x,&x_array); CHKERRQ(info);
249:   for (i=0; i< xm; i++) 
250:     x_array[i] = user.Vt1[i-gxs+xs];
251:   info = VecRestoreArray(x,&x_array); CHKERRQ(info);
252:   /* Set data structure */ 
253:   info = TaoAppSetInitialSolutionVec(my_app, x); CHKERRQ(info);

255:   /* Set routines for function and Jacobian evaluation */
256:   info = TaoSetOptions(my_app,tao); CHKERRQ(info);

258:   /* Iteratively solve the linear complementarity problems  */
259:   for (i = 1; i < user.mt; i++) {

261:     /* Solve the current version */
262:     info = TaoSolveApplication(my_app,tao);  CHKERRQ(info);

264:     /* Update Vt1 with the solution */
265:     info = DAGetLocalVector(user.da,&localX);CHKERRQ(info);
266:     info = DAGlobalToLocalBegin(user.da,x,INSERT_VALUES,localX); CHKERRQ(info);
267:     info = DAGlobalToLocalEnd(user.da,x,INSERT_VALUES,localX); CHKERRQ(info);
268:     info = VecGetArray(localX,&x_array); CHKERRQ(info);
269:     for (j = 0; j < gxm; j++) {
270:       user.Vt1[j] = x_array[j];
271:     }
272:     info = VecRestoreArray(x,&x_array); CHKERRQ(info);
273:     info = DARestoreLocalVector(user.da,&localX); CHKERRQ(info);
274:   }

276:   /* Free TAO data structures */

278:   info = TaoDestroy(tao); CHKERRQ(info);

280:   info = TaoAppDestroy(my_app);  CHKERRQ(info);

282:   /* Free PETSc data structures */
283:   info = VecDestroy(x); CHKERRQ(info);
284:   info = VecDestroy(c); CHKERRQ(info);
285:   info = MatDestroy(J); CHKERRQ(info);
286:   info = DADestroy(user.da); CHKERRQ(info);
287:   /* Free user-defined workspace */
288:   info = PetscFree(user.Vt1); CHKERRQ(info);
289:   info = PetscFree(user.c); CHKERRQ(info);
290:   info = PetscFree(user.d); CHKERRQ(info);

292:   /* Finalize TAO and PETSc */
293:   PetscFinalize();
294:   TaoFinalize();

296:   return 0;
297: }

299: /* -------------------------------------------------------------------- */
302: int ComputeVariableBounds(TAO_APPLICATION tao, Vec xl, Vec xu, void*ctx)
303: {
304:   AppCtx *user = (AppCtx *) ctx;
305:   int i,info;
306:   int    xs,xm;
307:   int ms = user->ms;
308:   PetscScalar sval=0.0,*xl_array,ub= TAO_INFINITY;

310:   /* Set the variable bounds */
311:   info = VecSet(xu, ub); CHKERRQ(info);
312:   info = DAGetCorners(user->da,&xs,PETSC_NULL,PETSC_NULL,&xm,PETSC_NULL,PETSC_NULL); CHKERRQ(info);

314:   info = VecGetArray(xl,&xl_array); CHKERRQ(info);
315:   for (i = 0; i < xm; i++){
316:     sval = (xs+i)*user->ds;
317:     xl_array[i] = PetscMax(user->strike - sval, 0);
318:   }
319:   info = VecRestoreArray(xl,&xl_array); CHKERRQ(info);

321:   if (xs==0){
322:     info = VecGetArray(xu,&xl_array); CHKERRQ(info);
323:     xl_array[0] = PetscMax(user->strike, 0);
324:     info = VecRestoreArray(xu,&xl_array); CHKERRQ(info);
325:   }
326:   if (xs+xm==ms){
327:     info = VecGetArray(xu,&xl_array); CHKERRQ(info);
328:     xl_array[xm-1] = 0;
329:     info = VecRestoreArray(xu,&xl_array); CHKERRQ(info);
330:   }

332:   return 0;
333: }
334: /* -------------------------------------------------------------------- */

338: /*  
339:     FormFunction - Evaluates gradient of f.             

341:     Input Parameters:
342: .   tao  - the TAO_SOLVER context
343: .   X    - input vector
344: .   ptr  - optional user-defined context, as set by TaoAppSetConstraintRoutine()
345:     
346:     Output Parameters:
347: .   F - vector containing the newly evaluated gradient
348: */
349: int FormConstraints(TAO_APPLICATION tao, Vec X, Vec F, void *ptr) 
350: {
351:   AppCtx *user = (AppCtx *) ptr;
352:   PetscScalar *x, *f;
353:   double *Vt1 = user->Vt1, *c = user->c, *d = user->d;
354:   double rate = user->rate;
355:   double dt = user->dt, ds = user->ds;
356:   int ms = user->ms;
357:   int i, info;
358:   int    xs,xm,gxs,gxm;
359:   Vec    localX,localF;
360:   double zero=0.0;

362:   info = DAGetLocalVector(user->da,&localX);CHKERRQ(info);
363:   info = DAGetLocalVector(user->da,&localF);CHKERRQ(info);
364:   info = DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX); CHKERRQ(info);
365:   info = DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX); CHKERRQ(info);
366:   info = DAGetCorners(user->da,&xs,PETSC_NULL,PETSC_NULL,&xm,PETSC_NULL,PETSC_NULL); CHKERRQ(info);
367:   info = DAGetGhostCorners(user->da,&gxs,PETSC_NULL,PETSC_NULL,&gxm,PETSC_NULL,PETSC_NULL); CHKERRQ(info);
368:   info = VecSet(F, zero);CHKERRQ(info);
369:   /* 
370:      The problem size is smaller than the discretization because of the
371:      two fixed elements (V(0,T) = E and V(Send,T) = 0.
372:   */

374:   /* Get pointers to the vector data */
375:   info = VecGetArray(localX, &x); CHKERRQ(info);
376:   info = VecGetArray(localF, &f); CHKERRQ(info);
377:   
378:   /* Left Boundary */
379:   if (gxs==0){ 
380:     f[0] = x[0]-user->strike;
381:   } else {
382:     f[0] = 0;
383:   }

385:   /* All points in the interior */
386:   /*  for (i=gxs+1;i<gxm-1;i++){ */
387:   for (i=1;i<gxm-1;i++){
388:     f[i] = (1.0/dt + rate)*x[i] - Vt1[i]/dt +
389:       (c[i]/(4*ds))*(x[i+1] - x[i-1] + Vt1[i+1] - Vt1[i-1]) +
390:       (d[i]/(2*ds*ds))*(x[i+1] -2*x[i] + x[i-1] + 
391:                         Vt1[i+1] - 2*Vt1[i] + Vt1[i-1]);
392:   }

394:   /* Right boundary */
395:   if (gxs+gxm==ms){ 
396:     f[xm-1]=x[gxm-1];
397:   } else {
398:     f[xm-1]=0;
399:   }

401:   /* Restore vectors */
402:   info = VecRestoreArray(localX, &x); CHKERRQ(info);
403:   info = VecRestoreArray(localF, &f); CHKERRQ(info);
404:   info = DALocalToGlobalBegin(user->da,localF,F); CHKERRQ(info);
405:   info = DALocalToGlobalEnd(user->da,localF,F); CHKERRQ(info);
406:   info = DARestoreLocalVector(user->da,&localX); CHKERRQ(info);
407:   info = DARestoreLocalVector(user->da,&localF); CHKERRQ(info);
408:   info = PetscLogFlops(24*(gxm-2)); CHKERRQ(info);
409:   /*
410:   info=VecView(F,PETSC_VIEWER_STDOUT_WORLD);
411:   */
412:   return 0;
413: }

415: /* ------------------------------------------------------------------- */
418: /*
419:    FormJacobian - Evaluates Jacobian matrix.

421:    Input Parameters:
422: .  tao  - the TAO_SOLVER context
423: .  X    - input vector
424: .  ptr  - optional user-defined context, as set by TaoSetJacobian()

426:    Output Parameters:
427: .  J    - Jacobian matrix
428: */
429: int FormJacobian(TAO_APPLICATION tao, Vec X, Mat *tJ, Mat *tJPre, MatStructure *flag, void *ptr)
430: { 
431:   AppCtx *user = (AppCtx *) ptr;
432:   Mat J = *tJ;
433:   double *c = user->c, *d = user->d;
434:   double rate = user->rate;
435:   double dt = user->dt, ds = user->ds;
436:   int ms = user->ms;
437:   PetscScalar val[3];
438:   int col[3];
439:   int i, info;
440:   int gxs,gxm;
441:   PetscTruth assembled;

443:   /* Set various matrix options */
444:   *flag=SAME_NONZERO_PATTERN;
445:   info = MatSetOption(J,MAT_IGNORE_OFF_PROC_ENTRIES); CHKERRQ(info);
446:   info = MatSetOption(J,MAT_COLUMNS_SORTED); CHKERRQ(info);
447:   info = MatSetOption(J,MAT_ROWS_SORTED); CHKERRQ(info);
448:   info = MatAssembled(J,&assembled); CHKERRQ(info);
449:   if (assembled){info = MatZeroEntries(J);  CHKERRQ(info);}


452:   info = DAGetGhostCorners(user->da,&gxs,PETSC_NULL,PETSC_NULL,&gxm,PETSC_NULL,PETSC_NULL); CHKERRQ(info);

454:   if (gxs==0){
455:     i = 0;
456:     col[0] = 0;
457:     val[0]=1.0;
458:     info = MatSetValues(J,1,&i,1,col,val,INSERT_VALUES); CHKERRQ(info);
459:   }
460:   for (i=1; i < gxm-1; i++) {
461:     col[0] = gxs + i - 1;
462:     col[1] = gxs + i;
463:     col[2] = gxs + i + 1;
464:     val[0] = -c[i]/(4*ds) + d[i]/(2*ds*ds);
465:     val[1] = 1.0/dt + rate - d[i]/(ds*ds);
466:     val[2] =  c[i]/(4*ds) + d[i]/(2*ds*ds);
467:     info = MatSetValues(J,1,&col[1],3,col,val,INSERT_VALUES); CHKERRQ(info);
468:   }
469:   if (gxs+gxm==ms){
470:     i = ms-1;
471:     col[0] = i;
472:     val[0]=1.0;
473:     info = MatSetValues(J,1,&i,1,col,val,INSERT_VALUES); CHKERRQ(info);
474:   }

476:   /* Assemble the Jacobian matrix */
477:   info = MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY); CHKERRQ(info);
478:   info = MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY); CHKERRQ(info);
479:   info = PetscLogFlops(18*(gxm)+5); CHKERRQ(info);
480:   return 0;
481: }