Actual source code: blackscholes.c
1: /**********************************************************************
2: American Put Options Pricing using the Black-Scholes Equation
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.
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:
15: dV/dt + 0.5(sigma**2)(S**alpha)(d2V/dS2) + (r - D)S(dV/dS) - rV = 0.
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.
21: To completely specify the problem, we need to impose some boundary
22: conditions. These are as follows:
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
28: where T is the exercise time time and E the strike price (price paid
29: for the contract).
31: An explicit formula for the value of an European option can be
32: found. See the references for examples.
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 exercise
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.
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)
46: where the variables are the same as before and we have the same boundary
47: conditions.
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.
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.
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 "petsctao.h" so we can use TAO solvers.
69: Include "petscdmda.h" so that we can use distributed meshes (DMs) for managing
70: the parallel mesh.
71: */
73: #include <petscdmda.h>
74: #include <petsctao.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: TaoCreate(); TaoDestroy();
94: Routines: TaoSetJacobianRoutine(); TaoAppSetConstraintRoutine();
95: Routines: TaoSetFromOptions();
96: Routines: TaoSolveApplication();
97: Routines: TaoSetVariableBoundsRoutine(); TaoSetInitialSolutionVec();
98: Processors: 1
99: T*/
101: /*
102: User-defined application context - contains data needed by the
103: application-provided call-back routines, FormFunction(), and FormJacobian().
104: */
106: typedef struct {
107: PetscReal *Vt1; /* Value of the option at time T + dt */
108: PetscReal *c; /* Constant -- (r - D)S */
109: PetscReal *d; /* Constant -- -0.5(sigma**2)(S**alpha) */
111: PetscReal rate; /* Interest rate */
112: PetscReal sigma, alpha, delta; /* Underlying asset properties */
113: PetscReal strike, expiry; /* Option contract properties */
115: PetscReal es; /* Finite value used for maximum asset value */
116: PetscReal ds, dt; /* Discretization properties */
117: PetscInt ms, mt; /* Number of elements */
119: DM dm;
120: } AppCtx;
122: /* -------- User-defined Routines --------- */
124: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
125: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
126: PetscErrorCode ComputeVariableBounds(Tao, Vec, Vec, void*);
128: int main(int argc, char **argv)
129: {
130: Vec x; /* solution vector */
131: Vec c; /* Constraints function vector */
132: Mat J; /* jacobian matrix */
133: PetscBool flg; /* A return variable when checking for user options */
134: Tao tao; /* Tao solver context */
135: AppCtx user; /* user-defined work context */
136: PetscInt i, j;
137: PetscInt xs,xm,gxs,gxm;
138: PetscReal sval = 0;
139: PetscReal *x_array;
140: Vec localX;
142: /* Initialize PETSc, TAO */
143: PetscInitialize(&argc, &argv, (char *)0, help);
145: /*
146: Initialize the user-defined application context with reasonable
147: values for the American option to price
148: */
149: user.rate = 0.04;
150: user.sigma = 0.40;
151: user.alpha = 2.00;
152: user.delta = 0.01;
153: user.strike = 10.0;
154: user.expiry = 1.0;
155: user.mt = 10;
156: user.ms = 150;
157: user.es = 100.0;
159: /* Read in alternative values for the American option to price */
160: PetscOptionsGetReal(NULL,NULL, "-alpha", &user.alpha, &flg);
161: PetscOptionsGetReal(NULL,NULL, "-delta", &user.delta, &flg);
162: PetscOptionsGetReal(NULL,NULL, "-es", &user.es, &flg);
163: PetscOptionsGetReal(NULL,NULL, "-expiry", &user.expiry, &flg);
164: PetscOptionsGetInt(NULL,NULL, "-ms", &user.ms, &flg);
165: PetscOptionsGetInt(NULL,NULL, "-mt", &user.mt, &flg);
166: PetscOptionsGetReal(NULL,NULL, "-rate", &user.rate, &flg);
167: PetscOptionsGetReal(NULL,NULL, "-sigma", &user.sigma, &flg);
168: PetscOptionsGetReal(NULL,NULL, "-strike", &user.strike, &flg);
170: /* Check that the options set are allowable (needs to be done) */
172: user.ms++;
173: DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.ms,1,1,NULL,&user.dm);
174: DMSetFromOptions(user.dm);
175: DMSetUp(user.dm);
176: /* Create appropriate vectors and matrices */
178: DMDAGetCorners(user.dm,&xs,NULL,NULL,&xm,NULL,NULL);
179: DMDAGetGhostCorners(user.dm,&gxs,NULL,NULL,&gxm,NULL,NULL);
181: DMCreateGlobalVector(user.dm,&x);
182: /*
183: Finish filling in the user-defined context with the values for
184: dS, dt, and allocating space for the constants
185: */
186: user.ds = user.es / (user.ms-1);
187: user.dt = user.expiry / user.mt;
189: PetscMalloc1(gxm,&(user.Vt1));
190: PetscMalloc1(gxm,&(user.c));
191: PetscMalloc1(gxm,&(user.d));
193: /*
194: Calculate the values for the constant. Vt1 begins with the ending
195: boundary condition.
196: */
197: for (i=0; i<gxm; i++) {
198: sval = (gxs+i)*user.ds;
199: user.Vt1[i] = PetscMax(user.strike - sval, 0);
200: user.c[i] = (user.delta - user.rate)*sval;
201: user.d[i] = -0.5*user.sigma*user.sigma*PetscPowReal(sval, user.alpha);
202: }
203: if (gxs+gxm==user.ms) {
204: user.Vt1[gxm-1] = 0;
205: }
206: VecDuplicate(x, &c);
208: /*
209: Allocate the matrix used by TAO for the Jacobian. Each row of
210: the Jacobian matrix will have at most three elements.
211: */
212: DMCreateMatrix(user.dm,&J);
214: /* The TAO code begins here */
216: /* Create TAO solver and set desired solution method */
217: TaoCreate(PETSC_COMM_WORLD, &tao);
218: TaoSetType(tao,TAOSSILS);
220: /* Set routines for constraints function and Jacobian evaluation */
221: TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);
222: TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);
224: /* Set the variable bounds */
225: TaoSetVariableBoundsRoutine(tao,ComputeVariableBounds,(void*)&user);
227: /* Set initial solution guess */
228: VecGetArray(x,&x_array);
229: for (i=0; i< xm; i++)
230: x_array[i] = user.Vt1[i-gxs+xs];
231: VecRestoreArray(x,&x_array);
232: /* Set data structure */
233: TaoSetSolution(tao, x);
235: /* Set routines for function and Jacobian evaluation */
236: TaoSetFromOptions(tao);
238: /* Iteratively solve the linear complementarity problems */
239: for (i = 1; i < user.mt; i++) {
241: /* Solve the current version */
242: TaoSolve(tao);
244: /* Update Vt1 with the solution */
245: DMGetLocalVector(user.dm,&localX);
246: DMGlobalToLocalBegin(user.dm,x,INSERT_VALUES,localX);
247: DMGlobalToLocalEnd(user.dm,x,INSERT_VALUES,localX);
248: VecGetArray(localX,&x_array);
249: for (j = 0; j < gxm; j++) {
250: user.Vt1[j] = x_array[j];
251: }
252: VecRestoreArray(x,&x_array);
253: DMRestoreLocalVector(user.dm,&localX);
254: }
256: /* Free TAO data structures */
257: TaoDestroy(&tao);
259: /* Free PETSc data structures */
260: VecDestroy(&x);
261: VecDestroy(&c);
262: MatDestroy(&J);
263: DMDestroy(&user.dm);
264: /* Free user-defined workspace */
265: PetscFree(user.Vt1);
266: PetscFree(user.c);
267: PetscFree(user.d);
269: PetscFinalize();
270: return 0;
271: }
273: /* -------------------------------------------------------------------- */
274: PetscErrorCode ComputeVariableBounds(Tao tao, Vec xl, Vec xu, void*ctx)
275: {
276: AppCtx *user = (AppCtx *) ctx;
277: PetscInt i;
278: PetscInt xs,xm;
279: PetscInt ms = user->ms;
280: PetscReal sval=0.0,*xl_array,ub= PETSC_INFINITY;
282: /* Set the variable bounds */
283: VecSet(xu, ub);
284: DMDAGetCorners(user->dm,&xs,NULL,NULL,&xm,NULL,NULL);
286: VecGetArray(xl,&xl_array);
287: for (i = 0; i < xm; i++) {
288: sval = (xs+i)*user->ds;
289: xl_array[i] = PetscMax(user->strike - sval, 0);
290: }
291: VecRestoreArray(xl,&xl_array);
293: if (xs==0) {
294: VecGetArray(xu,&xl_array);
295: xl_array[0] = PetscMax(user->strike, 0);
296: VecRestoreArray(xu,&xl_array);
297: }
298: if (xs+xm==ms) {
299: VecGetArray(xu,&xl_array);
300: xl_array[xm-1] = 0;
301: VecRestoreArray(xu,&xl_array);
302: }
304: return 0;
305: }
306: /* -------------------------------------------------------------------- */
308: /*
309: FormFunction - Evaluates gradient of f.
311: Input Parameters:
312: . tao - the Tao context
313: . X - input vector
314: . ptr - optional user-defined context, as set by TaoAppSetConstraintRoutine()
316: Output Parameters:
317: . F - vector containing the newly evaluated gradient
318: */
319: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec F, void *ptr)
320: {
321: AppCtx *user = (AppCtx *) ptr;
322: PetscReal *x, *f;
323: PetscReal *Vt1 = user->Vt1, *c = user->c, *d = user->d;
324: PetscReal rate = user->rate;
325: PetscReal dt = user->dt, ds = user->ds;
326: PetscInt ms = user->ms;
327: PetscInt i, xs,xm,gxs,gxm;
328: Vec localX,localF;
329: PetscReal zero=0.0;
331: DMGetLocalVector(user->dm,&localX);
332: DMGetLocalVector(user->dm,&localF);
333: DMGlobalToLocalBegin(user->dm,X,INSERT_VALUES,localX);
334: DMGlobalToLocalEnd(user->dm,X,INSERT_VALUES,localX);
335: DMDAGetCorners(user->dm,&xs,NULL,NULL,&xm,NULL,NULL);
336: DMDAGetGhostCorners(user->dm,&gxs,NULL,NULL,&gxm,NULL,NULL);
337: VecSet(F, zero);
338: /*
339: The problem size is smaller than the discretization because of the
340: two fixed elements (V(0,T) = E and V(Send,T) = 0.
341: */
343: /* Get pointers to the vector data */
344: VecGetArray(localX, &x);
345: VecGetArray(localF, &f);
347: /* Left Boundary */
348: if (gxs==0) {
349: f[0] = x[0]-user->strike;
350: } else {
351: f[0] = 0;
352: }
354: /* All points in the interior */
355: /* for (i=gxs+1;i<gxm-1;i++) { */
356: for (i=1;i<gxm-1;i++) {
357: f[i] = (1.0/dt + rate)*x[i] - Vt1[i]/dt + (c[i]/(4*ds))*(x[i+1] - x[i-1] + Vt1[i+1] - Vt1[i-1]) +
358: (d[i]/(2*ds*ds))*(x[i+1] -2*x[i] + x[i-1] + Vt1[i+1] - 2*Vt1[i] + Vt1[i-1]);
359: }
361: /* Right boundary */
362: if (gxs+gxm==ms) {
363: f[xm-1]=x[gxm-1];
364: } else {
365: f[xm-1]=0;
366: }
368: /* Restore vectors */
369: VecRestoreArray(localX, &x);
370: VecRestoreArray(localF, &f);
371: DMLocalToGlobalBegin(user->dm,localF,INSERT_VALUES,F);
372: DMLocalToGlobalEnd(user->dm,localF,INSERT_VALUES,F);
373: DMRestoreLocalVector(user->dm,&localX);
374: DMRestoreLocalVector(user->dm,&localF);
375: PetscLogFlops(24.0*(gxm-2));
376: /*
377: info=VecView(F,PETSC_VIEWER_STDOUT_WORLD);
378: */
379: return 0;
380: }
382: /* ------------------------------------------------------------------- */
383: /*
384: FormJacobian - Evaluates Jacobian matrix.
386: Input Parameters:
387: . tao - the Tao context
388: . X - input vector
389: . ptr - optional user-defined context, as set by TaoSetJacobian()
391: Output Parameters:
392: . J - Jacobian matrix
393: */
394: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat J, Mat tJPre, void *ptr)
395: {
396: AppCtx *user = (AppCtx *) ptr;
397: PetscReal *c = user->c, *d = user->d;
398: PetscReal rate = user->rate;
399: PetscReal dt = user->dt, ds = user->ds;
400: PetscInt ms = user->ms;
401: PetscReal val[3];
402: PetscInt col[3];
403: PetscInt i;
404: PetscInt gxs,gxm;
405: PetscBool assembled;
407: /* Set various matrix options */
408: MatSetOption(J,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
409: MatAssembled(J,&assembled);
410: if (assembled) MatZeroEntries(J);
412: DMDAGetGhostCorners(user->dm,&gxs,NULL,NULL,&gxm,NULL,NULL);
414: if (gxs==0) {
415: i = 0;
416: col[0] = 0;
417: val[0]=1.0;
418: MatSetValues(J,1,&i,1,col,val,INSERT_VALUES);
419: }
420: for (i=1; i < gxm-1; i++) {
421: col[0] = gxs + i - 1;
422: col[1] = gxs + i;
423: col[2] = gxs + i + 1;
424: val[0] = -c[i]/(4*ds) + d[i]/(2*ds*ds);
425: val[1] = 1.0/dt + rate - d[i]/(ds*ds);
426: val[2] = c[i]/(4*ds) + d[i]/(2*ds*ds);
427: MatSetValues(J,1,&col[1],3,col,val,INSERT_VALUES);
428: }
429: if (gxs+gxm==ms) {
430: i = ms-1;
431: col[0] = i;
432: val[0]=1.0;
433: MatSetValues(J,1,&i,1,col,val,INSERT_VALUES);
434: }
436: /* Assemble the Jacobian matrix */
437: MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
438: MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
439: PetscLogFlops(18.0*(gxm)+5);
440: return 0;
441: }
443: /*TEST
445: build:
446: requires: !complex
448: test:
449: args: -tao_monitor -tao_type ssils -tao_gttol 1.e-5
450: requires: !single
452: test:
453: suffix: 2
454: args: -tao_monitor -tao_type ssfls -tao_max_it 10 -tao_gttol 1.e-5
455: requires: !single
457: test:
458: suffix: 3
459: args: -tao_monitor -tao_type asils -tao_subset_type subvec -tao_gttol 1.e-5
460: requires: !single
462: test:
463: suffix: 4
464: args: -tao_monitor -tao_type asils -tao_subset_type mask -tao_gttol 1.e-5
465: requires: !single
467: test:
468: suffix: 5
469: args: -tao_monitor -tao_type asils -tao_subset_type matrixfree -pc_type jacobi -tao_max_it 6 -tao_gttol 1.e-5
470: requires: !single
472: test:
473: suffix: 6
474: args: -tao_monitor -tao_type asfls -tao_subset_type subvec -tao_max_it 10 -tao_gttol 1.e-5
475: requires: !single
477: test:
478: suffix: 7
479: args: -tao_monitor -tao_type asfls -tao_subset_type mask -tao_max_it 10 -tao_gttol 1.e-5
480: requires: !single
482: TEST*/