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double volatility : volatility of short rate double inityield : initial yield guess for Newton-Raphson method [out]: void **********************************************************************************/ void HullWhiteLog::buildHWLog(vector<double> zero, double a, int N, double T, double volatility, double inityield) { int i, j, k; int jmax = 0; // max upward branching level int jmin = 0; // min downward branching level double P[20] = {00}; // sum of Arrow-Debreu securities at time // step i double pu1[20][20] = {00}; // up probability at node i,j double pm1[20][20] = {00}; // middle probability at node i,j double pd1[20][20] = {00}; // down probability at node i,j double sum = 00; // sum of discounted Arrow-Debreu securities // at branch jmax double sum1= 00; // sum of discounted Arrow-Debreu securities double sum2 = 00; // sum of rst derivatives of Arrow-Debreu // securities double alpha[20] = {0.

2

Theorem 9.5 (Bounds). Let S be a subset of R2 satisfying: (A 0) S is compact, S and its complement are connected, and the parameterization : [0, L] S of the boundary S by its arc length is of class C 1 except perhaps at a nite number of points where is only continuous. Let X be a real-valued Gaussian process de ned on some neighborhood of S, satisfying assumptions (A1) to (A3). Then, for every real u, using the notation of Theorem 9.9 gives us P{M > u} P{Y (0) > u} +

0}; // the value of r(i*dt) r*(i*dt) double alpha1[20][20] = {00}; // calibrated shift parameter at node i,j double alpha2 = 00; // computed Newton-Raphson shift param value double alpha3 = 00; // updates alpha2 double r[20][20] = {00}; // short rate at node i, j double error = 00; // computed error in Newton-Raphson method double B[20] = {00}; // zero bond price at time step i double Q[20][20] = {00}; // the present value of a security that pays // off $1 if node (i,j) is reached, else 0 const double tolerance = 000001; // error tolerance level double dt = T/N; // time step size (typically, 05 or 1 year) double dr = volatility*sqrt(3*dt); // state size step jmax = (int) ceil(0.

(13.27) for vT j-cap and quotes them annualized and in percentages. It is often assumed that the same average cap volatility vT j-cap is used for all caplets concurring with the Tj -maturity cap. However, if the same caplets concur to a different cap, such as a Tj+1-maturity cap, their average volatility is changed. This seems to create inconsistencies since the same caplets are linked to different volatilities when concurring to different caps. To correct this inconsistency and to correctly quote cap prices, it is necessary that the following relationship be satis ed:

1835/(a*dt)); jmin = -jmax; // rst stage of HW procedure for (i = 0; i < N; i++) { for (j = i; j >= -i; j--) { // normal branching a if ((j != jmax) && (j != jmin)) { pu[j] = 0167 + 05*(a*a*j*j*dt*dt a*j*dt); pm[j] = 0666 a*a*j*j*dt*dt; pd[j] = 0167 + 05*(a*a*j*j*dt*dt + a*j*dt); } else if (j == jmin) { // up branching if a == 0 pu[j] = 0167 + 05*(a*a*j*j*dt*dt + a*j*dt);.

Joule/Kelvin or, in dB,

10 2

(4.76)

(12.89)

(10)

When you create a chart, the number formatting on the chart is linked to the worksheet cells that contain the numbers. For example, the values displayed on the chart s value axis or data labels use the same number format as the values used to create the chart. If you like, you can apply number formats (including custom number formats) to values that appear in charts. Generally, you can double-click any part of a chart that displays a number. This brings up the appropriate Format dialog box. Click the Number tab and specify the desired number format. You can choose from a built-in format, or use a custom number format. To reestablish links between a chart s number formats and the worksheet number formatting, select the Linked to source check box on the Number tab of the Format dialog box.

In a Web services implementation model, SOAP can be implemented as a client, as a server application, or both, as follows: