\form#0:\[ dx_i = \mu(t,x_i) dt + \sigma(t,x_i) dW_t \]
\form#1:$ r = f(t,x) $
\form#2:\[ dr_t = (\theta(t) - \alpha(t) r_t)dt + \sigma(t) dW_t \]
\form#3:$ \alpha $
\form#4:$ \sigma $
\form#5:\[ d\ln{r_t} = (\theta(t) - \alpha \ln{r_t})dt + \sigma dW_t \]
\form#6:\[ dr_t = (\theta(t) - k r_t)dt + \sigma \sqrt{r_t} dW_t \]
\form#7:\[ V = \sqrt{\sum_{i=1}^{n} \frac{(T_i - M_i)^2}{M_i}}, \]
\form#8:$ T_i $
\form#9:$ M_i $
\form#10:$ x = x(t, r) $
\form#11:\[ dx_t = \mu(t,x)dt + \sigma(t,x)dW_t \]
\form#12:\[ \frac{\partial P}{\partial t} + \mu \frac{\partial P}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 P}{\partial x^2} = r(t,x)P \]
\form#13:$ y $
\form#14:\[ dy_t = \mu(t, y_t) dt + \sigma(t, y_t) dW_t. \]
\form#15:$ p_u, p_m $
\form#16:$ p_d $
\form#17:\[ p_{u}y_{i+1,k+1}+p_{m}y_{i+1,k}+p_{d}y_{i+1,k-1}=E_{i,j} \]
\form#18:\[ p_u y_{i+1,k+1}^2 + p_m y_{i+1,k}^2 + p_d y_{i+1,k-1}^2 = V^2_{i,j}+E_{i,j}^2, \]
\form#19:$ E_{i,j}=\mathbf{E}\left( y(t_{i+1})|y(t_{i})=y_{i,j}\right) $
\form#20:$ V_{i,j}^{2}=\mathbf{Var}\{y(t_{i+1})|y(t_{i})=y_{i,j}\} $
\form#21:$ V_{i,j}=V_{i} $
\form#22:$ y_{i+1} $
\form#23:$ V_{i}\sqrt{3} $
\form#24:\[ p_{u} = \frac{1}{6}+\frac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} + \frac{E_{i,j}-y_{i+1,k}}{2\sqrt{3}V_{i}}, \]
\form#25:\[ p_{m} = \frac{2}{3}-\frac{(E_{i,j}-y_{i+1,k})^{2}}{3V_{i}^{2}}, \]
\form#26:\[ p_{d} = \frac{1}{6}+\frac{(E_{i,j}-y_{i+1,k})^{2}}{6V_{i}^{2}} - \frac{E_{i,j}-y_{i+1,k}}{2\sqrt{3}V_{i}}. \]
\form#27:$ \mathbf{P}_{0} $
\form#28:\[ \mathbf{P}_{\mathbf{i}}=\mathbf{P}_{0}+\lambda \mathbf{e}_{\mathbf{i}}, \]
\form#29:$ \lambda $
\form#30:\[ \mathbf{d_i} = -\nabla f(\mathbf{x_i})+\frac{\left\Vert \nabla f(\mathbf{x_i})\right\Vert ^{2}}{\left\Vert \nabla f(\mathbf{x_{i-1}})\right\Vert ^{2}}\mathbf{d_{i-1}}, \]
\form#31:\[ \mathbf{d}_{0} = -\nabla f(\mathbf{x}_{0}). \]
\form#32:$ N $
\form#33:$ [x_0,x_1,\dots] $
\form#34:$ p $
\form#35:$ x_i $
\form#36:$ p(x_i) $
\form#37:$ f $
\form#38:$ [f(x_0),f(x_1),\dots] $
\form#39:$ m $
\form#40:$ [x_0,x_m,x_{2m},\dots] $
\form#41:$ P $
\form#42:\[ P = N \times T \times \min(a L + b, C). \]
\form#43:\[ P = N \times T \times \max(a L + b, F). \]
\form#44:\[ P = N \times T \times \min(\max(a L + b, F), C). \]
\form#45:$ T $
\form#46:$ L $
\form#47:$ a $
\form#48:$ b $
\form#49:$ C $
\form#50:$ F $
\form#51:\[ R = \min(a L + b, C) = (a L + b) + \min(C - b - \xi |a| L, 0) \]
\form#52:$ \xi = sgn(a) $
\form#53:\[ R = (a L + b) + |a| \min(\frac{C - b}{|a|} - \xi L, 0) \]
\form#54:\[ D_{\mathrm{simple}} = \frac{\sum t_i c_i B(t_i)}{\sum c_i B(t_i)} \]
\form#55:$ c_i $
\form#56:$ i $
\form#57:$ t_i $
\form#58:$ B(t_i) $
\form#59:\[ D_{\mathrm{modified}} = -\frac{1}{P} \frac{\partial P}{\partial y} \]
\form#60:\[ D_{\mathrm{Macaulay}} = \left( 1 + \frac{y}{N} \right) D_{\mathrm{modified}} \]
\form#61:\[ C = \frac{1}{P} \frac{\partial^2 P}{\partial y^2} \]
\form#62:$ 1+rt $
\form#63:$ (1+r)^t $
\form#64:$ e^{rt} $
\form#65:$ r $
\form#66:$ D_1$
\form#67:$ D_2 > D_1 $
\form#68:$ V_1 $
\form#69:$ V_2 $
\form#70:\[ V = V_1 - V_2. \]
\form#71:$ LGD = (1-r)\,L $
\form#72:$ t_i, i=1, ..., N$
\form#73:$ E_i = E_{t_i}\,\left[Pay(LGD)\right] $
\form#74:$ Pay(LGD) $
\form#75:$ t_i$
\form#76:\[ Pay(L) = min (D_1, LGD) - min (D_2, LGD) = \left\{ \begin{array}{lcl} \displaystyle 0 &;& LGD < D_1 \\ \displaystyle LGD - D_1 &;& D_1 \leq LGD \leq D_2 \\ \displaystyle D_2 - D_1 &;& LGD > D_2 \end{array} \right. \]
\form#77:\[ V_1 \:=\: \sum_{i=1}^N (E_i - E_{i-1}) \cdot d_i \]
\form#78:$ d_i$
\form#79:$ D_2 - D_1. $
\form#80:$ E_i $
\form#81:$ t_i, $
\form#82:\[ V_2 = m \, \cdot \sum_{i=1}^N \,(D_2 - D_1 - E_i) \cdot \Delta_{i-1,i}\,d_i \]
\form#83:$ \Delta_{i-1, i}$
\form#84:$ t_{i-1}$
\form#85:$ t_i.$
\form#86:$ E_{i+1} < E_i. $
\form#87:$Q_i(t)$
\form#88:\[ Y_i = a_i\,M+\sqrt{1-a_i^2}\:Z_i \]
\form#89:$M$
\form#90:$Z_i$
\form#91:$-1\leq a_i \leq 1$
\form#92:$Y_i$
\form#93:$Y_j$
\form#94:$a_i a_j$
\form#95:$F_Y(y)$
\form#96:$y$
\form#97:$t$
\form#98:$F_Y(y)=Q_i(t)$
\form#99:$y=F_Y^{-1}(Q_i(t))$
\form#100:$F_Z(z)$
\form#101:\[ Prob \,(Y_i < y|M) = F_Z \left( \frac{y-a_i\,M}{\sqrt{1-a_i^2}}\right) \qquad \mbox{or} \qquad Prob \,(t_i < t|M) = F_Z \left( \frac{F_Y^{-1}(Q_i(t))-a_i\,M} {\sqrt{1-a_i^2}} \right) \]
\form#102:$ M, Z_i $
\form#103:$ Y $
\form#104:\[ F_Y(y) = Prob\,(Y<y) = \int_{-\infty}^\infty\,\int_{-\infty}^{\infty}\: D_Z(z)\,D_M(m) \quad \Theta \left(y - a\,m - \sqrt{1-a^2}\,z\right)\,dm\,dz, \qquad \Theta (x) = \left\{ \begin{array}{ll} 1 & x \geq 0 \\ 0 & x < 0 \end{array}\right. \]
\form#105:$ D_Z(z) $
\form#106:$ D_M(m) $
\form#107:$ Z$
\form#108:$ M, $
\form#109:\[ F(y) = Prob \,(Y < y) = \int_{-\infty}^\infty D_M(m)\,dm\: \int_{-\infty}^{g(y,a,m)} D_Z(z)\,dz, \qquad g(y,a,m) = \frac{y - a\cdot m}{\sqrt{1-a^2}}, \qquad a < 1 \]
\form#110:\[ F(y) = Prob \,(Y < y) = \int_{-\infty}^\infty D_Z(z)\,dz\: \int_{-\infty}^{h(y,a,z)} D_M(m)\,dm, \qquad h(y,a,z) = \frac{y - \sqrt{1 - a^2}\cdot z}{a}, \qquad a > 0. \]
\form#111:$ F_Y(y) $
\form#112:\[ \hat p(m) = F_Z \left( \frac{F_Y^{-1}(p)-a\,m}{\sqrt{1-a^2}}\right) \]
\form#113:\[ \hat p_i(m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}} \right) \]
\form#114:$ \rho(m) $
\form#115:\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, F_Z \left( \frac{F_Y^{-1}(p)-a\,m}{\sqrt{1-a^2}}\right) \]
\form#116:\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i (m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}} \right) \]
\form#117:\[ \int_{-\infty}^\infty\,dm\,\rho(m)\, f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i = F_Z \left( \frac{F_Y^{-1}(p_i)-a\,m}{\sqrt{1-a^2}}\right) \]
\form#118:$ M, Z,$
\form#119:$ \phi(x) = \exp(-x^2/2) / \sqrt{2\pi}. $
\form#120:$ Z_i $
\form#121:$ D_Z $
\form#122:$ M $
\form#123:$ D_M $
\form#124:$ N_z $
\form#125:$ N_m $
\form#126:$ \nu $
\form#127:$ \nu / (\nu - 2) $
\form#128:$ Z $
\form#129:$ \sqrt{(N_z - 2) / N_z} $
\form#130:$ \sqrt{(N_m - 2) / N_m}, $
\form#131:$ \sqrt{(N_z - 2) / N_z}.$
\form#132:$ \sqrt{(N_m - 2) / N_m}. $
\form#133:\[ \begin{array}{rcl} P_t &=& \exp(p_t + X_t + Y_t) \\ dX_t &=& -\alpha X_tdt + \sigma_x dW_t^x \\ dY_t &=& -\beta Y_{t-}dt + J_tdN_t \\ \omega(J) &=& \eta e^{-\eta J} \\ G_t &=& \exp(g_t + U_t) \\ dU_t &=& -\kappa U_tdt + \sigma_udW_t^u \\ \rho &=& \mathrm{corr} (dW_t^x, dW_t^u) \end{array} \]
\form#134:$ x $
\form#135:\[ dx = a (b(t) - x_t) dt + \sigma dW_t. \]
\form#136:\[ \begin{array}{rcl} S &=& exp(X_t + Y_t) \\ dX_t &=& \alpha(\mu(t)-X_t)dt + \sigma dW_t \\ dY_t &=& -\beta Y_{t-}dt + J_tdN_t \\ \omega(J)&=& \eta_u e^{-\eta_u J} \end{array} \]
\form#137:\[ \begin{array}{rcl} dE(t) &=& \left[ \frac{\partial}{\partial t} \mu(t) +\theta_1 \left(\mu(t)-E(t^-)\right)\right]dt +\sigma dW(t) + h(E(t^-))dJ(t) \\ \mu(t)&=& \alpha + \beta t +\gamma \cos(\epsilon+2\pi t) +\delta \cos(\zeta + 4\pi t) \end{array} \]
\form#138:\[ d f(r_t) = (\theta(t) - \alpha f(r_t))dt + \sigma dW_t, \]
\form#139:$ alpha $
\form#140:$ sigma $
\form#141:\[ dx = a (level - x_t) dt + \sigma dW_t \]
\form#142:\[ db = \theta dt + \sigma dW_t \]
\form#143:\[ X(t) = B(T) \]
\form#144:$ K $
\form#145:\[ P_n(0,T) \max(y (N [(1+K)^{T}-1] - N \left[ \frac{I(T)}{I(0)} -1 \right]), 0) \]
\form#146:$ P_n(0,t) $
\form#147:$ t $
\form#148:$ I(t) $
\form#149:$P$
\form#150:$ P_{CleanFwd}(t) = P_{DirtyFwd}(t) - AI(t=deliveryDate) $
\form#151:$ AI $
\form#152:$ P_{DirtyFwd}(t) = \frac{P_{DirtySpot}(t) - SpotIncome(t)} {discountCurve->discount(t=deliveryDate)} $
\form#153:$ SpotIncome(t) = \sum_i \left( CF_i \times incomeDiscountCurve->discount(t_i) \right) $
\form#154:$ CF_i $
\form#155:$ P_{DirtyFwd}(t) $
\form#156:$ t>0 $
\form#157:$ t=0 $
\form#158:\[ \sum_{i=1}^{M} P_n(0,t_i) N K = \sum_{i=1}^{M} P_n(0,t_i) N \left[ \frac{I(t_i)}{I(t_i-1)} - 1 \right] \]
\form#159:$ t_M $
\form#160:\[ P_n(0,T) N [(1+K)^{T}-1] = P_n(0,T) N \left[ \frac{I(T)}{I(0)} -1 \right] \]
\form#161:\[ \rho_{i,j}=e^{(-\beta \|i-j\|)} \]
\form#162:\[ \sigma_i(t)=k_i*((a*(T_{i}-t)+d)*e^{-b(T_{i}-t)}+c) \]
\form#163:\[ \rho_{i,j}=rho + (1-rho)*e^{(-\beta \|i-j\|)} \]
\form#164:\[ \sigma_i(t)=(a*(T_{i}-t)+d)*e^{-b(T_{i}-t)}+c \]
\form#165:$ a_{0} = value, a_{i}=a_{i-1}+increment $
\form#166:$ \forall i : v_i = v_i + x $
\form#167:$ \forall i : v_i = v_i \times w_i $
\form#168:$ B_{i,n}(x) $
\form#169:\[ B_{i,n}(x) \equiv \left( \begin{array}{c} n \\ i \end{array} \right) x^i (1-x)^{n-i} \]
\form#170:$ (p+1) $
\form#171:$ N_{i,p}(x), i = 0,1,2 \ldots n $
\form#172:$ n+1 $
\form#173:$ p+n+2 $
\form#174:$ (x_0, x_1 \ldots x_{n+p+1}) $
\form#175:$ p=1 $
\form#176:$ p=2 $
\form#177:$ p=3 $
\form#178:\[ \begin{array}{rcl} N_{i,0}(x) &=& 1 \textrm{\ if\ } x_{i} \leq x < x_{i+1} \\ &=& 0 \textrm{\ otherwise} \\ N_{i,p}(x) &=& N_{i,p-1}(x) \frac{(x - x_{i})}{ (x_{i+p-1} - x_{i})} + N_{i+1,p-1}(x) \frac{(x_{i+p} - x)}{(x_{i+p} - x_{i+1})} \end{array} \]
\form#179:\[ \mathrm{close}(x,y,n) \equiv |x-y| \leq \varepsilon |x| \wedge |x-y| \leq \varepsilon |y| \]
\form#180:$ \varepsilon $
\form#181:$ n $
\form#182:\[ \mathrm{close}(x,y,n) \equiv |x-y| \leq \varepsilon |x| \vee |x-y| \leq \varepsilon |y| \]
\form#183:\[ \Gamma(z) = \int_0^{\infty}t^{z-1}e^{-t}dt \]
\form#184:$ k $
\form#185:\[ f(x) = \frac {\Gamma\left(\frac{n+1}{2}\right)} {\sqrt{n\pi} \, \Gamma\left(\frac{n}{2}\right)}\: \frac {1} {\left(1+\frac{x^2}{n}\right)^{(n+1)/2}} \]
\form#186:\[ F(x) = \int_{-\infty}^x\,f(y)\,dy = \frac{1}{2}\, +\,\frac{1}{2}\,sgn(x)\, \left[ I\left(1,\frac{n}{2},\frac{1}{2}\right) - I\left(\frac{n}{n+y^2}, \frac{n}{2},\frac{1}{2}\right)\right] \]
\form#187:$ I(z; a, b) $
\form#188:\[ P_{k+1}(x)=(x-\alpha_k) P_k(x) - \beta_k P_{k-1}(x) \]
\form#189:\[ \mu_0 = \int{w(x)dx} \]
\form#190:\[ \int_{0}^{\inf} f(x) \mathrm{d}x \]
\form#191:\[ w(x;s)=x^s \exp{-x} \]
\form#192:\[ s > -1 \]
\form#193:\[ \int_{-\inf}^{\inf} f(x) \mathrm{d}x \]
\form#194:\[ w(x;\mu)=|x|^{2\mu} \exp{-x*x} \]
\form#195:\[ \mu > -0.5 \]
\form#196:\[ \int_{-1}^{1} f(x) \mathrm{d}x \]
\form#197:\[ w(x;\alpha,\beta)=(1-x)^\alpha (1+x)^\beta \]
\form#198:\[ w(x)=1/cosh(x) \]
\form#199:\[ w(x)=1 \]
\form#200:\[ w(x)=(1-x^2)^{-1/2} \]
\form#201:\[ w(x)=(1-x^2)^{1/2} \]
\form#202:\[ w(x)=(1-x^2)^{\lambda-1/2} \]
\form#203:$ \epsilon $
\form#204:\[ \int_{a}^{b} f \mathrm{d}x = \frac{1}{2} f(x_{0}) + f(x_{1}) + f(x_{2}) + \dots + f(x_{N-1}) + \frac{1}{2} f(x_{N}) \]
\form#205:$ x_0 = a $
\form#206:$ x_N = b $
\form#207:$ x_i = a+i \Delta x $
\form#208:$ \Delta x = (b-a)/N $
\form#209:$ N \times M $
\form#210:$ v_{ij} $
\form#211:$ j $
\form#212:$ S $
\form#213:$ S S^T = M. $
\form#214:\[ S = U \cdot D \cdot U^T \, ,\]
\form#215:$ \cdot $
\form#216:$ ^T $
\form#217:$ angles $
\form#218:$ t0 $
\form#219:$ epsilon $
\form#220:$ \beta $
\form#221:$ [0,1] $
\form#222:$ d $
\form#223:\[ f(x + t \cdot d) - f(x) \leq -\alpha t f'(x+t \cdot d) \]
\form#224:\[ f(x+\frac{t}{\beta} \cdot d) - f(x) > -\frac{\alpha}{\beta} t f'(x+t \cdot d) \]
\form#225:$ d_i = - f'(x_i) + c_i*d_{i-1} $
\form#226:$ c_i = ||f'(x_i)||^2/||f'(x_{i-1})||^2 $
\form#227:$ d_1 = - f'(x_1) $
\form#228:\[ min \{ r(x) : x in R^n \} \]
\form#229:$ r(x) = |f(x)|^2 $
\form#230:$ f(x) $
\form#231:$ R^n $
\form#232:$ R^m $
\form#233:\[ f = (f_1, ..., f_m) \]
\form#234:$ f_i(x) = b_i - \phi(x,t_i) $
\form#235:$ phi $
\form#236:\[ grad r(x) = f'(x)^t.f(x) \]
\form#237:$ = - f'(x) $
\form#238:$ (0, 1)^N. $
\form#239:$ |f(x)| < \epsilon $
\form#240:$ |x-\xi| < \epsilon $
\form#241:$ \xi $
\form#242:$ x_\mathrm{min} $
\form#243:$ x_\mathrm{max} $
\form#244:$ f(x_\mathrm{min}) \leq 0 \leq f(x_\mathrm{max}) $
\form#245:$ f(x_\mathrm{max}) \leq 0 \leq f(x_\mathrm{min}) $
\form#246:$ 2^{n-1} $
\form#247:$ L^2 $
\form#248:\[ \frac{N}{N-1} \times \frac{ \sum_{i=1}^{N} \theta \times x_i^{2}}{ \sum_{i=1}^{N} w_i} \]
\form#249:$ \theta $
\form#250:\[ \frac{\sum w_i (min(0, x_i-target))^2 }{\sum w_i}. \]
\form#251:\[ y = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} \exp (-u^2/2) du \]
\form#252:\[ \mathrm{E}\left[ x \;|\; x < \mathrm{VaR}(p) \right], \]
\form#253:\[ \langle x \rangle = \frac{\sum w_i x_i}{\sum w_i}. \]
\form#254:\[ \sigma^2 = \frac{N}{N-1} \left\langle \left( x-\langle x \rangle \right)^2 \right\rangle. \]
\form#255:$ \epsilon = \sigma/\sqrt{N}. $
\form#256:\[ \frac{N^2}{(N-1)(N-2)} \frac{\left\langle \left( x-\langle x \rangle \right)^3 \right\rangle}{\sigma^3}. \]
\form#257:\[ \frac{N^2(N+1)}{(N-1)(N-2)(N-3)} \frac{\left\langle \left(x-\langle x \rangle \right)^4 \right\rangle}{\sigma^4} - \frac{3(N-1)^2}{(N-2)(N-3)}. \]
\form#258:$ \mathcal{R} $
\form#259:\[ \mathrm{E}\left[f \;|\; \mathcal{R}\right] = \frac{\sum_{x_i \in \mathcal{R}} f(x_i) w_i}{ \sum_{x_i \in \mathcal{R}} w_i}. \]
\form#260:$ \bar{x} $
\form#261:\[ y = \frac{\sum_{x_i < \bar{x}} w_i}{ \sum_i w_i} \]
\form#262:$ (0-1]. $
\form#263:\[ y = \frac{\sum_{x_i > \bar{x}} w_i}{ \sum_i w_i} \]
\form#264:\[ \frac{N}{N-1} \left\langle \left( x-\langle x \rangle \right)^2 \right\rangle. \]
\form#265:\[ \frac{N}{N-1} \mathrm{E}\left[ (x-\langle x \rangle)^2 \;|\; x < \langle x \rangle \right]. \]
\form#266:\[ \frac{N}{N-1} \mathrm{E}\left[ x^2 \;|\; x < 0\right]. \]
\form#267:\[ \frac{N}{N-1} \mathrm{E}\left[ (x-t)^2 \;|\; x < t \right]. \]
\form#268:\[ \mathrm{E}\left[ \Theta \;|\; (-\infty,\infty) \right] \]
\form#269:\[ \Theta(x) = \left\{ \begin{array}{ll} 1 & x < t \\ 0 & x \geq t \end{array} \right. \]
\form#270:\[ \mathrm{E}\left[ t-x \;|\; x<t \right] \]
\form#271:$ u $
\form#272:$ v = Lu $
\form#273:$ Lu' = u $
\form#274:$ u' $
\form#275:$ D_{-} $
\form#276:\[ \frac{\partial u_{i}}{\partial x} \approx \frac{u_{i}-u_{i-1}}{h} = D_{-} u_{i} \]
\form#277:$ D_{+} $
\form#278:\[ \frac{\partial u_{i}}{\partial x} \approx \frac{u_{i+1}-u_{i}}{h} = D_{+} u_{i} \]
\form#279:$ D_{+}D_{-} $
\form#280:\[ \frac{\partial^2 u_{i}}{\partial x^2} \approx \frac{u_{i+1}-2u_{i}+u_{i-1}}{h^2} = D_{+}D_{-} u_{i} \]
\form#281:$ D_{0} $
\form#282:\[ \frac{\partial u_{i}}{\partial x} \approx \frac{u_{i+1}-u_{i-1}}{2h} = D_{0} u_{i} \]
\form#283:\[ \begin{array}{rcl} dY_t &=& -\beta Y_{t-}dt + J_tdN_t \\ \omega(J)&=&\frac{1}{\eta_u}e^{-\frac{1}{\eta_u}J} \end{array} \]
\form#284:$t1 <= t2$
\form#285:$ \mu \Delta t $
\form#286:$ a(t) = a $
\form#287:$ a(t) = 0 $
\form#288:$ a(t) = a_i if t_{i-1} \geq t < t_i $
\form#289:$ A(t,T) $
\form#290:$ B(t,T) $
\form#291:\[ P(t, T, r_t) = A(t,T)e^{ -B(t,T) r_t}. \]
\form#292:\[ d\ln r_t = (\theta(t) - \alpha \ln r_t)dt + \sigma dW_t, \]
\form#293:\[ r_t = e^{\varphi(t) + x_t} \]
\form#294:$ \varphi(t) $
\form#295:$ x_t $
\form#296:\[ dr_t = k(\theta - r_t)dt + \sqrt{r_t}\sigma dW_t . \]
\form#297:$ y_t $
\form#298:\[ dy_t=\left[ (\frac{k\theta }{2}+\frac{\sigma ^2}{8})\frac{1}{y_t}- \frac{k}{2}y_t \right] d_t+ \frac{\sigma }{2}dW_{t} \]
\form#299:\[ dr_t = (\theta(t) - \alpha r_t)dt + \sqrt{r_t}\sigma dW_t . \]
\form#300:\[ r_t = \varphi(t) + y_t^2 \]
\form#301:\[ \varphi(t) = f(t) - \frac{2k\theta(e^{th}-1)}{2h+(k+h)(e^{th}-1)} - \frac{4 x_0 h^2 e^{th}}{(2h+(k+h)(e^{th}-1))^1}, \]
\form#302:$ f(t) $
\form#303:$ h = \sqrt{k^2 + 2\sigma^2} $
\form#304:\[ dr_t = (\theta(t) - \alpha r_t)dt + \sigma dW_t \]
\form#305:\[ r_t = \varphi(t) + x_t \]
\form#306:\[ \varphi(t) = f(t) + \frac{1}{2}[\frac{\sigma(1-e^{-at})}{a}]^2, \]
\form#307:\[ dr_t = a(b - r_t)dt + \sigma dW_t , \]
\form#308:\[ r_t = f(t, x_t, y_t) \]
\form#309:\[ x_t = \mu_x(t, x_t)dt + \sigma_x(t, x_t) dW_t^x \]
\form#310:\[ y_t = \mu_y(t,y_t)dt + \sigma_y(t, y_t) dW_t^y \]
\form#311:$ W^x $
\form#312:$ W^y $
\form#313:\[ dW^x_t dW^y_t = \rho dt \]
\form#314:$ \rho $
\form#315:\[ dr_t = \varphi(t) + x_t + y_t \]
\form#316:\[ dx_t = -a x_t dt + \sigma dW^1_t, x_0 = 0 \]
\form#317:\[ dy_t = -b y_t dt + \sigma dW^2_t, y_0 = 0 \]
\form#318:$ dW^1_t dW^2_t = \rho dt $
\form#319:\[ \varphi(t) = f(t) + \frac{1}{2}(\frac{\sigma(1-e^{-at})}{a})^2 + \frac{1}{2}(\frac{\eta(1-e^{-bt})}{b})^2 + \rho\frac{\sigma(1-e^{-at})}{a}\frac{\eta(1-e^{-bt})}{b}, \]
\form#320:$ t \geq 0 $
\form#321:\[ \begin{array}{rcl} dS(t, S) &=& (r-d) S dt +\sqrt{v} S dW_1 \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ dr(t) &=& (\theta(t) - a r) dt + \eta dW_3 \\ dW_1 dW_2 &=& \rho dt \\ dW_1 dW_3 &=& 0 \\ dW_2 dW_3 &=& 0 \\ \end{array} \]
\form#322:\[ \begin{array}{rcl} dS(t, S) &=& (r-d-\lambda m) S dt +\sqrt{v} S dW_1 + (e^J - 1) S dN \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ dW_1 dW_2 &=& \rho dt \end{array} \]
\form#323:$ \omega(J) $
\form#324:\[ \omega(J) = \frac{1}{\sqrt{2\pi \delta^2}} \exp\left[-\frac{(J-\nu)^2}{2\delta^2}\right] \]
\form#325:\[ \begin{array}{rcl} \omega(J)&=& p\frac{1}{\eta_u}e^{-\frac{1}{\eta_u}J} 1_{J>0} + q\frac{1}{\eta_d}e^{\frac{1}{\eta_d}J} 1_{J<0} \\ p + q &=& 1 \end{array} \]
\form#326:\[ \begin{array}{rcl} dS(t, S) &=& (r-d-\lambda m) S dt +\sqrt{v} S dW_1 + (e^J - 1) S dN \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ d\lambda(t) &=& \kappa_\lambda(\theta_\lambda-\lambda) dt \\ dW_1 dW_2 &=& \rho dt \end{array} \]
\form#327:\[ \begin{array}{rcl} dS(t, S) &=& (r-d-\lambda m) S dt +\sqrt{v} S dW_1 + (e^J - 1) S dN \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ dW_1 dW_2 &=& \rho dt \\ \omega(J) &=& \frac{1}{\sqrt{2\pi \delta^2}} \exp\left[-\frac{(J-\nu)^2}{2\delta^2}\right] \end{array} \]
\form#328:\[ dS(t, S) = (r(t) - q(t) - \frac{\sigma(t, S)^2}{2}) dt + \sigma dW_t. \]
\form#329:\[ dS(t, S) = (r(t) - \frac{\sigma(t, S)^2}{2}) dt + \sigma dW_t. \]
\form#330:\[ dS(t, S) = \frac{\sigma(t, S)^2}{2} dt + \sigma dW_t. \]
\form#331:\[ dS(t, S) = (r(t) - r_f(t) - \frac{\sigma(t, S)^2}{2}) dt + \sigma dW_t. \]
\form#332:$ \mu(t_0 + \Delta t, \mathbf{x}_0) \Delta t $
\form#333:$ \mu(t_0 + \Delta t, x_0) \Delta t $
\form#334:$ \sigma(t_0 + \Delta t, \mathbf{x}_0) \sqrt{\Delta t} $
\form#335:$ \sigma(t_0 + \Delta t, x_0) \sqrt{\Delta t} $
\form#336:$ \sigma(t_0 + \Delta t, \mathbf{x}_0)^2 \Delta t $
\form#337:$ \sigma(t_0 + \Delta t, x_0)^2 \Delta t $
\form#338:$ \mu(t_0, \mathbf{x}_0) \Delta t $
\form#339:$ \mu(t_0, x_0) \Delta t $
\form#340:$ \sigma(t_0, \mathbf{x}_0) \sqrt{\Delta t} $
\form#341:$ \sigma(t_0, x_0) \sqrt{\Delta t} $
\form#342:$ \sigma(t_0, \mathbf{x}_0)^2 \Delta t $
\form#343:$ \sigma(t_0, x_0)^2 \Delta t $
\form#344:\[ dS(t, S)= \mu S dt + \sigma S dW_t. \]
\form#345:\[ \begin{array}{rcl} dS(t, S) &=& \mu S dt + \sqrt{v} S dW_1 \\ dv(t, S) &=& (\omega + (\beta + \alpha * q_{2} + \gamma * q_{3} - 1) v) dt + (\alpha \sigma_{12} + \gamma \sigma_{13}) v dW_1 + \sqrt{\alpha^{2} (\sigma^{2}_{2} - \sigma^{2}_{12}) + \gamma^{2} (\sigma^{2}_{3} - \sigma^{2}_{13}) + 2 \alpha \gamma (\sigma_{23} - \sigma_{12} \sigma_{13})} v dW_2 \ \ N = normalCDF(\lambda) \\ n &=& \exp{-\lambda^{2}/2} / \sqrt{2 \pi} \\ q_{2} &=& 1 + \lambda^{2} \\ q_{3} &=& \lambda n + N + \lambda^2 N \\ \sigma^{2}_{2} = 2 + 4 \lambda^{4} \\ \sigma^{2}_{3} = \lambda^{3} n + 5 \lambda n + 3N + \lambda^{4} N + 6 \lambda^{2} N -\\lambda^{2} n^{2} - N^{2} - \lambda^{4} N^{2} - 2 \lambda n N - 2 \lambda^{3} nN - 2 \lambda^{2} N^{2} \ \ \sigma_{12} = -2 \lambda \\ \sigma_{13} = -2 n - 2 \lambda N \\ \sigma_{23} = 2N + \sigma_{12} \sigma_{13} \\ \end{array} \]
\form#346:\[ \begin{array}{rcl} dS(t, S) &=& \mu S dt + \sqrt{v} S dW_1 \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ dW_1 dW_2 &=& \rho dt \end{array} \]
\form#347:\[ dx = a (r - x_t) dt + \sigma dW_t. \]
\form#348:\[ dx = a (b - x_t) dt + \sigma \sqrt{x_t} dW_t. \]
\form#349:\[ d\mathrm{x}_t = \mu(t, x_t)\mathrm{d}t + \sigma(t, \mathrm{x}_t) \cdot d\mathrm{W}_t. \]
\form#350:\[ dx_t = \mu(t, x_t)dt + \sigma(t, x_t)dW_t. \]
\form#351:$ \mu(t, \mathrm{x}_t) $
\form#352:$ \sigma(t, \mathrm{x}_t) $
\form#353:$ E(\mathrm{x}_{t_0 + \Delta t} | \mathrm{x}_{t_0} = \mathrm{x}_0) $
\form#354:$ \Delta t $
\form#355:$ S(\mathrm{x}_{t_0 + \Delta t} | \mathrm{x}_{t_0} = \mathrm{x}_0) $
\form#356:$ V(\mathrm{x}_{t_0 + \Delta t} | \mathrm{x}_{t_0} = \mathrm{x}_0) $
\form#357:\[ E(\mathrm{x}_0,t_0,\Delta t) + S(\mathrm{x}_0,t_0,\Delta t) \cdot \Delta \mathrm{w} \]
\form#358:$ E $
\form#359:$ \mathrm{x} + \Delta \mathrm{x} $
\form#360:$ \mu(t, x_t) $
\form#361:$ \sigma(t, x_t) $
\form#362:$ E(x_{t_0 + \Delta t} | x_{t_0} = x_0) $
\form#363:$ S(x_{t_0 + \Delta t} | x_{t_0} = x_0) $
\form#364:$ V(x_{t_0 + \Delta t} | x_{t_0} = x_0) $
\form#365:\[ E(x_0,t_0,\Delta t) + S(x_0,t_0,\Delta t) \cdot \Delta w \]
\form#366:$ x + \Delta x $
\form#367:$ p(t) $
\form#368:\[ S(t) = 1 - \int_0^t p(\tau) d\tau. \]
\form#369:$ h(t) $
\form#370:\[ S(t) = \exp\left( - \int_0^t h(\tau) d\tau \right). \]
\form#371:$ S(t) $
\form#372:$ p(t) = -\frac{d}{dt} S(t). $
\form#373:\[ f(T-t) = [ a + b(T-t) ] e^{-c(T-t)} + d \]
\form#374:\[ f(u) \]
\form#375:\[ f(0) \]
\form#376:\[ f(\inf) \]
\form#377:\[ f(T-t)f(S-t) \]
\form#378:\[ \int_{t1}^{t2} f(T-t)f(S-t)dt \]
\form#379:\[ \sqrt{ \frac{\int_{tMin}^{tMax} f^2(T-u)du}{tMax-tMin} } \]
\form#380:\[ \frac{\int_{tMin}^{tMax} f^2(T-u)du}{tMax-tMin} \]
\form#381:\[ f(T-t) \]
\form#382:\[ f(T-t)f(T-t) \]
\form#383:\[ f(T-u)f(S-u) \]
\form#384:\[ \int f(T-t)f(S-t)dt \]
\form#385:\[ \int_0^T \sigma_L^2(t)dt = \sigma_B^2 T \]
\form#386:$ \sigma_L(t) $
\form#387:$ \sigma_B(T) $
\form#388:\[ \sigma_L(t) = \sqrt{\frac{\mathrm{d}}{\mathrm{d}t}\sigma_B^2(t)t} \]
\form#389:$ d(t) $
\form#390:$ b_i(t) $
\form#391:\[ d(t) = \sum_{i=0} c_i b_i(t) \]
\form#392:$ Ax = b $
\form#393:$ T=0 $
\form#394:\[ z(t) = \int_0^t f(\tau) d\tau \]
\form#395:$ d(t) = \exp \left( -z(t) t \right) $
\form#396:\[ d(t) = \sum_{i=1}^9 c_i \exp^{-kappa i t} \]
\form#397:$ \kappa $
\form#398:$ d(t) = \exp^{-r t}, $
\form#399:$r$
\form#400:\[ r \equiv c_0 + (c_0 + c_1)*(1 - exp^{-\kappa*t}/(\kappa t) - c_2 exp^{ - \kappa t}. \]
\form#401:\[ r \equiv c_0 + (c_0 + c_1)(\frac {1 - exp^{-\kappa t}}{\kappa t}) - c_2exp^{ - \kappa t} + c_3{(\frac{1 - exp^{-\kappa_1 t}}{\kappa_1 t} -exp^{-\kappa_1 t})}. \]
\form#402:$ N_{i,3}(t) $
\form#403:\[ d(t) = \sum_{i=0}^{n} c_i * N_{i,3}(t) \]