\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:$ D_{0} $
\form#33:$ D_{+} $
\form#34:$ D_{-} $
\form#35:$ D_{+}D_{-} $
\form#36:$ N $
\form#37:$ [x_0,x_1,\dots] $
\form#38:$ p $
\form#39:$ x_i $
\form#40:$ p(x_i) $
\form#41:$ f $
\form#42:$ [f(x_0),f(x_1),\dots] $
\form#43:$ m $
\form#44:$ [x_0,x_m,x_{2m},\dots] $
\form#45:\[ D_{\mathrm{simple}} = \frac{\sum t_i c_i B(t_i)}{\sum c_i B(t_i)} \]
\form#46:$ c_i $
\form#47:$ i $
\form#48:$ t_i $
\form#49:$ B(t_i) $
\form#50:\[ D_{\mathrm{modified}} = -\frac{1}{P} \frac{\partial P}{\partial y} \]
\form#51:$ P $
\form#52:\[ D_{\mathrm{Macaulay}} = \left( 1 + \frac{y}{N} \right) D_{\mathrm{modified}} \]
\form#53:\[ C = \frac{1}{P} \frac{\partial^2 P}{\partial y^2} \]
\form#54:$ r $
\form#55:$ L $
\form#56:$ u $
\form#57:$ v = Lu $
\form#58:$ Lu' = u $
\form#59:$ u' $
\form#60:\[ \frac{\partial u_{i}}{\partial x} \approx \frac{u_{i}-u_{i-1}}{h} = D_{-} u_{i} \]
\form#61:\[ \frac{\partial u_{i}}{\partial x} \approx \frac{u_{i+1}-u_{i}}{h} = D_{+} u_{i} \]
\form#62:\[ \frac{\partial^2 u_{i}}{\partial x^2} \approx \frac{u_{i+1}-2u_{i}+u_{i-1}}{h^2} = D_{+}D_{-} u_{i} \]
\form#63:\[ \frac{\partial u_{i}}{\partial x} \approx \frac{u_{i+1}-u_{i-1}}{2h} = D_{0} u_{i} \]
\form#64:$P$
\form#65:$ P_{CleanFwd}(t) = P_{DirtyFwd}(t) - AI(t=deliveryDate) $
\form#66:$ AI $
\form#67:$ P_{DirtyFwd}(t) = \frac{P_{DirtySpot}(t) - SpotIncome(t)} {discountCurve->discount(t=deliveryDate)} $
\form#68:$ SpotIncome(t) = \sum_i \left( CF_i \times incomeDiscountCurve->discount(t_i) \right) $
\form#69:$ CF_i $
\form#70:$ P_{DirtyFwd}(t) $
\form#71:$ t>0 $
\form#72:$ t=0 $
\form#73:$ 1+rt $
\form#74:$ (1+r)^t $
\form#75:$ e^{rt} $
\form#76:$ \mu \Delta t $
\form#77:\[ f(T-t) = [ a + b(T-t) ] e^{-c(T-t)} + d \]
\form#78:\[ f(u) \]
\form#79:\[ f(T-u) \]
\form#80:\[ f(T-u)f(T-u) \]
\form#81:\[ f(T-u)f(S-u) \]
\form#82:\[ \sqrt{ \int_{tMin}^{tMax} f^2(T-u)du }\]
\form#83:\[ \int_{tMin}^{tMax} f^2(T-u)du \]
\form#84:\[ \int_{tMin}^{tMax} f(T-u)f(S-u)du \]
\form#85:\[ \int f(T-t)f(S-t)dt \]
\form#86:$ a_{0} = value, a_{i}=a_{i-1}+increment $
\form#87:$ \forall i : v_i = v_i + x $
\form#88:$ \forall i : v_i = v_i \times w_i $
\form#89:$ x $
\form#90:\[ \mathrm{close}(x,y,n) \equiv |x-y| \leq \varepsilon |x| \wedge |x-y| \leq \varepsilon |y| \]
\form#91:$ \varepsilon $
\form#92:$ n $
\form#93:\[ \mathrm{close}(x,y,n) \equiv |x-y| \leq \varepsilon |x| \vee |x-y| \leq \varepsilon |y| \]
\form#94:$ 2^{n-1} $
\form#95:$ L^2 $
\form#96:\[ \Gamma(z) = \int_0^{\infty}t^{z-1}e^{-t}dt \]
\form#97:\[ P_{k+1}(x)=(x-\alpha_k) P_k(x) - \beta_k P_{k-1}(x) \]
\form#98:\[ \mu_0 = \int{w(x)dx} \]
\form#99:\[ \int_{0}^{\inf} f(x) \mathrm{d}x \]
\form#100:\[ w(x;s)=x^s \exp{-x} \]
\form#101:\[ s > -1 \]
\form#102:\[ \int_{-\inf}^{\inf} f(x) \mathrm{d}x \]
\form#103:\[ w(x;\mu)=|x|^{2\mu} \exp{-x*x} \]
\form#104:\[ \mu > -0.5 \]
\form#105:\[ \int_{-1}^{1} f(x) \mathrm{d}x \]
\form#106:\[ w(x;\alpha,\beta)=(1-x)^\alpha (1+x)^\beta \]
\form#107:\[ w(x)=1/cosh(x) \]
\form#108:\[ w(x)=1 \]
\form#109:\[ w(x)=(1-x^2)^{-1/2} \]
\form#110:\[ w(x)=(1-x^2)^{1/2} \]
\form#111:\[ w(x)=(1-x^2)^{\lambda-1/2} \]
\form#112:\[ \frac{N}{N-1} \times \frac{ \sum_{i=1}^{N} \theta \times x_i^{2}}{ \sum_{i=1}^{N} w_i} \]
\form#113:$ \theta $
\form#114:\[ \frac{\sum w_i (min(0, x_i-target))^2 }{\sum w_i}. \]
\form#115:\[ y = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} \exp (-u^2/2) du \]
\form#116:\[ \mathrm{E}\left[ x \;|\; x < \mathrm{VaR}(p) \right], \]
\form#117:\[ \langle x \rangle = \frac{\sum w_i x_i}{\sum w_i}. \]
\form#118:\[ \sigma^2 = \frac{N}{N-1} \left\langle \left( x-\langle x \rangle \right)^2 \right\rangle. \]
\form#119:$ \epsilon = \sigma/\sqrt{N}. $
\form#120:\[ \frac{N^2}{(N-1)(N-2)} \frac{\left\langle \left( x-\langle x \rangle \right)^3 \right\rangle}{\sigma^3}. \]
\form#121:\[ \frac{N^2(N+1)}{(N-1)(N-2)(N-3)} \frac{\left\langle \left(x-\langle x \rangle \right)^4 \right\rangle}{\sigma^4} - \frac{3(N-1)^2}{(N-2)(N-3)}. \]
\form#122:$ \mathcal{R} $
\form#123:\[ \mathrm{E}\left[f \;|\; \mathcal{R}\right] = \frac{\sum_{x_i \in \mathcal{R}} f(x_i) w_i}{ \sum_{x_i \in \mathcal{R}} w_i}. \]
\form#124:$ \bar{x} $
\form#125:\[ y = \frac{\sum_{x_i < \bar{x}} w_i}{ \sum_i w_i} \]
\form#126:$ (0-1]. $
\form#127:\[ y = \frac{\sum_{x_i > \bar{x}} w_i}{ \sum_i w_i} \]
\form#128:\[ \frac{N}{N-1} \left\langle \left( x-\langle x \rangle \right)^2 \right\rangle. \]
\form#129:$ \epsilon $
\form#130:$ M $
\form#131:$ N \times M $
\form#132:$ v_{ij} $
\form#133:$ j $
\form#134:$ k $
\form#135:$ S $
\form#136:$ S S^T = M. $
\form#137:\[ \frac{N}{N-1} \mathrm{E}\left[ (x-\langle x \rangle)^2 \;|\; x < \langle x \rangle \right]. \]
\form#138:\[ \frac{N}{N-1} \mathrm{E}\left[ x^2 \;|\; x < 0\right]. \]
\form#139:\[ \frac{N}{N-1} \mathrm{E}\left[ (x-t)^2 \;|\; x < t \right]. \]
\form#140:\[ \mathrm{E}\left[ \Theta \;|\; (-\infty,\infty) \right] \]
\form#141:\[ \Theta(x) = \left\{ \begin{array}{ll} 1 & x < t \\ 0 & x \geq t \end{array} \right. \]
\form#142:\[ \mathrm{E}\left[ t-x \;|\; x<t \right] \]
\form#143:$ a $
\form#144:$ b $
\form#145:\[ \int_{a}^{b} f \mathrm{d}x = \frac{1}{2} f(x_{0}) + f(x_{1}) + f(x_{2}) + \dots + f(x_{N-1}) + \frac{1}{2} f(x_{N}) \]
\form#146:$ x_0 = a $
\form#147:$ x_N = b $
\form#148:$ x_i = a+i \Delta x $
\form#149:$ \Delta x = (b-a)/N $
\form#150:\[ S = U \cdot D \cdot U^T \, ,\]
\form#151:$ \cdot $
\form#152:$ ^T $
\form#153:$ \beta $
\form#154:$ [0,1] $
\form#155:$ d $
\form#156:$ t $
\form#157:\[ f(x + t \cdot d) - f(x) \leq -\alpha t f'(x+t \cdot d) \]
\form#158:\[ f(x+\frac{t}{\beta} \cdot d) - f(x) > -\frac{\alpha}{\beta} t f'(x+t \cdot d) \]
\form#159:$ d_i = - f'(x_i) + c_i*d_{i-1} $
\form#160:$ c_i = ||f'(x_i)||^2/||f'(x_{i-1})||^2 $
\form#161:$ d_1 = - f'(x_1) $
\form#162:\[ min \{ r(x) : x in R^n \} \]
\form#163:$ r(x) = |f(x)|^2 $
\form#164:$ f(x) $
\form#165:$ R^n $
\form#166:$ R^m $
\form#167:\[ f = (f_1, ..., f_m) \]
\form#168:$ f_i(x) = b_i - \phi(x,t_i) $
\form#169:$ phi $
\form#170:\[ grad r(x) = f'(x)^t.f(x) \]
\form#171:$ = - f'(x) $
\form#172:$ t \geq 0 $
\form#173:\[ \begin{array}{rcl} dS(t, S) &=& (r-d-\lambda m) S dt +\sqrt{v} S dW_1 + (e^J - 1) S dN \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ dW_1 dW_2 &=& \rho dt \end{array} \]
\form#174:$ \omega(J) $
\form#175:\[ \omega(J) = \frac{1}{\sqrt{2\pi \delta^2}} \exp\left[-\frac{(J-\nu)^2}{2\delta^2}\right] \]
\form#176:\[ \begin{array}{rcl} \omega(J)&=& p\frac{1}{\eta_u}e^{-\frac{1}{\eta_u}J} 1_{J>0} + q\frac{1}{\eta_d}e^{\frac{1}{\eta_d}J} 1_{J<0} \\ p + q &=& 1 \end{array} \]
\form#177:\[ \begin{array}{rcl} dS(t, S) &=& (r-d-\lambda m) S dt +\sqrt{v} S dW_1 + (e^J - 1) S dN \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ d\lambda(t) &=& \kappa_\lambda(\theta_\lambda-\lambda) dt \\ dW_1 dW_2 &=& \rho dt \end{array} \]
\form#178:\[ dS(t, S) = (r(t) - q(t) - \frac{\sigma(t, S)^2}{2}) dt + \sigma dW_t. \]
\form#179:\[ dS(t, S) = (r(t) - \frac{\sigma(t, S)^2}{2}) dt + \sigma dW_t. \]
\form#180:\[ dS(t, S) = \frac{\sigma(t, S)^2}{2} dt + \sigma dW_t. \]
\form#181:\[ dS(t, S) = (r(t) - r_f(t) - \frac{\sigma(t, S)^2}{2}) dt + \sigma dW_t. \]
\form#182:$ \mu(t_0, \mathbf{x}_0) \Delta t $
\form#183:$ \mu(t_0, x_0) \Delta t $
\form#184:$ \sigma(t_0, \mathbf{x}_0) \sqrt{\Delta t} $
\form#185:$ \sigma(t_0, x_0) \sqrt{\Delta t} $
\form#186:$ \sigma(t_0, \mathbf{x}_0)^2 \Delta t $
\form#187:$ \sigma(t_0, x_0)^2 \Delta t $
\form#188:\[ dS(t, S)= \mu S dt + \sigma S dW_t. \]
\form#189:\[ \begin{array}{rcl} dS(t, S) &=& \mu S dt + \sqrt{v} S dW_1 \\ dv(t, S) &=& \kappa (\theta - v) dt + \sigma \sqrt{v} dW_2 \\ dW_1 dW_2 &=& \rho dt \end{array} \]
\form#190:\[ dx = -a x_t dt + \sigma dW_t. \]
\form#191:\[ dx = a (b - x_t) dt + \sigma \sqrt{x_t} dW_t. \]
\form#192:$ (0, 1)^N. $
\form#193:\[ \rho_{i,j}=e^{(-\beta \|i-j\|)} \]
\form#194:\[ \sigma_i(t)=k_i*((a*(T_{i}-t)+d)*e^{-b(T_{i}-t)}+c) \]
\form#195:\[ \rho_{i,j}=rho + (1-rho)*e^{(-\beta \|i-j\|)} \]
\form#196:\[ \sigma_i(t)=(a*(T_{i}-t)+d)*e^{-b(T_{i}-t)}+c \]
\form#197:$ A(t,T) $
\form#198:$ B(t,T) $
\form#199:\[ P(t, T, r_t) = A(t,T)e^{ -B(t,T) r_t}. \]
\form#200:\[ d\ln r_t = (\theta(t) - \alpha \ln r_t)dt + \sigma dW_t, \]
\form#201:$ alpha $
\form#202:$ sigma $
\form#203:\[ r_t = e^{\varphi(t) + x_t} \]
\form#204:$ \varphi(t) $
\form#205:$ x_t $
\form#206:\[ dr_t = k(\theta - r_t)dt + \sqrt{r_t}\sigma dW_t . \]
\form#207:$ y_t $
\form#208:\[ dy_t=\left[ (\frac{k\theta }{2}+\frac{\sigma ^2}{8})\frac{1}{y_t}- \frac{k}{2}y_t \right] d_t+ \frac{\sigma }{2}dW_{t} \]
\form#209:\[ dr_t = (\theta(t) - \alpha r_t)dt + \sqrt{r_t}\sigma dW_t . \]
\form#210:\[ r_t = \varphi(t) + y_t^2 \]
\form#211:\[ \varphi(t) = f(t) - \frac{2k\theta(e^{th}-1)}{2h+(k+h)(e^{th}-1)} - \frac{4 x_0 h^2 e^{th}}{(2h+(k+h)(e^{th}-1))^1}, \]
\form#212:$ f(t) $
\form#213:$ h = \sqrt{k^2 + 2\sigma^2} $
\form#214:\[ dr_t = (\theta(t) - \alpha r_t)dt + \sigma dW_t \]
\form#215:\[ r_t = \varphi(t) + x_t \]
\form#216:\[ \varphi(t) = f(t) + \frac{1}{2}[\frac{\sigma(1-e^{-at})}{a}]^2, \]
\form#217:\[ dr_t = a(b - r_t)dt + \sigma dW_t , \]
\form#218:$ a(t) = a $
\form#219:$ a(t) = 0 $
\form#220:$ a(t) = a_i if t_{i-1} \geq t < t_i $
\form#221:\[ r_t = f(t, x_t, y_t) \]
\form#222:\[ x_t = \mu_x(t, x_t)dt + \sigma_x(t, x_t) dW_t^x \]
\form#223:\[ y_t = \mu_y(t,y_t)dt + \sigma_y(t, y_t) dW_t^y \]
\form#224:$ W^x $
\form#225:$ W^y $
\form#226:\[ dW^x_t dW^y_t = \rho dt \]
\form#227:$ \rho $
\form#228:\[ dr_t = \varphi(t) + x_t + y_t \]
\form#229:\[ dx_t = -a x_t dt + \sigma dW^1_t, x_0 = 0 \]
\form#230:\[ dy_t = -b y_t dt + \sigma dW^2_t, y_0 = 0 \]
\form#231:$ dW^1_t dW^2_t = \rho dt $
\form#232:\[ \varphi(t) = f(t) + \frac{1}{2}(\frac{\sigma(1-e^{-at})}{a})^2 + \frac{1}{2}(\frac{\eta(1-e^{-bt})}{b})^2 + \rho\frac{\sigma(1-e^{-at})}{a}\frac{\eta(1-e^{-bt})}{b}, \]
\form#233:$ |f(x)| < \epsilon $
\form#234:$ |x-\xi| < \epsilon $
\form#235:$ \xi $
\form#236:$ x_\mathrm{min} $
\form#237:$ x_\mathrm{max} $
\form#238:$ f(x_\mathrm{min}) \leq 0 \leq f(x_\mathrm{max}) $
\form#239:$ f(x_\mathrm{max}) \leq 0 \leq f(x_\mathrm{min}) $
\form#240:\[ d\mathrm{x}_t = \mu(t, x_t)\mathrm{d}t + \sigma(t, \mathrm{x}_t) \cdot d\mathrm{W}_t. \]
\form#241:\[ dx_t = \mu(t, x_t)dt + \sigma(t, x_t)dW_t. \]
\form#242:$ \mu(t, \mathrm{x}_t) $
\form#243:$ \sigma(t, \mathrm{x}_t) $
\form#244:$ E(\mathrm{x}_{t_0 + \Delta t} | \mathrm{x}_{t_0} = \mathrm{x}_0) $
\form#245:$ \Delta t $
\form#246:$ S(\mathrm{x}_{t_0 + \Delta t} | \mathrm{x}_{t_0} = \mathrm{x}_0) $
\form#247:$ V(\mathrm{x}_{t_0 + \Delta t} | \mathrm{x}_{t_0} = \mathrm{x}_0) $
\form#248:\[ E(\mathrm{x}_0,t_0,\Delta t) + S(\mathrm{x}_0,t_0,\Delta t) \cdot \Delta \mathrm{w} \]
\form#249:$ E $
\form#250:$ \mathrm{x} + \Delta \mathrm{x} $
\form#251:$ \mu(t, x_t) $
\form#252:$ \sigma(t, x_t) $
\form#253:$ E(x_{t_0 + \Delta t} | x_{t_0} = x_0) $
\form#254:$ S(x_{t_0 + \Delta t} | x_{t_0} = x_0) $
\form#255:$ V(x_{t_0 + \Delta t} | x_{t_0} = x_0) $
\form#256:\[ E(x_0,t_0,\Delta t) + S(x_0,t_0,\Delta t) \cdot \Delta w \]
\form#257:$ x + \Delta x $
\form#258:\[ \int_0^T \sigma_L^2(t)dt = \sigma_B^2 T \]
\form#259:$ \sigma_L(t) $
\form#260:$ \sigma_B(T) $
\form#261:$ T $
\form#262:\[ \sigma_L(t) = \sqrt{\frac{\mathrm{d}}{\mathrm{d}t}\sigma_B^2(t)t} \]