MathsGee Answers is Zero-Rated (You do not need data to access) on: Telkom | Dimension Data | Rain | MWEB
First time here? Checkout the FAQs!
x


To add value to MathsGee Answers, you can now support those users that are helping you with their answers, questions, votes and comments. Simply click on their name and DONATE an amount to say "Thank you"



Network: Global | Joburg Libraries | MOOCs | StartUpTribe | Zim Invest | Donate

MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB

0 like 0 dislike
9 views
Mathematical Finance: What are Affine Jump Diffusion (AJD) Models?
in Data Science by Wooden (1,884 points) | 9 views

1 Answer

0 like 0 dislike
Best answer
Mathematical Finance: What are Affine Jump Diffusion (AJD) Models?

The state vector $X_t$ follows a Markov process solving the SDE \begin{equation*} dX_t=\mu(X_t)dt +\sigma(X_t)dW_t + dZ_t \end{equation*} where $W$ is an adapted Brownian, and $Z$ is a pure jump process with intensity $\lambda$. The moment generating function of the jump sizes is $\theta(c)=EQ(\exp(cJ))$. Impose an affine structure on $\mu,\sigma\sigma^T,\lambda$ and the discount rate $R$, possibly time dependent: \begin{equation*} \mu(x)=K_0+K_1x \quad (\sigma(x)\sigma(x)^T)_{ij}=(H_0)_{ij}+(H_1)_{ij}x \quad \lambda(x)=L_0+L_1x \quad R(x)=R_0+R_1x \end{equation*} Given $X_0$, the risk neutral coefficients ($K,H,L,\theta,R$) completely determine the discounted risk neutral distribution of $X$. Introduce the transform function \begin{equation*} \psi(u,X_0,T)=EQ\left[\left.\exp\left(-\int_0^{T}R(X_s)ds\right)e^{u^TX_T}\right|F_0\right]=e^{\alpha(0,u)+\beta(0,u)^Tx_0} \end{equation*} where $\alpha$ and $\beta$ solve the Ricatti ODEs subject to $\alpha(T,u)=0,\beta(T,u)=u$: \begin{align*} -\dot{\beta}(t,u)=&K_1^T\beta(t,u)+\tfrac{1}{2}\beta(t,u)^TH_1\beta(t,u)+L_1(\theta(\beta(t,u))-1)-R_1\\ -\dot{\alpha}(t,u)=&K_0^T\beta(t,u)+\tfrac{1}{2}\beta(t,u)^TH_0\beta(t,u)+L_0(\theta(\beta(t,u))-1)-R_0 \end{align*}
by Wooden (1,884 points)

Related questions

0 like 0 dislike
1 answer
0 like 0 dislike
0 answers
0 like 0 dislike
1 answer

MathsGee Answers is a global, STEM-focused Q&A platform where you can ask people from all over the world educational questions for improved outcomes.



MathsGee Supporting City of Joburg

Registered Members Online
MathsGee Tools

Math Worksheet Generator

Math Algebra Solver

Trigonometry Simulations

Vectors Simulations

Matrix Arithmetic Simulations

Matrix Transformations Simulations

Quadratic Equations Simulations

Probability & Statistics Simulations

PHET Simulations

Visual Statistics

Interactive Courseware

ZeroEd Search Engine

Article Rewriter Tool

Word Counter Tool

Other Tools

Big Blue Button | STEM Gender Equality | ZOOM | Slack | eBook