Suppose ${\left(M_k\right)}_{k=0}^n$is a Markov Chain occupying states $E_1,E_2,E_3$ for $n\in \mathbb{N}$, $k\in \left\{0,1,\cdots ,n\right\}$ and further assume that $p_{ij}=P\left(M_k=E_j|M_{k-1}=E_i\right)>0$ is the probability of transitioning from state $E_i$ to $E_j$ in one step. Since $p_{ij}$ is strictly positive, ${\left(M_k\right)}_{k=0}^n$ is ergodic. Define $X_n={\displaystyle {\sum }_{k=1}^{n}1{\text{l}}_{E_1}}$, $Y_n={\displaystyle {\sum }_{k=1}^{n}1{\text{l}}_{E_2}}$, $Z_n={\displaystyle {\sum }_{k=1}^{n}1{\text{l}}_{E_3}}$ to be the occupation times of state $E_1,E_2,E_3$ during the time-period $\left\{1,\cdots ,n\right\}$ where $1{\text{l}}_{E_i}$ is the indicator function on state $E_i$. Then given $X_n=x,Y_n=y,Z_n=z$, $x+y+z=n$ and $0\le {\pi }_i\le 1$ the probability that $M_0=E_i$ with ${\pi }_1+{\pi }_2+{\pi }_3=1$, the probability mass function of our occupation time in ${\left(M_k\right)}_{k=0}^n$ reads

$\begin{array}{l}{p}_n\left(x,y,z\right)=P\left(X_n=x,Y_n=y,Z_n=z\right)=p_n\left(x,y,n-x-y\right)\\ =F\left(x,y,n-x-y\right)+\left({\pi }_1{d}_1+{\pi }_3{d}_8\right)F\left(x,y-1,n-x-y\right)\\ +\left({\pi }_1{d}_2+{\pi }_2{d}_5\right)F\left(x,y,n-x-y-1\right)\\ +\left({\pi }_2{d}_4+{\pi }_3{d}_7\right)F\left(x-1,y,n-x-y\right)+{\pi }_3{d}_{9}F\left(x-1,y-1,n-x-y\right)\\ +{\pi }_2{d}_{6}F\left(x-1,y,n-x-y-1\right)+{\pi }_1{d}_{3}F\left(x,y-1,n-x-y-1\right)\end{array}$ (1)

where

$d_1=p_{12}-p_{22}$

$d_2=p_{13}-p_{33}$

$d_3=p_{22}{p}_{33}-p_{23}{p}_{32}+p_{12}{p}_{23}+p_{13}{p}_{32}-p_{12}{p}_{33}-p_{13}{p}_{22}$

$d_4=p_{21}-p_{11}$

$d_5=p_{23}-p_{33}$

$d_6=p_{13}{p}_{21}-p_{11}{p}_{23}-p_{13}{p}_{31}+p_{23}{p}_{31}+p_{11}{p}_{33}-p_{21}{p}_{33}$

$d_7=p_{31}-p_{11}$

$d_8=p_{32}-p_{22}$

$d_9=-p_{12}{p}_{21}+p_{11}{p}_{22}+p_{12}{p}_{31}-p_{22}{p}_{31}-p_{11}{p}_{32}+p_{21}{p}_{32}$

${\nu }_{st}=\frac{p_{12}{p}_{21}}{p_{11}{p}_{22}};{\nu }_{su}=\frac{p_{13}{p}_{31}}{p_{11}{p}_{33}};{\nu }_{tu}=\frac{p_{23}{p}_{32}}{p_{22}{p}_{33}};$

${\nu }_{stu}=\frac{-p_{13}{p}_{22}{p}_{31}+p_{12}{p}_{23}{p}_{31}+p_{13}{p}_{21}{p}_{32}-p_{11}{p}_{23}{p}_{32}-p_{12}{p}_{21}{p}_{33}}{p_{33}{p}_{22}{p}_{11}}$

$\begin{array}{c}F\left(x,y,z\right)={\displaystyle \underset{k=0}{\overset{n}{\sum }}{\displaystyle \underset{j=0}{\overset{n}{\sum }}{\displaystyle \underset{i=0}{\overset{n}{\sum }}{\displaystyle \underset{l=0}{\overset{Min\left(x+i,y+j,z+k\right)}{\sum }}\left(\begin{array}{c}x+i\\ l\end{array}\right)\left(\begin{array}{c}y+j\\ l\end{array}\right)\left(\begin{array}{c}z+k\\ l\end{array}\right)\left(\begin{array}{c}l\\ k\end{array}\right)\left(\begin{array}{c}l-k\\ j\end{array}\right)\left(\begin{array}{c}l-k-j\\ i\end{array}\right)}}}}\\ \ast \left[{\nu }_{st}^k{\nu }_{su}^j{\nu }_{tu}^i{\nu }_{stu}^{l-k-j-i}{p}_{11}^x{p}_{22}^y{p}_{33}^z\right]\end{array}$

The proof and derivation of the formula above may be found in Evans [3] .

Suppose $M\left(t\right)$ is a continuous time Markov Chain ranging over states $E_1,E_2,E_3$ and denote by ${\pi }_i$ the probability of starting in $E_i$ at time 0. Define ${\lambda }_{ij}$ to be the transition rate from $E_i$ to $E_j$. Then for $0\le u<t$, the occupation times of $E_1,E_2,E_3$ by time $t$ are $X\left(t\right)={\displaystyle {\int }_0^{t}1{\text{l}}_{E_1}}\text{d}u$, $Y\left(t\right)={\displaystyle {\int }_0^{t}1{\text{l}}_{E_2}}\text{d}u$, $Z\left(t\right)={\displaystyle {\int }_0^{t}1{\text{l}}_{E_3}}\text{d}u$, where $X\left(t\right)+Y\left(t\right)+Z\left(t\right)=t$ and ${\pi }_1+{\pi }_2+{\pi }_3=1$, our probability density function of occupation times in $M\left(t\right)$ reads

$f\left(x,y,t\right)={\text{e}}^{-x\left({\lambda }_{12}+{\lambda }_{13}-{\lambda }_{31}-{\lambda }_{32}\right)-y\left({\lambda }_{21}+{\lambda }_{23}-{\lambda }_{31}-{\lambda }_{32}\right)-t\left({\lambda }_{31}+{\lambda }_{32}\right)}g\left(x,y,t-y-x\right)$

with

$\begin{array}{l}g\left(x,y,t-y-x\right)\\ ={\pi }_1{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{l=1}{\overset{m}{\sum }}{\displaystyle \underset{k=0}{\overset{m-l}{\sum }}{\displaystyle \underset{j=0}{\overset{m-l-k}{\sum }}b}}}}\left(m,l,k,j\right)\frac{x^{m-l}{y}^{m-k-1}{\left(t-x-y\right)}^{m-j-1}}{\left(m-l\right)!\left(m-k-1\right)!\left(m-j-1\right)!}\end{array}$

$+{\pi }_1{\displaystyle \underset{m=2}{\overset{\infty }{\sum }}{\displaystyle \underset{k=1}{\overset{m-1}{\sum }}{\displaystyle \underset{j=0}{\overset{m-k}{\sum }}b}}}\left(m,0,k,j\right)\frac{x^m{y}^{m-k-1}{\left(t-x-y\right)}^{m-j-1}}{\left(m\right)!\left(m-k-1\right)!\left(m-j-1\right)!}$

$+{\pi }_1{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{j=0}{\overset{m-1}{\sum }}b}}\left(m,0,0,j\right)\frac{x^m{y}^{m-1}{\left(t-x-y\right)}^{m-j-1}}{\left(m\right)!\left(m-1\right)!\left(m-j-1\right)!}$

$+{\pi }_1{\delta }_y{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,0,m,0\right)\frac{x^m{\left(t-x-y\right)}^{m-1}}{\left(m\right)!\left(m-1\right)!}$

$+{\pi }_1{\delta }_z{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,0,0,m\right)\frac{x^m{y}^{m-1}}{\left(m\right)!\left(m-1\right)!}$

$+{\pi }_1{\delta }_y{\delta }_z$

$+{\pi }_2{\displaystyle \underset{m=2}{\overset{\infty }{\sum }}{\displaystyle \underset{l=1}{\overset{m-1}{\sum }}{\displaystyle \underset{k=0}{\overset{m-l}{\sum }}{\displaystyle \underset{j=0}{\overset{m-l-k}{\sum }}b}}}}\left(m,l,k,j\right)\frac{x^{m-l-1}{y}^{m-k}{\left(t-x-y\right)}^{m-j-1}}{\left(m-l-1\right)!\left(m-k\right)!\left(m-j-1\right)!}$

$+{\pi }_2{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{k=1}{\overset{m}{\sum }}{\displaystyle \underset{j=0}{\overset{m-k}{\sum }}b}}}\left(m,0,k,j\right)\frac{x^{m-1}{y}^{m-k}{\left(t-x-y\right)}^{m-j-1}}{\left(m-1\right)!\left(m-k\right)!\left(m-j-1\right)!}$

$+{\pi }_2{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{j=0}{\overset{m-1}{\sum }}b}}\left(m,0,0,j\right)\frac{x^{m-1}{y}^m{\left(t-x-y\right)}^{m-j-1}}{\left(m-1\right)!\left(m\right)!\left(m-j-1\right)!}$

$+{\pi }_2{\delta }_x{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,m,0,0\right)\frac{y^m{(t-x-y)}^{m-1}}{\left(m\right)!\left(m-1\right)!}$

$+{\pi }_2{\delta }_z{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,0,0,m\right)\frac{x^{m-1}{y}^m}{\left(m-1\right)!\left(m\right)!}$

$+{\pi }_2{\delta }_x{\delta }_z$

$+{\pi }_3{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{l=0}{\overset{m-1}{\sum }}{\displaystyle \underset{k=0}{\overset{m-l-1}{\sum }}{\displaystyle \underset{j=0}{\overset{m-l-k}{\sum }}b}}}}\left(m,l,k,j\right)\frac{x^{m-l-1}{y}^{m-k-1}{\left(t-x-y\right)}^{m-j}}{\left(m-l-1\right)!\left(m-k-1\right)!\left(m-j\right)!}$

$+{\pi }_3{\displaystyle \underset{m=2}{\overset{\infty }{\sum }}{\displaystyle \underset{l=1}{\overset{m-1}{\sum }}b}}\left(m,l,m-l,0\right)\frac{x^{m-l-1}{y}^{l-1}{\left(t-x-y\right)}^m}{\left(m-l-1\right)!\left(l-1\right)!\left(m\right)!}$

$+{\pi }_3{\delta }_x{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,m,0,0\right)\frac{y^{m-1}{\left(t-x-y\right)}^m}{\left(m-1\right)!\left(m\right)!}$

$+{\pi }_3{\delta }_y{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,0,m,0\right)\frac{x^{m-1}{\left(t-x-y\right)}^m}{\left(m-1\right)!\left(m\right)!}$

$+{\pi }_3{\delta }_x{\delta }_y$

$+c_1{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{l=0}{\overset{m-1}{\sum }}{\displaystyle \underset{k=0}{\overset{m-l}{\sum }}{\displaystyle \underset{j=0}{\overset{m-l-k}{\sum }}b}}}}\left(m,l,k,j\right)\frac{x^{m-l-1}{y}^{m-k}{\left(t-x-y\right)}^{m-j}}{\left(m-l-1\right)!\left(m-k\right)!\left(m-j\right)!}$

$+c_1{\delta }_x{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,m,0,0\right)\frac{y^m{\left(t-x-y\right)}^m}{\left(m\right)!\left(m\right)!}$

$+c_1{\delta }_x$

$+c_2{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{l=1}{\overset{m}{\sum }}{\displaystyle \underset{k=0}{\overset{m-l}{\sum }}{\displaystyle \underset{j=0}{\overset{m-l-k}{\sum }}b}}}}\left(m,l,k,j\right)\frac{x^{m-l}{y}^{m-k-1}{\left(t-x-y\right)}^{m-j}}{\left(m-l\right)!\left(m-k-1\right)!\left(m-j\right)!}$

$+c_2{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{k=0}{\overset{m-1}{\sum }}{\displaystyle \underset{j=0}{\overset{m-k}{\sum }}b}}}\left(m,0,k,j\right)\frac{x^m{y}^{m-k-1}{\left(t-x-y\right)}^{m-j}}{\left(m\right)!\left(m-k-1\right)!\left(m-j\right)!}$ (2)

$+c_2{\delta }_y{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,0,m,0\right)\frac{x^m{\left(t-x-y\right)}^m}{\left(m\right)!\left(m\right)!}$

$+c_2{\delta }_y$

$+c_3{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{l=1}{\overset{m}{\sum }}{\displaystyle \underset{k=0}{\overset{m-l}{\sum }}{\displaystyle \underset{j=0}{\overset{m-l-k}{\sum }}b}}}}\left(m,l,k,j\right)\frac{x^{m-l}{y}^{m-k}{\left(t-x-y\right)}^{m-j-1}}{\left(m-l\right)!\left(m-k\right)!\left(m-j-1\right)!}$

$+c_3{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{k=1}{\overset{m}{\sum }}{\displaystyle \underset{j=0}{\overset{m-k}{\sum }}b}}}\left(m,0,k,j\right)\frac{x^m{y}^{m-k}{\left(t-x-y\right)}^{m-j-1}}{\left(m\right)!\left(m-k\right)!\left(m-j-1\right)!}$

$+c_3{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{j=0}{\overset{m-1}{\sum }}b}}\left(m,0,0,j\right)\frac{x^m{y}^m{\left(t-x-y\right)}^{m-j-1}}{\left(m\right)!\left(m\right)!\left(m-j-1\right)!}$

$+c_3{\delta }_z{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}b}\left(m,0,0,m\right)\frac{x^m{y}^m}{\left(m\right)!\left(m\right)!}$

$+c_3{\delta }_z$

$+c_4{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{\displaystyle \underset{l=0}{\overset{m}{\sum }}{\displaystyle \underset{k=0}{\overset{m-l}{\sum }}{\displaystyle \underset{j=0}{\overset{m-l-k}{\sum }}b}}}}\left(m,l,k,j\right)\frac{x^{m-l}{y}^{m-k}{\left(t-x-y\right)}^{m-j}}{\left(m-l\right)!\left(m-k\right)!\left(m-j\right)!}$

$+c_4$

$\begin{array}{l}{c}_1={\pi }_2{\lambda }_{23}+{\pi }_3{\lambda }_{32}\\ c_2={\pi }_1{\lambda }_{13}+{\pi }_3{\lambda }_{31}\\ c_3={\pi }_1{\lambda }_{12}+{\pi }_2{\lambda }_{21}\\ c_4={\pi }_1\left[{\lambda }_{12}{\lambda }_{13}+\left({\lambda }_{13}-{\lambda }_{23}\right)\left({\lambda }_{32}-{\lambda }_{12}\right)\right]+{\pi }_2\left[{\lambda }_{21}{\lambda }_{23}+\left({\lambda }_{31}-{\lambda }_{21}\right)\left({\lambda }_{23}-{\lambda }_{13}\right)\right]\\ +{\pi }_3\left[{\lambda }_{31}{\lambda }_{32}+\left({\lambda }_{12}-{\lambda }_{32}\right)\left({\lambda }_{31}-{\lambda }_{21}\right)\right]\end{array}$

$\begin{array}{l}b\left(m,l,k,j\right)\\ =\left(\begin{array}{c}m\\ l\end{array}\right)\left(\begin{array}{c}m-l\\ k\end{array}\right)\left(\begin{array}{c}m-l-k\\ j\end{array}\right){\left({\lambda }_{23}{\lambda }_{32}\right)}^l{\left({\lambda }_{13}{\lambda }_{31}\right)}^k{\left({\lambda }_{12}{\lambda }_{21}\right)}^j{\left({\lambda }_{12}{\lambda }_{23}{\lambda }_{31}+{\lambda }_{13}{\lambda }_{21}{\lambda }_{32}\right)}^{m-l-k-j}\end{array}$

and $\delta \left(x\right)={\delta }_x,\delta \left(y\right)={\delta }_y,\delta \left(t-x-y\right)={\delta }_z$ are Dirac delta functionals evaluated at $x,y,t-x-y$ respectively.

The proof and derivation of the formula above may be found in Evans [4] .

Suppose $r$ is the risk free interest rate of a bond in a given economy and denote by $S$and $\sigma$ the price and volatility of a risky asset. Then given a time to maturity $T$ and strike price $K$, the value of a European Call and Put option is given by

$C\left(S,T,K,\sigma ,r\right)=SN\left(a_1\right)-K{\text{e}}^{-rt}N\left(a_2\right)$ (3)

and

$P\left(S,T,K,\sigma ,r\right)=K{\text{e}}^{-rt}N\left(-a_2\right)-SN\left(-a_1\right)$ (4)

respectively where

$N\left(T\right)={{\displaystyle \int }}_{-\infty }^T\frac{{\text{e}}^{\frac{-x^2}{2}}}{\sqrt{2\pi }}\text{d}x$, $a_1=\frac{\ln \left(S/K\right)+rT+\left({\sigma }^2/2\right)T}{\sigma \sqrt{T}}$, $a_2=a_1-\sigma \sqrt{T}$

The derivation of $C\left(S,T,K,\sigma ,r\right)$ may be found in Black [1] and $P\left(S,T,K,\sigma ,r\right)$ is obtained from $C\left(S,T,K,\sigma ,r\right)$ by the principle of Put-Call parity.

$P=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{8}& \frac{3}{8}\\ \frac{1}{5}& \frac{12}{25}& \frac{8}{25}\\ \frac{3}{10}& \frac{6}{10}& \frac{1}{10}\end{array}\right],\Pi =\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{4}& \frac{1}{4}\end{array}\right]$

the expected value of our European Call Option

$\begin{array}{l}E\left[C\left(100,T,100,\sigma \left(T\right),0.04\right)\right]\\ ={\displaystyle \underset{x=0}{\overset{T}{\sum }}{\displaystyle \underset{y=0}{\overset{T-x}{\sum }}C}}\left(100,T,100,\sigma \left(T\right),0.04\right)p_T\left(x,y,T-x-y\right)\end{array}$

and our European Put Option

$\begin{array}{l}E\left[P\left(100,T,100,\sigma \left(T\right),0.04\right)\right]\\ ={\displaystyle \underset{x=0}{\overset{T}{\sum }}{\displaystyle \underset{y=0}{\overset{T-x}{\sum }}P}}\left(100,T,100,\sigma \left(T\right),0.04\right)p_T\left(x,y,T-x-y\right)\end{array}$

is given by Table 1 and Table 2 respectively for $T\in \left\{1,2,\cdots ,12\right\}$. Our choice of parameters are all fractions as many software packages can handle their mul- tiplication and sum with no numerical precision loss whatsoever.

Define $R=\left({\lambda }_{ij}\right)$ to be the instantaneous transition rate matrix and denote by $\Pi$ the initial distribution of states, then for

$R=\left[\begin{array}{ccc}-9& 3& 6\\ 4& -6& 2\\ 1& 3& -4\end{array}\right],\Pi =\left[1,0,0\right]$

the expected value of our European Call Option

$\begin{array}{l}E\left[C\left(100,T,100,\sigma \left(T\right),0.04\right)\right]\\ ={\displaystyle {\int }_0^T{\displaystyle {\int }_0^{T-x}C}}\left(100,T,100,\sigma \left(T\right),0.04\right)f\left(x,y,T\right)\text{d}y\text{d}x\end{array}$

and our European Put Option

$\begin{array}{l}E\left[P\left(100,T,100,\sigma \left(T\right),0.04\right)\right]\\ ={\displaystyle {\int }_0^T{\displaystyle {\int }_0^{T-x}P}}\left(100,T,100,\sigma \left(T\right),0.04\right)f\left(x,y,T\right)\text{d}y\text{d}x\end{array}$

is given by Table 3 and Table 4 respectively for $T\in \left\{1,2,\cdots ,12\right\}$. We would like to note that our choice to use integer values of $T$ for (2) is arbitrary and that one may of course set $T$ equal to any positive constant. Furthermore, we truncate (2) by setting the upper limit of m to be 40. Our choice of upper bound is based off the heuristic that for the given parameters, the pdf almost integrates to 1 over its domain.