October 13th, 2023, 8:02 am

Friends, I have been able to write a first prototype program about the conditional density research that works well. In the new program, I used monte carlo to create generate joint paths of stochastic volatility and asset. Then I divided the volatility density into forty subdivisions and found conditional density for asset associated with each subdivision. I, then, found the univariate Z-series of each of these asset densities associated with different volatility grid points/subdivisions. The coefficients of all of these subdivisions are different and slightly noisy. These coefficients can be represented as c0(1:N), c1(1:N), c2(1:N) and so on when there are total N grid points for volatility Then I took each coefficient across all asset Z-series , for example c0(1:N) and then c1(1:N) and continued to c5(1:N), ( I took only six coefficients since terminal time was only two years and I thought that 5th order Z-series will suffice for short maturity.)

Please note that these coefficients of asset Z-series are conditional on stochastic volatility grid. Finally I constructed a Z-series of each of these six coefficients of asset Z-series as a function of stochastic volatility as

[$]c_0(Z_v) \, = \, c_{00} \, + \, c_{01} \, Z_v \,+ \, c_{02} \, {Z_v}^2 \, + \, c_{03} \, {Z_v}^3 \, + \, c_{04} \, {Z_v}^4 \, + \, c_{05} \, {Z_v}^5 \,[$]

[$]c_1(Z_v) \, = \, c_{10} \, + \, c_{11} \, Z_v \,+ \, c_{12} \, {Z_v}^2 \, + \, c_{13} \, {Z_v}^3 \, + \, c_{14} \, {Z_v}^4 \, + \, c_{15} \, {Z_v}^5 \,[$]

[$]c_2(Z_v) \, = \, c_{20} \, + \, c_{21} \, Z_v \,+ \, c_{22} \, {Z_v}^2 \, + \, c_{23} \, {Z_v}^3 \, + \, c_{24} \, {Z_v}^4 \, + \, c_{25} \, {Z_v}^5 \,[$]

[$]c_3(Z_v) \, = \, c_{30} \, + \, c_{31} \, Z_v \,+ \, c_{32} \, {Z_v}^2 \, + \, c_{33} \, {Z_v}^3 \, + \, c_{34} \, {Z_v}^4 \, + \, c_{35} \, {Z_v}^5 \,[$]

[$]c_4(Z_v) \, = \, c_{40} \, + \, c_{41} \, Z_v \,+ \, c_{42} \, {Z_v}^2 \, + \, c_{43} \, {Z_v}^3 \, + \, c_{44} \, {Z_v}^4 \, + \, c_{45} \, {Z_v}^5 \,[$]

[$]c_5(Z_v) \, = \, c_{50} \, + \, c_{51} \, Z_v \,+ \, c_{52} \, {Z_v}^2 \, + \, c_{53} \, {Z_v}^3 \, + \, c_{54} \, {Z_v}^4 \, + \, c_{55} \, {Z_v}^5 \,[$]

Equations (A) -- parametric form of Z-series coefficients conditional on volatility

Z-series of asset, in stochastic volatility model, conditional on stochastic volatility will then be given as

[$]X(Z_x,Z_v) \, = \, c_{0}(Z_v) \, + \, c_{1}(Z_v) \, Z_x \,+ \, c_{2}(Z_v) \, {Z_x}^2 \, + \, c_{3}(Z_v) \, {Z_x}^3 \, + \, c_{4}(Z_v) \, {Z_x}^4 \, + \, c_{5}(Z_v) \, {Z_x}^5 \,[$]

In order to find Z-series of asset conditional on a particular stochastic volatility from Equations (A), we only need to substitute the value of [$]Z_v[$] associated with that stochastic volatility subdivision on the stochastic volatility grid. I still have to see if the same algorithm would work for correlated SV models.

Finally, I compared the conditional asset density constructed from original noisy conditional Z-series that was calculated separately for each of the stochastic volatility grid points with the conditional asset density constructed from the above smooth universal expansion(after substituting the conditional value of Z_v associated with particular volatility subdivision in Equations(A)) of each coefficient as given by Equations(A) and found that match between conditional densities from both Z-series, numerically calculated and parametrically calculated, across all of the volatility grid points was very good.

I will be posting some graphs showing this comparison later today.

I want to continue making more changes and will post the program at the end of weekend or possibly on Monday.

You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal