Stochastic Volatility Model

4.1.1 Stochastic Volatility Model

Under the risk-neutral measure Q, the equation (3.1) can be rewritten as following dy_t =

where µ is the drift term which is a constant. The stochastic volatility v_tfollows a mean-reverting square root process with long term level ˜θ = _κ+h^κθ

2, the speed of adjustment is

˜

κ = κ + h₂, the variation coefficient of the diffusion volatility is σ_v, h₁ is the price risk premium and h2 is the volatility risk premium. Moreover, dW_y,t^Q and dW_v,t^Q, are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dW_y,t^Q ∼ N (0, dt), dW_v,t^Q ∼ N (0, dt) and Corr

dW_y,t^Q, dW_v,t^Q



= ρdt. The detail proof are shown in the Appendix A.

4.1.2 Stochastic Volatility Model with Merton Jumps

Under risk-neutral measure Q, the equation (3.5) can be rewritten as following dy_t=

where µ is the drift term which is a constant. The stochastic volatility v_tfollows a mean-reverting square root process with long term level ˜θ = _κ+h^κθ

2, the speed of adjustment is

˜

κ = κ + h₂, the variation coefficient of the diffusion volatility is σ_v, h₁ is the price risk premium and h₂ is the volatility risk premium. Moreover, dW_y,t^Q and dW_v,t^Q, are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dW_y,t^Q ∼ N (0, dt), dW_v,t^Q ∼ N (0, dt) and Corr

dW_y,t^Q, dW_v,t^Q

= ρdt. The return jump dJ_y,t^Q = ξ_y,tdN_y,t^Q, the jump size ξ_y,t is a random variable, dN_y,t^Q is used to depict the number of the return jump which follows a Poisson process with the intensity λ^Q_y which is a constant, i.e., dN_y,t^Q ∼ P oisson λ^Q_ydt
, and M_J^Q

y is the compensator of the return jump and the average jump amplitude, i.e., M_J^Q_y = E^Q The risk-neutral parameters of Merton jumps are

λ^Q = λye^{γh3+12h23δ2}, γ^Q = γ + h3δ², δ^Q= δ

‧

The detail proof are shown in the Appendix B.

4.1.3 Stochastic Volatility Model with Independent Merton Jumps

Under the risk-neutral measure Q, the equation (3.9) can be rewritten as following dy_t=

where µ is the drift term which is a constant. The stochastic volatility vtfollows a mean-reverting square root process with long term level ˜θ = _κ+h^κθ

2, the speed of adjustment is

˜

κ = κ + h2, the variation coefficient of the diffusion volatility is σv, h1 is the price risk premium and h₂ is the volatility risk premium. Moreover, dW_y,t^Q and dW_v,t^Q, are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dW_y,t^Q ∼ N (0, dt), dW_v,t^Q ∼ N (0, dt) and Corr

dN_y,t^Q is used to depict the number of the return jump which follows a Poisson process with the intensity λ^Q_y which is a constant, i.e., dN_y,t^Q ∼ P oisson λ^Q_ydt
, and M_J^Q

y is the compensator of the return jump and the average jump amplitude, i.e., M_J^Q

y = E^P

The compound Poisson process of volatility jump, dJ_v,t = ξ_v,tdN_v,t = d PNv,t

j=1 ξ_v,t^j  , where the volatility jump size ξ_v,t is a random variable which follows the exponential distribution with mean µ_v, i.e. ξ_v,t∼ exp (µ_v), dN_v,t is used to depict the number of the volatility jump which follows a Poisson process with the intensity λ_v which is a constant, i.e., dN_v,t ∼ P oisson (λ_vdt). The risk-neutral parameters of Merton jumps are

λ^Q = λ_ye^{γh3+12h23δ2}, γ^Q = γ + h₃δ², δ^Q= δ The detail proof are shown in the Appendix C.

4.1.4 Stochastic Volatility Model with Correlated Merton Jumps

Under risk-neutral measure Q, the equation (3.13) can be rewritten as following dy_t=

‧

where µ is the drift term which is a constant. The stochastic volatility v_tfollows a mean-reverting square root process with long term level ˜θ = _κ+h^κθ

2, the speed of adjustment is

˜

κ = κ + h₂, the variation coefficient of the diffusion volatility is σ_v, h₁ is the price risk premium and h₂ is the volatility risk premium. Moreover, dW_y,t^Q and dW_v,t^Q, are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dW_y,t^Q ∼ N (0, dt), dW_v,t^Q ∼ N (0, dt) and Corr

y is the compensator of the return jump and the average jump amplitude, i.e., M_J^Q_y = E^Q the volatility jump size ξv,tis a random variable which follows the exponential distribution with mean µ^Q_v, i.e. ξ_v,t ∼ exp µ^Q_v
, dN_t^Q is used to depict the number of the volatility jump which follows a Poisson process with the intensity λ^Q which is a constant, i.e., dN_t^Q ∼ P oisson λ^Qdt
. The risk-neutral parameters of Merton jumps and volatility jumps are

λ^Q = λ_y· e^{h3γ+12h23δ2}

1 − h₃ρ_Jµ_v, γ^Q = γ + ρ_Jξ_v,t+ h₃δ², δ^Q = δ, µ^Q_v = µ_v 1 − h₃ρ_Jµ_v The detail proof are shown in the Appendix D.

4.1.5 Stochastic Volatility Model with Double-Exponential Jumps

Under risk-neutral measure Q, the equation (3.17) can be rewritten as following dy_t=

where µ is the drift term which is a constant. The stochastic volatility v_tfollows a mean-reverting square root process with long term level ˜θ = _κ+h^κθ

2, the speed of adjustment is

˜

κ = κ + h₂, the variation coefficient of the diffusion volatility is σ_v, h₁ is the price risk

‧

premium and h₂ is the volatility risk premium. Moreover, dW_y,t^Q and dW_v,t^Q, are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dW_y,t^Q ∼ N (0, dt), dW_v,t^Q ∼ N (0, dt) and Corr

dW_y,t^Q, dW_v,t^Q

= ρdt. The return jump dJ_y,t^Q = ξ_y,tdN_y,t^Q, the jump size ξ_y,t is a random variable, dN_y,t^Q is used to depict the number of the return jump which follows a Poisson process with the intensity λ^Q_y which is a constant, i.e., dN_y,t^Q ∼ P oisson λ^Q_ydt
, and M_J^Q

y is the compensator of the return jump and the average jump amplitude, i.e., M_J^Q_y = E^P

The risk-neutral parameters of Double-Exponential jumps are λ^Q = λ_y

The detail proof are shown in the Appendix E.

4.1.6 Stochastic Volatility Model with Independent Double-Exponential Jumps

Under risk-neutral measure Q, the equation (3.21) can be rewritten as following dy_t=

where µ is the drift term which is a constant. The stochastic volatility v_tfollows a mean-reverting square root process with long term level ˜θ = _κ+h^κθ

2, the speed of adjustment is

˜

κ = κ + h₂, the variation coefficient of the diffusion volatility is σ_v, h₁ is the price risk premium and h₂ is the volatility risk premium. Moreover, dW_y,t^Q and dW_v,t^Q, are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dW_y,t^Q ∼ N (0, dt), dW_v,t^Q ∼ N (0, dt) and Corr

dW_y,t^Q, dW_v,t^Q

= ρdt. The return jump dJ_y,t^Q = ξ_y,tdN_y,t^Q, the jump size ξ_y,t is a random variable, dN_y,t^Q is used to depict the number of the return

‧

jump which follows a Poisson process with the intensity λ^Q_y which is a constant, i.e., dN_y,t^Q ∼ P oisson λ^Q_ydt
, and M_J^Q

y is the compensator of the return jump and the average jump amplitude, i.e., M_J^Q

y = E^P 

The compound Poisson process of volatility jump, dJ_v,t = ξ_v,tdN_v,t = d PNv,t

j=1 ξ_v,t^j  , where the volatility jump size ξ_v,t is a random variable which follows the exponential distribution with mean µ_v, i.e. ξ_v,t∼ exp (µ_v), dN_v,t is used to depict the number of the volatility jump which follows a Poisson process with the intensity λ_v which is a constant, i.e., dN_v,t ∼ P oisson (λ_vdt). The risk-neutral parameters of double-exponential jumps and volatility jumps are

λ^Q = λ_y

The detail proof are shown in the Appendix F.

4.1.7 Stochastic Volatility Model with Correlated Double-Exponential