# SV models with leverage effect

The leverage effect is the phenomenon that volatility tends to rise following a drop in returns. In SV models the leverage effect is modeled by letting the noise terms in the two equations be correlated. SV models with leverage effect can be written as:

X_{t} = σ_{X} exp(h_{t}/2)ε_{t}, (1)

h_{t+1} = ϕh_{t} + ση_{t}, (2)

where the pairs (ε_{t}, η_{t}) are iid with E(ε_{t}) = E(η_{t}) = 0, Var(ε_{t}) = Var(η_{t}) = 1 and corr(ε_{t}, η_{t}) = ρ.

If ρ < 0, which is the standard case, a drop in returns at time t then tends to give increased volatility at time t+1.

Due to correlated noise terms, the structure in SV models with leverage effect is (slightly) more complicated than in the models without leverage effect. It is therefore also more difficult to find an expression for the joint density function, p(**X**, **h**|θ), in this case.

SV models on the form (1-2) can be represented graphically as:

All paths to X_{t} and h_{t+1} goes via/through h_{t}, so the pair (X_{t}, h_{t+1}) is conditionally independent of previous variables given h_{t}.

This leads to the following expression for the joint density:

p(X,h|θ) = p(h_{1}|θ) Πp(X_{t}, h_{t+1}|h_{t}, θ)

This expression may be further simplified by factorizing p(Xt, ht+1|ht, θ). This can be done in two ways:

a. We can use that p(X_{t}, h_{t+1}|h_{t}, θ) = p(X_{t}|h_{t+1}, h_{t}, θ)p(h_{t+1}|h_{t}, θ) and write the joint density as p(X,h|θ) = p(h_{1}|θ) Πp(X_{t}|h_{t+1}, h_{t}, θ)p(h_{t+1}|h_{t}, θ)

This is represented in the folowing graph

b. Alternatively we can use p(X_{t}, h_{t+1}|h_{t}, θ) = p(X_{t}|h_{t}, θ)p(h_{t+1}|X_{t}, h_{t}, θ), which gives the following expression p(X,h|θ) = p(h_{1}|θ) Πp(X_{t}|h_{t}, θ)p(h_{t+1}|X_{t}, h_{t}, θ). The graphical representation for this form is given by

The two forms should be equivalent, so it should in principle be possible to use both. However, as we shall see, it might be reasons to prefer one over the other.

**The Gaussian leverage model**

In the Gaussian leverage model it is assumed that the pairs (ε_{t}, η_{t}) are iid bi-variate normally distributed, with standard normal marginals. This is the most popular leverage model and it is a discrete time version of models used in option pricing.

Then ε_{t}|η_{t} ̴ N(ρη_{t}, 1-ρ^{2}) and we can write ε_{t} = ρη_{t} + sqrt(1-ρ^{2})w_{t} where w_{t} is standard normal and η_{t} and w_{t} are independent. Noting that η_{t} = (h_{t+1} – ϕh_{t})/σ, the model can be written as:

X_{t} = σ_{X} exp(h_{t}/2)ρ(h_{t+1} – ϕh_{t})/σ + σ_{X} exp(h_{t}/2) sqrt(1-ρ^{2})w_{t},

h_{t+1} = ϕh_{t} + ση_{t},

where w_{t} and η_{t} are iid N(0,1).

On this form we may use the formulation a) for the joint density function, and we see that

h_{1} ̴ N(0,σ^{2}/(1-ϕ^{2})), h_{t+1}|h_{t} ̴ N(ϕh_{t}, σ^{2}) and

X_{t}|h_{t+1}, h_{t} ̴ N(σ_{X}ρ exp(h_{t}/2) (h_{t+1} – ϕh_{t})/σ, σ_{X}^{2} exp(h_{t}) (1-ρ^{2})). Thus it is easy to find an expression for log p(**X**, **h**|θ) here. See **sdv_lev_1.tpl** for how this can be done.

Alternatively we may use the other version. Noting that η_{t}|ε_{t} ̴ N(ρε_{t}, 1-ρ^{2}), we may write η_{t} = ρε_{t} + sqrt(1-ρ^{2}) v_{t}, where v_{t} ̴ N(0,1) and vt is independent of ε_{t}. Then, using that ε_{t} = Xt exp(-h_{t}/2)/σ_{X}, the model may be written as:

X_{t} = σ_{X} exp(h_{t}/2)ε_{t} ,

h_{t+1} = ϕh_{t} + σρX_{t} exp(-h_{t}/2)/σ_{X} + σ sqrt(1-ρ2) v_{t} ,

where ε_{t} and v_{t} are iid N(0,1) by assumption.

Here it is seen that h_{t+1}|(X_{t}, h_{t}, θ) ̴ N(ϕh_{t} + σρX_{t} exp(-h_{t}/2)/σX, σ2(1 - ρ2)) and

X_{t}|(h_{t},θ) ̴ N(0, σ_{X}^{2} exp(h_{t})), so we can easily find an expression for log p(**X**, **h**|θ), see **sdv_lev_2.tpl** for how this can be done.

The two specifications for the Gaussian leverage model should give the same results. Comparing the par files **sdv_lev_1.par** and **sdv_lev_2.par**, we see that the results are practically identical, as they should. However, it seems that sdv_lev_1 runs somewhat faster and that the difference in run time is increasing in the size of the data set. This suggests that it might be preferable to use the parametrization given in sdv_lev_1, at least for large data sets. This version may be less intuitive than the other and is less commonly used, but it might actually be preferable because of the run time issue. **Leverage models with heavier tails and/or skewness**

The moments of returns in the Gaussian leverage model are the same as in the basic SV model. In order to model both leverage effect and heavier tails and/or skewness, the following specification is used:

X_{t} = σ_{X} exp(h_{t}/2)ε_{t} ,

h_{t+1} = ϕh_{t} + σρX_{t} exp(-h_{t}/2)/σ_{X} + σ sqrt(1-ρ^{2}) v_{t},

where v_{t} ̴ N(0, 1)and ε_{t} has some standardized continuous distribution. This looks like the formulation used to set up sdv_lev2, but here ε_{t} is not necessarily normally distributed. In the SV_lev_t model a standardized t-distribution is used for ε_{t}. This not only gives heavier tails in the returns, but also some tail thickness in the volatility process. This might actually be a favorable property. In SV_lev_st ε_{t }follows a skewed t-distribution, which captures skewness in returns, but also gives skewness in the volatility process. If ε_{t }has negative skewness and ρ also is negative, which is the usual case, then there is positive skewness in the volatility process. This seems like a reasonable property, since big positive “jumps” in volatility are more likely to occur than large negative ones.

For models on this form h_{t+1}|(X_{t}, h_{t}, θ) ̴ N(ϕh_{t} + σρX_{t} exp(-h_{t}/2)/σ_{X}, σ^{2}(1 – ρ^{2})) no matter which distribution we choose for ε_{t}. When ε_{t} is not normally distributed, the distribution of h_{1}|θ is unknown, so strictly speaking we cannot find an exact expression for p(**X**,**h**|θ) here. However it is still the case that E[h_{1}] = 0 and Var(h_{1}) = σ^{2}/(1-ϕ^{2}), and by assuming that h_{1}|θ ̴ N(0,σ^{2}/(1-ϕ^{2})), we find an approximate expression for the joint density. Since p(h_{1}|θ) is only a minor contributor to p(**X**,**h**|θ), the error is small. The distribution of X_{t}|h_{t} depends on the distribution used for ε_{t}, as can be seen in the tpl-files, **sdv_t_lev.tpl** and **sdv_st_lev.tpl**.