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Quantitative Finance · Glossário

O que é Path-dependent volatility model?

Definition 12.9 Derivatives and Volatility · Capítulo 12 — Rough Volatility and Forward-Variance Models

A path-dependent volatility model makes the instantaneous volatility a function of past returns. In the Guyon–Lekeufack form it is

σt=β0+β1R1,t+β2R2,t,R1,t=∫−∞tK1(t−s) dSsSs,R2,t=∫−∞tK2(t−s) (dSsSs)2 ⁣/dt,\sigma_t=\beta_0+\beta_1R_{1,t}+\beta_2\sqrt{R_{2,t}},\qquad R_{1,t}=\int_{-\infty}^tK_1(t-s)\,\frac{dS_s}{S_s},\qquad R_{2,t}=\int_{-\infty}^tK_2(t-s)\,\Bigl(\frac{dS_s}{S_s}\Bigr)^2\!\Big/dt,

a trend feature R1R_1 (a weighted sum of past returns) and an activity feature R2R_2 (a weighted sum of past squared returns), with decaying kernels K1,K2K_1,K_2 that integrate to one and β1<0\beta_1<0.

Exemplos

Example 12.10 (One large day)

Take β0=0.04\beta_0=0.04, β1=−0.06\beta_1=-0.06, β2=0.65\beta_2=0.65 and exponential kernels with rates κ1=25\kappa_1=25 and κ2=15\kappa_2=15 a year (illustrative values). With no trend the calm level is the fixed point σ∗=β0/(1−β2)=11.4%\sigma^*=\beta_0/(1-\beta_2)=11.4\%. A day of −4%-4\% lifts volatility to 22.0%. The excess halves in ten trading days, and a month (21 trading days) later volatility is still 13.8%. A day of +4%+4\% first lowers volatility, to 10.6%, because the trend term dominates. The activity term decays more slowly, so volatility then drifts up to 12.4% before settling (Figure 12.5, left).

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