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Quantitative Finance · Glossário

O que é Schwartz–Smith model?

Também chamado de: Schwartz--Smith model

Definition 16.1 Rates, Credit, XVA and Risk · Capítulo 16 — Commodity and Energy Derivatives

The Schwartz–Smith model writes the log spot price as the sum of a short-term deviation χt\chi_t, an Ornstein–Uhlenbeck process reverting to zero at speed κ\kappa with volatility σs\sigma_s, and a long-term level ξt\xi_t, a Brownian motion with volatility σl\sigma_l, the two correlated by ρ\rho. Its forward curve moves as

dF(t,T)F(t,T)=σse−κ(T−t) dWt1+σl dWt2,d⟨W1,W2⟩=ρ dt,\frac{dF(t,T)}{F(t,T)} = \sigma_se^{-\kappa(T-t)}\,dW^1_t+\sigma_l\,dW^2_t,\qquad d\langle W^1,W^2\rangle = \rho\,dt,

so that ln⁡F(t,T)=ln⁡F(0,T)+e−κ(T−t)χt+ξt−12V(t,T)\ln F(t,T) = \ln F(0,T)+e^{-\kappa(T-t)}\chi_t+\xi_t-\tfrac12V(t,T) with VV the variance. Seasonality sits in the initial curve F(0,T)F(0,T), which the model fits exactly.

Exemplos

Example 16.2 (Calibration)

An illustrative Henry Hub-like curve runs from 2.60 dollars for April to 3.50 for January. Options expiring half a month before each delivery are quoted at implied volatilities falling from 62.0% (May) to 43.0% (March), synthetic. With ρ=30%\rho = 30\% fixed, a Levenberg–Marquardt fit gives κ=1.50\kappa = 1.50, σs=60.0%\sigma_s = 60.0\% and σl=20.0%\sigma_l = 20.0\%.

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