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Quantitative Finance · शब्दावली

Quadratic-exponential scheme क्या है?

Definition 23.1 Derivatives and Volatility · अध्याय 23 — Monte Carlo Pricers in Practice

The quadratic-exponential scheme (QE) simulates Heston’s variance over a step by matching the mean mm and variance s2s^2 of its exact conditional law: when ψ=s2/m2\psi=s^2/m^2 is small it draws a(b+Z)2a(b+Z)^2, a scaled non-central square of a normal; when ψ\psi is large, a mixture of a mass at zero and an exponential. The log-price then uses the integrated variance implied by the start and end variances.

Bias of a one-year at-the-money call in a Heston model whose variance can reach zero, by time steps, against the Fourier price (400 000 paths). Euler’s bias decays slowly; QE’s stays within the statistical error at every step count. Data: the tutorial.
Figure 23.1. Bias of a one-year at-the-money call in a Heston model whose variance can reach zero, by time steps, against the Fourier price (400 000 paths). Euler’s bias decays slowly; QE’s stays within the statistical error at every step count. Data: the tutorial.

उदाहरण

Example 23.2 (Euler against QE)

A one-year at-the-money call in chapter 10’s base Heston model (v0=vˉ=0.04v_0=\bar v=0.04, κ=1.5\kappa=1.5, η=0.6\eta=0.6, ρ=−0.7\rho=-0.7) is worth 6.751 by Fourier inversion. With 400 000 paths and full-truncation Euler, the bias is 0.857 with 4 steps a year, 0.373 with 8, 0.140 with 16 and 0.041 with 32. With QE the price is within one standard error (0.014) of the exact value at every step count, from 4 steps a year (Figure 23.1). QE saves a factor of ten or more in steps, and so in run time.

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