Simulating Bergomi Forward Variance and Spot Volatility

Summary

The document asks how to simulate spot prices in Bergomi’s two-factor stochastic volatility model, where spot dynamics depend on the instantaneous forward variance _t^t and the model evolves forward variance across maturities. The central issue is notation: the maturity label T and current time t are separate arguments, so evaluating at T=t gives instantaneous variance, while the decay factors govern how shocks affect forward variances at other maturities. The post asks whether simulation requires tracking multiple maturities and how to recover the variance process used in the spot equation.

It provides the model equations and references Bergomi’s book and paper, but offers no answer, implementation procedure, calibration details, or numerical evidence. As a result, it identifies an important modeling and discretization question rather than establishing a complete simulation method. The discussion is useful for understanding the distinction between instantaneous variance and the forward variance curve, but readers need the cited source or further derivation to specify a practical discretization.

Key ideas

• In the Bergomi model, spot volatility is driven by instantaneous forward variance evaluated at the current time.
• The forward variance process has both a current-time argument and a maturity argument.
• The maturity-dependent exponential factors describe the evolution of forward variance across maturities.
• The document poses, but does not resolve, how to discretize and simulate the forward variance curve.

Tags

# Simulate Spot Process with Forward Variance (Bergomi)


# Simulate Spot Process with Forward Variance (Bergomi)












I am reading Bergomi's book (Stochastic Volatility Modeling), and in section 8.7 The two-factor model (page 326), the following dynamics are given:

\begin{align} dS_t &= \sqrt{\xi_t^t}\,S_t\,dW_t^S\\ d\xi_t^T &= 2\,\nu\,\xi_t^T\,\alpha_\theta\,\Big( (1-\theta)\,e^{-k_1(T-t)}\,dW_t^1 + \theta \,e^{-k_2(T-t)}\,dW_t^2 \Big) \end{align}

In section 7.3.1 Simulating the N-factor model (page 223), it is described that the spot process $S$ over 1 time interval can be discretized as \begin{equation} \delta \ln S \ =\ ...\ +\ \sqrt{\xi_t^t}\, \delta W^S \end{equation}

But how do we simulate $\xi_t^t$ (which specifically uses $t=T$)? Do we simulate multiple $\xi_t^T$ for various $T$? But then how do we get back values for $\xi_t^t$?

I am probably struggling to understand the notation, because $\xi_t^t$ would suggest that $e^{-k_1(T-t)} = e^{-k_1(t-t)} = 1$, since $t=T$. So parameter $k_1$ becomes irrelevant, which doesn't make sense.

In general, I just want to simulate a variance process $V_t$ such that I can also simulate the spot process as per $dS_t = \sqrt{V_t}\,S_t\,dW_t^S$. But I am struggling to tie the dynamics of $\xi_t^T$ back to $V_t$.

Any help on how to understood this notation, how to simulate $\xi_t^t$, and how to get $V_t$ from $\xi_t^T$ would be greatly appreciated.

References:

- Paper Smile Dynamics II by Lorenzo Bergomi

- Book Stochastic Volatility Modeling by Lorenzo Bergomi

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.