First-Order Expansion of Forward Variance in the Bergomi Model
Summary
The document explains how to obtain a first-order approximation for forward variance in a Bergomi volatility model. Starting from a stochastic differential equation in which forward variance is driven by Brownian shocks and scaled by a small parameter, the derivation first writes the state as its initial value plus the accumulated stochastic integral. It then substitutes that expression back into the dynamics.
At first order, terms that contribute an additional power of the scaling parameter are dropped. The variance-dependent coefficients are therefore evaluated at their initial state, leaving a stochastic integral with fixed initial variance and loading functions. Integrating this approximation recovers the stated linear perturbation around initial forward variance. This is a local expansion in the scaling parameter, rather than an exact solution of the full nonlinear dynamics. The document presents an algebraic derivation but gives no calibration, numerical example, or discussion of approximation error.
Key ideas
- Write the stochastic differential equation in integral form before expanding it.
- Substitution expresses the state dependence inside the volatility loading functions.
- A first-order approximation drops terms with higher powers of the scaling parameter.
- The resulting stochastic integral uses the initial forward variance and loadings.
- The expansion is approximate and does not establish its error for particular model parameters.
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# Bergomi Volatility Model
# Bergomi Volatility Model
I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating respectively over $\xi_{t}^{T}$ and $W_{t}^{k}$.
$\begin{array}{l} \text {The dynamics are represented as : }\left\{\begin{array}{l} d S_{t}^{\omega}=(r-q) S_{t}^{\omega} d t+\sqrt{\xi_{t}^{t}} S_{t}^{\omega} d Z_{t} \\ d \xi_{t}^{T}=\omega \xi_{t}^{T} \sum_{k} \lambda_{k t}^{T}\left(\xi_{t}\right) d W_{t}^{k} \end{array}\right.\\ \text { At order } 1(\text { Using one factor}) \text { in } \omega: \quad \xi_{t}^{T}=\xi_{0}^{T}\left(1+\omega \int_{0}^{t} \sum_{k}\left(\lambda_{k \tau}^{T}\right)_{0} d W_{\tau}^{k}\right) \end{array}$
With The instantaneous variance of the spot process such $\xi_{t}^{t}$, $S_t$ the stock price, $w$ a scaling factor,$d W_{\tau}^{k}$ correlated standard brownian motions.
Could someone help me to understand this step thank you.
## Answer by Antoine Conze (score 4, accepted)
https://quant.stackexchange.com/a/64228
SDE for forward variance: $$ d \xi_t^T = w \xi_t^T \sum_k \lambda_{kt}^T(\xi_t^T) dW_t^k $$
Integrate: $$ \xi_t^T = \xi_0^T + w \int_0^t \xi_{\tau}^T \sum_k \lambda_{k{\tau}}^T(\xi_{\tau}^T) dW_{\tau}^k $$
Plug into RHS of SDE: $$ d \xi_t^T = w \left(\xi_0^T + w \int_0^t \xi_{\tau}^T \sum_k \lambda_{k{\tau}}^T(\xi_{\tau}^T) dW_{\tau}^k\right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \sum_k \lambda_{kt}^T\left(\xi_0^T + w \int_0^t \xi_{\tau}^T \sum_k \lambda_{k{\tau}}^T(\xi_{\tau}^T) dW_{\tau}^k\right) dW_t^k $$
Expand at first order in $w$: $$ d \xi_t^T \approx w \xi_0^T \sum_k \lambda_{kt}^T\left(\xi_0^T\right) dW_t^k $$
Integrate: $$ \xi_t^T \approx \xi_0^T + \int_0^t w \xi_0^T \sum_k \lambda_{k{\tau}}^T\left(\xi_0^T\right) dW_{\tau}^k = \xi_0^T \left(1 + w \int_0^t \sum_k \lambda_{k{\tau}}^T\left(\xi_0^T\right) dW_{\tau}^k \right) $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.