Abstract
The pricing of exotic options with a payoff involving asset prices at different times requires a model capable of explaining the covariance of underlying securities. Assuming that asset returns are ruled by a Brownian motion with drift is convenient for mathematical developments. However, this model does not replicate the time dependence observed for some asset classes, as underlined by Willinger et al. (Finance Stoch 3:1–13, 1999). This point is one of the main motivations justifying the study of fractional Brownian motion (fBm) seen in Chap. 6. Gaussian fields offer a natural extension of fBm in which the marginal distribution is Gaussian with various covariance structures.
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Notes
- 1.
Except for the Ornstein–Uhlenbeck process, which is a Markov process that may be reformulated as a homogeneous Gaussian field. But in general, a time Gaussian field is not a Markov process.
- 2.
Here the yield \(y_{t}^{(j)}\) is defined as the cumulative return up to time t of the jth asset.
- 3.
Note that Σ may eventually be replaced by a time-dependent matrix Σ(t) in order to replicate seasonality effects in the covariance. In this framework, the pricing by simulations would still be possible.
- 4.
- 5.
Note that ν(⋅) can be any measure on \(\mathbb {R}\), not necessarily a probability measure (e.g., Lebesgue measure).
- 6.
As mentioned in Sect. 7.2, homogeneous fields have constant asymptotic variance. This feature makes them more suitable for the modeling of commodities or interest rates.
- 7.
The discount rate is set to r = 5% and c 1 = c 2 = 0.
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Hainaut, D. (2022). Gaussian Fields for Asset Prices. In: Continuous Time Processes for Finance. Bocconi & Springer Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-031-06361-9_7
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