Local Volatility and Stochastic Volatility Models Without Closed-Form Densities
Summary
The discussion asks which Itô diffusion models used in finance lack a known terminal distribution, despite familiar examples such as geometric Brownian motion and the Cox–Ingersoll–Ross process having tractable distributions. The answer points to local volatility models whose state- and time-dependent diffusion coefficient is fitted to market data. A complex fitted surface, including one approximated with a truncated Fourier representation, generally offers no closed-form terminal law.
The answer also broadens the setting to a two-dimensional process containing both an asset price and its variance. The Heston stochastic volatility model is given as an example where the terminal price density may not be available in closed form, even though a transform such as its Laplace transform can be. The responses are illustrative rather than a detailed catalogue or proof: the availability of a transform or numerical methods can still make pricing feasible, and “unknown” here means lacking a convenient closed-form distribution.
Key ideas
- A fitted local volatility surface can produce a diffusion with no closed-form terminal distribution.
- Approximating a complex local volatility surface with Fourier terms does not generally yield a tractable density.
- A two-dimensional asset-price and variance process is another common source of intractable terminal distributions.
- The Heston model is cited as having no closed-form terminal price density while allowing a closed-form transform in some cases.
- Lack of a closed-form density does not rule out numerical pricing methods.
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# Itô diffusion processes in finance with unknown distribution at a terminal value
# Itô diffusion processes in finance with unknown distribution at a terminal value
In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes Monte Carlo simulations viable if not necessary even for the calculation of European options.
However, all models used in mathematical finance that come immediately to my mind are processes for which the distribution is known, such as geometric Brownian motion and the Cox-Ingersoll-Ross process.
What would be examples for diffusion processes, preferably widely used, of the above form for which the distribution of $X_T$ is unknown?
## Answer by Brian B (score 1, accepted)
https://quant.stackexchange.com/a/9610
Any of a wide variety of local vol models, where (from your equation) $b(\cdot,\cdot)$ is some fitted surface, are unlikely to have closed-form solutions for the terminal distribution. Indeed it's well-known that these models tend to have very unusual forward term structures of volatility.
As a specific example, take $b(\cdot,\cdot)$ to be an approximation derived from the first few terms of the 2-d Fourier representation of some high-resolution local vol fit $\tilde{b}(\cdot,\cdot)$.
Our SDE becomes $$ dX_t=\mu(t)dt+dB_t\sum_{\vec{k}} \omega_{\vec{k}} \exp\left( i\vec{c}_{\vec{k}}\cdot \binom{t}{X_t} \right) $$
I'm pretty sure there will be no closed form.
## Answer by Richi Wa (score 3)
https://quant.stackexchange.com/a/9581
If you allow $X_t$ to be two dimensional then a model with a stock price $X_t^1$ and its variance process $X_t^2$ (stochastic volatility) would fit your definition.
In such cases to my knowledge we often don't have a closed form of the density of $X_T^1$ but in some cases we have a closed form of the Laplace transform.
An example is the Heston model.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.