Heston Transition Probabilities When Volatility Is Unobserved

Summary

The document asks whether the Heston model’s characteristic function can give the conditional distribution of a future asset price when the current price is known but the current variance is not. It explains that, when both current price and variance are specified, the characteristic function can be used to recover the future distribution, for example through an inversion formula for the cumulative probability.

The key limitation is that the Heston transition law depends on both state variables. Knowing the price alone does not specify the current variance, so the conditional probability requires additional information about variance, such as its conditional distribution given the observed price, and integration over that uncertainty. The document poses this issue but provides no solution, data, or worked calculation for constructing that distribution. Its scope is therefore a focused modeling question rather than a complete estimation method.

Key ideas

• The Heston model’s characteristic function can be inverted to obtain a future price distribution when the current state is specified.
• The current state includes both asset price and variance.
• Conditioning only on price leaves uncertainty about variance that must be modeled or integrated out.
• The document raises this hidden-state problem but does not give a method for resolving it.

Tags

# Transition densities in the Heston model


# Transition densities in the Heston model












Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u s_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the process $S$ at time $t$, i.e.: $$ \mathbb{P}(S_T > x) = \frac{1}{2} + \frac{1}{\pi}\int_{0}^{\infty} Re\bigg(\frac{e^{-i u \log(x)} \Phi_{T,0}(u)}{i u }\bigg) du $$ (can be found in Carr Madan FFT approach to option pricing for example).

So, once we know the characteristic function of the process, we have all the information on the distribution and on the density of the process at a given time $t$, since we need only the information of the initial price $S_0$, and the initial volatility $V_0$ (here I am using an easy possible example, the Heston model). $$ \frac{dS_t}{S_t} = r dt + \sqrt{V_t} dW_t$$$$ dV_t = \kappa ( \theta - V_t) + \sqrt{V_t} dB_t$$ $$d<W,B>_t = \rho $$ Is it possible, using the characteristic function, to compute the following transition probability? $$ \mathbb{P}(S_T \leq x | S_t = y) $$ Of course, in the condition where I know $S_t$ and $V_t$ it's easy using the characteristic function, but I know only $S_t$ and I don't have any information on $V_t$.

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