Recovering an Intermediate Return Distribution by Deconvolution
Summary
The document considers whether two observed option-implied distributions, at an early expiry and a later expiry, can reveal the return distribution over the interval between them. It expresses the total log return as the sum of an early-period return and a subsequent-period return, then uses characteristic functions to relate the distribution of that sum to the distributions of its components.
Under an independence assumption, the characteristic function of the later increment is obtained by dividing the total-return characteristic function by that of the early increment. An inverse Fourier transform then yields the increment’s density, and the response notes that characteristic functions may be computed numerically for affine jump-diffusion models with time-varying parameters. The independence assumption is essential to this factorization; the document does not establish it for a general Markov diffusion. It also does not resolve conditional distributions given the early return, or discuss numerical stability when the denominator is small or zero.
Key ideas
- Independence makes the characteristic function of a sum equal to the product of the component characteristic functions.
- The later-period return characteristic function can be recovered by dividing the total-return function by the early-period function.
- An inverse Fourier transform can convert the recovered characteristic function into a density.
- The method does not identify conditional distributions from the marginal expiry distributions alone.
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Full text
# Early expiry options/deconvolution
# Early expiry options/deconvolution
Take a standard Markov setting (with assumptions as needed)
$$ dX_t = \mu(t,X_t)dt + \sigma(t,X_t) dW_t $$
Assume, you can get the distribution (i.e. option prices) for $t_1$ (the early expiry) and $t_2$. What I am really after is what can be said about $X$ in in that interval. I was thinking we can get the distribution of $\log X_{t_2} = \log X_{t_1} + \log X_{t_2-t_1}$ via deconvolution of the characteristic functions.
Is there anything else we can say, maybe about the conditional distributions? This is probably a basic question. Would appreciate if someone could point me towards the right approach here.
## Answer by Kermittfrog (score 1, accepted)
https://quant.stackexchange.com/a/68227
I think that this is a nice question. I have thought about this for a bit, but can only offer the following so far - maybe it helps the discussion.
Let's call the period 1 log return $x_1$, the period 2 log return $x_2$, and the joint return $y=x_1+x_2$.
If we assume independence of $x_1$ and $x_2$, we get for the characteristic function $\phi(t)$
$$ \begin{align} \phi_y(t)&\equiv\mathrm{E}\left(e^{ity}\right)\\ &=\mathrm{E}\left(e^{it\left(x_1+x_2\right)}\right)\\ &=\mathrm{E}\left(e^{itx_1}\right)\mathrm{E}\left(e^{itx_2}\right) \\&=\phi_{x_1}(t)\phi_{x_2}(t) \\ \Rightarrow \quad\quad\quad \phi_{x_2}(t)&=\frac{\phi_y(t)}{\phi_{x_1}(t)}\\ \Rightarrow \quad\quad\quad f(x_2)&=\frac{1}{2\pi}\int e^{-itx_2}\frac{\phi_y(t)}{\phi_{x_1}(t)}\mathrm{d}t\ \end{align} $$
which can be performed numerically. For $X_t$ following an affine jump diffusion with possibly time dependent parameters, the characteristic function(s) can be obtained numerically.
I do not know what else we can say about the conditional distribution $f(x_2|x_1)$, though.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.