Wiener Process Interpolation Variance

Article Quant Q&A · Author: Markov

Summary

The document examines an interpolation rule for a Wiener process whose volatility is piecewise constant between known endpoint values. The proposed rule scales the endpoint increment by the square root of a ratio involving accumulated variance terms. The accepted answer checks whether this construction matches the variance that the process definition implies for an intermediate-time increment.

For constant volatility on the interval, the increment variance from the left endpoint to an intermediate time is volatility squared times the elapsed time. The answer points out that the stated denominator does not generally produce this result and proposes using the interval’s variance, volatility squared times its full duration, as the denominator. Under that choice, the interpolated increment has the theoretically expected variance throughout the interval. The explanation addresses variance consistency for this specific setup; it does not provide a survey of interpolation methods for stochastic volatility or discuss other distributional properties.

Key ideas

For piecewise constant volatility, increment variance over an interval equals volatility squared times its duration.
An interpolation rule can be checked by calculating the variance of its intermediate-time increment.
The proposed denominator should represent the full interval variance for the stated setup.
Matching variance does not by itself compare every property of alternative interpolation methods.

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Statistics Volatility Derivatives pricing

Full text

# What kind of interpolation is this?


# What kind of interpolation is this?












I have Wiener process $W_t=\int_0^t\sigma(t)dB(t)$ where $B(t)$ - Brownian Motion and $\sigma(t)$ - piecewise constant function. I also take $t_k<t<t_{k+1}$ where I know the values of $W_{t_k}$ and $W_{t_{k+1}}$. I found implementation of some kind of interpolation but I don't understand how it is determined. It works as follows:

- $D = \sigma^2(t_{k+1})\times t_{k+1} - \sigma^2(t_{k})\times t_{k}$

- $N=\sigma^2(t)\times t -\sigma^2(t_k)\times t_k=\sigma^2(t_k)\times (t-t_k)$

- $W_t = \sqrt{N/D}\times W_{t_{k+1}} + (1-\sqrt{N/D})\times W_{t_k}$

And generally I would like to know what are the popular methods of interpolation for Wiener Process with stochastic\piecewise constant volatility.

## Answer by Kurt G. (score 2, accepted)

https://quant.stackexchange.com/a/69936

I don't understand why they not just use $$\tag{1} D=\sigma^2(t_k)(t_{k+1}-t_k) $$ which leads to the theoretically correct variance of $W_t-W_{t_k}$.

Rewriting (3) gives for the increment over the interval $[t_k,t]$ $$ W_t-W_{t_k}=\sqrt{N/D}\,(W_{t_{k+1}}-W_{t_k})\,. $$ This has a variance of $$\tag{2} \mathbb E\Big[(W_t-W_{t_k})^2\Big]=\frac{N}{D}\sigma^2(t_k)(t_{k+1}-t_k )=\frac{\sigma^2(t_k)(t-t_k)}{\sigma^2(t_{k+1})\,t_{k+1}-\sigma^2(t_k)\,t_k}\sigma^2(t_k)(t_{k+1}-t_k )\,. $$ From $W_t=\int_0^t\sigma(s)\,dB_s$ we should theoretically get $$\tag{3} \mathbb E\Big[(W_t-W_{t_k})^2\Big]=\int_{t_k}^t\sigma^2(s)\,ds=\sigma^2(t_k)(t-t_k)\,. $$ The last equals sign follows from the assumption of piecewise constancy of $\sigma\,.$

Obviously if (1) is used instead then (2) and (3) agree for all $t\in[t_k,t_{k+1}]\,.$

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.

Checking Wiener Process Interpolation Against Its Increment Variance

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