Empirical Benchmarks for Option Pricing Models
Summary
The document raises a research concern: option pricing models may be proposed without being tested against market data or compared with established alternatives. It notes that affordable options data can be difficult to obtain, while arguing that empirical assessment is important. As examples, it contrasts a well-known GARCH option pricing paper described as lacking an empirical section with a GARCH study that reports its data and compares its model with alternatives.
The request focuses especially on exchange-traded equity, ETF, and index options, including both American and European contracts, and welcomes papers that either test a pricing model or benchmark existing models. The only suggested lead is a study comparing Ho–Lee and Black–Derman–Toy approaches for Eurodollar futures options. The document does not summarize that study's methods, data, or findings, and provides no broader bibliography, so it serves mainly as a research starting point rather than evidence about model performance.
Key ideas
- Empirical option pricing studies should describe their market data and compare models against benchmarks.
- Access to affordable options data can make empirical evaluation difficult.
- The stated research interest centers on exchange-traded equity, ETF, and index options.
- A cited lead compares Ho–Lee and Black–Derman–Toy models for Eurodollar futures options, but no findings are given.
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Full text
# Fractional Brownian Motion's Covariance Proof
# Fractional Brownian Motion's Covariance Proof
Let's have the non independent Brownian motion such :
$B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$
Rewriting it as(For $r2 > r1 $) :
$\begin{aligned} B_{H}(r)=& \frac{1}{A(H)} \int_{-\infty}^{0}\left\{(r-s)^{H-1 / 2}-(-s)^{H-1 / 2}\right\} \mathrm{d} B(s) \\ &+\frac{1}{A(H)} \int_{0}^{r}(r-s)^{H-1 / 2} \mathrm{~d} B(s) \end{aligned}$
$E((B_H(r2) - B_H(r1))^2) = E(\frac{1}{A(H)^2} (\int_{-\infty}^{0}\left\{(r2-s)^{H-1 / 2}-(r1-s)^{H-1 / 2}\right\} \mathrm{d} B(s) \\) + \int_{0}^{r1}\left\{(r2-s)^{H-1 / 2}-(r1-s)^{H-1 / 2}\right\} \mathrm{d} B(s) + \int_{r1}^{r2}\left\{(r2-s)^{H-1 / 2}\right\} \mathrm{d} B(s))^2)$
Trough Ito's isometry we have :
$\implies \frac{1}{A(H)^2} (\int_{-\infty}^{r1}\left\{(r2-s)^{H-1 / 2}-(r1-s)^{H-1 / 2}\right\}^2 \mathrm{d} s \\)+ \int_{r1}^{r2}\left\{(r2-s)^{2H -1}\right\} \mathrm{d} s))$
$\implies \frac{1}{A(H)^2} (\int_{-\infty}^{r1}\left\{(r2-s)^{H-1 / 2}-(r1-s)^{H-1 / 2}\right\}^2 \mathrm{d} s \\)+ \frac{(r2-r1)^{2H}}{2H})$
How can we rewrite the first integral,using a variable change such :
$\frac{1}{A(H)^2} (\int_{-\infty}^{r1}\left\{(r2-s)^{H-1 / 2}-(r1-s)^{H-1 / 2}\right\}^2 \mathrm{d} s \\)+ \frac{(r2-r1)^{2H}}{2H})$
becomes :
$\frac{(r2-r1)^{2H}}{A(H)^2} \left(\int_{-\infty}^{0}\left((1-u)^{H-\frac{1}{2}}-(u)^{H-\frac{1}{2}}\right)^{2} d u+\frac{1}{2 H}\right)$
Thank you for your help.
**(For brevity I considered $r2 > r1$,which may not always be the case in a covariance matrix )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.