bartmelman/volatilitymodels

Comparing different volatility models

Market Description

• The price discovery process of stocks mean that stocks can be mean reverting.
• In aggregate, stocks increase in value

Stock Returns

• The returns of stocks are asymmetricly distributed.
• The average stock returns are positive

Difference between Stocks

Process of continuous incrementing stock returns vs $$ \begin{aligned} \textrm{profit 1 day} & = r\ \textrm{profit n days} & = \lim_{\epsilon \to 0} n \epsilon \ & = \ln{1+r} \end{aligned} $$

Proof $$ \begin{aligned} (1+\epsilon)^n & = 1 + r\ n & = \frac{\ln{1+r}}{\ln{1+\epsilon}}\ \lim_{\epsilon \to 0} n \epsilon & = \frac{\epsilon \ln{1+r}}{\ln{1+\epsilon}}\ & = \ln{1+r} \end{aligned} $$

Return adjusted profit $$ \begin{aligned} profit & = \frac{r-\ln{1+r}}{|r|} \end{aligned}
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Hold or rebalance

Hold $$ profit = r_{t-1} + $$

Rebalancing $$

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Ideal stock

• the expected profit based on the volatility should be higher than the volatility (mean reversion)
• compare 1 month ahead vol versus daily vol

Stock algorithm

• Rebalance portfolio with top 100 stocks

$$ \begin{aligned} R_t & = \sum_{i=1}^n w_i R_i \quad \textrm{s.t.} \quad \bm{w}' \iota = 1 \end{aligned} $$