Implied Probability Density Function of Stock Returns

Dive into the intricate world of options pricing where we elegantly derive the distribution of stock moves as implied by option prices. This post is essential for options traders, as it contrasts personal views against the odds the market has already priced in. We explore risk-neutral probabilities, dissecting how option prices reflect pseudo-probabilities and delve into the mathematical details of deriving the probability density function $\phi$ and cumulative distribution function $\Phi$ from the options market. $$\begin{aligned} C(S, t, K, T) &= e^{-r\tau} \left[\int_K^\infty S_T \, \phi(S, t, S_T, T) \, dS_T - \int_K^\infty K \, \phi(S, t, S_T, T) \, dS_T\right] \\ \implies \frac{\partial C}{\partial K} &= -e^{-r \tau} \int_K^\infty \phi(S, t, S_T, T) \, dS_T \\ &= -e^{-r \tau} \Phi(S, t, K, T) \end{aligned}$$ Additionally, we enhanced this post with practical Python code snippets, including a section on using Gaussian filters and cubic interpolation to create smooth distributions from discrete market data. Ultimately, the post reveals intriguing insights into market sentiment, like the skewed nature of the PDF and briefly dive into betting in different parts of the distribution.

Gamma Exposure

Back in 2017, SqueezeMetrics wrote a white paper detailing the impact of the options market on financial assets. Specifically, the paper describes that, at times, the options activity within an underlying asset, like a company’s stock, can exert significant influence on the price of the underlying. This phenomenon...

The Variance Day Model

• Volatility is defined as the standard deviation of price returns.
• Variance is the square of the standard deviation.
• Variance is proportional to time as \(\sigma^2 \tau\)

In a previous post, we’ve established that

• Time passes too fast over weekends in a calendar day model
• Time passes...
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The Concept of Variance Time

Tick-tock, tick-tock. The rhythmic sound of a clock’s second hand moving steadily around its face is a sound that has become synonymous with our perception of time. We’ve been conditioned to think of time as a linear progression, measured in days, hours, minutes, and seconds.

While our day-to-day lives is...