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.

Unlocking High Performance in Python$:$ Vectorization, Memory Management, JIT, and Cython

Explore the art of optimizing Python code for performance when handling large datasets or computationally intensive tasks. This post covers everything from vectorized operations using NumPy and Pandas, understanding Python's memory management, to advanced techniques like JIT compilation and Cython. Whether you're comparing data structures or employing JIT for complex computations, this guide offers insights into achieving significant performance improvements. We also include a practical Jupyter Notebook for hands-on experience.

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...

January 2023 Update

Is Santa Claus real?

In the September update, we said that Santa was coming back this year for a variety of reasons. After a bout of mostly dampened volatility starting October 7th, the S&P 500 has seen the largest 4 day rally since 2022. And, over the course...

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...

October 2023 Update

Act of God

Despite what we hear from many prominent traders like Ray Dalio and Paul Tudor Jones, we don’t believe that the events on October 7th will lead to a wider regional war. None of the big players in the region (Iran, Saudi Arabia, & Iranian proxies like...

September 2023 Update

If I said June and it doesn’t happen

It doesn’t mean that there must be higher odds in July. August isn’t more probable. Flows are tied to windows. If they don’t transpire during those windows, it actually tells you something. The probability of something happening is conditional on some...

July 2023 Update

It’s not 2008 and you are not in the Big Short part II

Since our last post, despite the bearish macro backdrop, the market is up 15%, erasing losses since the Russian invasion in Feb2022 while interest rates hiked at historic pace from 0.25% to 5.25%. We stated very...

March 2023 Update

It’s not 2008 and you are not in the Big Short

Contrary to popular beliefs, the feds telegraphed this, understood in real time what was happening with SVB and Signature bank. They were really quick, as soon as the decline happened, they come in and secure depositors. We believe...