In one of our discussions with our advisor Rob Navin, we were exploring Monte Carlo methods for pricing derivatives. 🧵

As we dug deeper into the math, Rob pointed out a subtle but profound insight:

"Given a finite set of Monte Carlo paths with a well-defined standard deviation, the equivalent continuous-time model (diffusion) can always be represented by a unique PDE that depends only on drift and volatility because of the central limit theorem”

These two parameters (drift μ, volatility σ) define the first two moments of the underlying process. So whether you approach it by simulating paths (Monte Carlo) or solving equations (PDE), you’re really describing the same process - just from two different angles. From Monte Carlo Paths to a PDE: Why Drift & Vol Alone Define the Limit

Start with a discrete-time Markov chain on a time grid t_k, with increments

ΔX_k = X_{k+1} - X_k.

If we run enough Monte Carlo paths and compute local conditional moments, something deep happens…

Dispersion Trading in Practice: The “Dirty” Version 🧵

Everyone loves the textbook dispersion trade: short index vol, long single stocks, vega-neutral.

But the people who actually make money run the dirty version.

Here’s why theory dies in backtests and practice wins in P&L In theory, dispersion trading isolates the correlation risk embedded in index options.

The clean (academic) setup:

Short index volatility (e.g., SPX options)

Long component stock volatility (options on the S&P constituents)

Sized vega-neutral across legs so that portfolio exposure to overall volatility level cancels out.

Formally, for an equal-weighted index of NNN stocks:

Funding risk in option pricing, why Rho (interest rate sensitivity) can sometimes dominate Vega (volatility sensitivity), and how interest rate spreads eat into
edge. 🧵 with simulation example.

Another "nugget" from our webinar with @KrisAbdelmessih
1. Option Values Depend on Funding

When you trade options, you’re not operating in a frictionless Black–Scholes world.

If you buy a call and short the stock (classic delta hedge), your broker will pay you interest on the short-sale proceeds — but usually at a discounted rate below the official benchmark (say, SOFR).

If you sell a call and buy the stock to hedge, you must borrow cash to finance your long stock position — and your broker will charge you above the benchmark.

This spread between what you earn on shorts and what you pay on longs is your funding cost, and it directly affects the option’s fair value.

In one of our webinar with @KrisAbdelmessih he discussed "How to extract the implied probability distribution of a stock’s future price using call spreads and butterflies" Long Thread🧵 Why Use Options to Infer Distribution?

The stock price alone tells you the market’s consensus expected value. But that hides important information:

Uncertainty (volatility)

Skew (asymmetry)

Tails (crash/rally risk)

Options, however, are priced off the entire distribution of future prices. By reverse-engineering option prices, especially call spreads or butterflies, we can back out what the market thinks about the future not just the average, but the entire shape.

Important points on Delta Hedging that every Volatility trader should consider 🧵

1/n

Delta Hedging and Matching Maturity

To hedge an option's delta effectively, use a forward with the same maturity. Using different maturities or stock introduces dividend risk, as forwards exclude interim dividend benefits.

This distinction helps prevent potential miscalculations in P&L.
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Join us for an exclusive Derivative Pricing Masterclass with Dr. Rob Navin – Founder & CEO of Real Time Risk Systems and had co-founded an alternative asset fund that grew from $150M to $1.1B AUM. A Ph.D. in Theoretical Particle Physics from Caltech.

Time -7:30 PM IST/2:00 PM GMT/ 10:00 AM EST

Date - 24h August 2025

Register Now - lnkd.in/gKamMDnn

Use the Coupon code "RTRS20" to get a 20% early bird off.

Key takeaways you’ll get:
How overnight grid computation can power real-time intraday pricing
Using vector-valued Taylor series expansions for speed and accuracy
Keeping “architecture purity” to avoid performance bottlenecks
Why this approach lets quants innovate without slowing down production
Real-world examples, including complex instruments like convertible bonds

(picture credit- Trading Volatility by Colin Bennett) 2/n
Dividend Risk in Forward Contracts

A forward contract equates to holding stock but shorting dividends payable before maturity.

This matters because a stock drops by the dividend value on the ex-date, leading to a dividend risk equal to the option's delta if you hedge with mismatched instruments.

Impact of cross sectional Factors on Implied and realized volatility of equity options 🧵

Reference paper Link - papers.ssrn.com/sol3/papers.cf… Stock Characteristics and Implied Volatility

Beta:

Implied volatility increases with beta, suggesting that market participants view high-beta stocks as riskier.

Important point on Delta Hedging that every Volatility trader should consider🧵

Do read Trading Volatility by Colin Bennett, if you are want to learn more about Vol trading

Subscribe to our Youtube Channel - youtube.com/@QuantInsider Delta Hedging and Matching Maturity

To hedge an option's delta effectively, use a forward with the same maturity. Using different maturities or stock introduces dividend risk, as forwards exclude interim dividend benefits.

This distinction helps prevent potential miscalculations in P&L.

Carry Trade 101 🧵

Subscribe to our Youtube Channel - youtube.com/@QuantInsider 1/12

Core Idea

“Borrow low, lend high.” A carry trade earns the yield differential (carry) between two assets or currencies after financing costs.

Implementing the Kelly Criterion in Continuous-Time 🧵

Subscribe to our Youtube Channel for more content of Quantitative Finance - youtube.com/@QuantInsider 1. Market set-up

Consider a money–market account earning a constant short rate r and mmm risky assets whose (ex-dividend) prices follow an Itô diffusion

Traders still talk about “Black-Scholes” vol, but almost nobody pushes the textbook equation into production to value real positions. Here’s why. 🧵 Textbook Assumptions and How the Real World Breaks Them

Constant volatility

Implied volatility is anything but constant. In 2024, 1-month ATM SPX vol averaged 15 %, while 20-delta puts printed 24 %—a 60 % “smile” markup.

Must know 18 Options and other Derivatives related Bloomberg function keys 🧵

Core Options Functions

OMON (Option Monitor)

The Option Monitor (OMON) function on Bloomberg provides real-time pricing, market data, and derived data for call and put options. It also allows you to analyze options and export data to Excel.

• Usage: Enter [Ticker] [EQUITY] OMON

OSA (Option Scenario Analysis)

Generates profit/loss tables and graphs by allowing you to input various option and underlying stock positions.

OV (Option Valuation)

• Provides pricing, volatility, and Greek data using models like Black‑Scholes, Trinomial, Roll‑Geske, etc.

The OV function on Bloomberg helps you value options by providing price and volatility data. You can use the OV function to calculate the OPM value for a loaded option

OVX (Exotic Option Valuation)
• Values exotic option types (e.g., chooser, binary, lookback) once the underlying is loaded.
The OVX function on Bloomberg is used to value

OPX function on the Bloomberg terminal helps traders analyze options expirations. It allows users to view options by expiration date, and sort them by volume and strike value. II. Supplementary Analytics Functions

DES (Description)
• Provides detailed descriptive information on a selected option.

QRM (Trade Recap)
• Shows a recap of recent trades for the option.

TSM (Trade Summary Matrix)
• Displays a matrix summarizing recent trade data.

GIP (Intraday Price Graph)
• Plots an intraday graph for the option.

GPO (Bar Chart)
• Generates a bar chart view of option pricing data.

Volatility and Correlation Analysis (VCA)
The Volatility and Correlation Analysis (VCA) tool on the Bloomberg Terminal can be used to analyze skew for major stock indexes. This tool allows users to analyze volatility across various securities.

HIVG (Historical Implied Volatility Graph)
• Graphs historical trends in the option’s implied volatility.

Mathematical Solution for Hedging When Implied Volatility is Stochastic 🧵

Implied volatility is a crucial factor in the pricing and hedging of options. Unlike historical volatility, implied volatility is derived from market prices of options and reflects the market's expectations of future volatility. When implied volatility is stochastic, meaning it changes unpredictably over time, hedging strategies become more complex.

When implied volatility is stochastic, traditional hedging techniques must be expanded to account for the additional risk factor.

Here's a detailed explanation of the mathematical approach to hedging under stochastic implied volatility conditions, primarily focusing on delta and vega hedging.

Directional Options trading by creating delta exposure is not as simple as it seems.

Let’s break this down step by step for a clear understanding.🧵

(Do Subscribe to our Youtube channel - ) for more content of Quant trading and Research. youtube.com/@QuantInsider Why Black-Scholes Delta Isn't Always Accurate?

The Black-Scholes model, a common tool in option pricing, assumes a smooth and constant volatility across all strike prices and maturities. However, in real-world markets:

Detailed overview of different Methods for "Change Point Detection" for identifying changes in market regimes, volatility shifts, or other significant events.🧵

Statistical Tests: 1.Cumulative Sum (CUSUM):
(Subscribe to our Youtube channel - ) youtube.com/@QuantInsider Use Case: Quickly detects shifts in mean returns or volatility. Ideal for simple, real-time signals where efficiency is key.

Why 0.50 Delta ≠ ATM? 🧵

A common misconception is that an at-the-money (ATM) option always has a delta of 0.50. However, this isn't always the case due to factors like the distribution of potential asset returns and the skewness of that distribution. In the Black-Scholes model:

Delta for a Call Option:

Here:

S: Current stock price
K: Strike price
T: Time to expiration
r: Risk-free interest rate
σ: Volatility

Volatility traders have Vega exposure, and the magnitude of this exposure determines the slope of their P&L. In contrast, for directional traders, the P&L slope is driven by their Delta exposure.

So understanding the sources of Vega convexity is crucial for vol traders 🧵 Volatility Smile and Term Structure:

Implied volatility typically varies across strike prices (volatility smile/skew) and maturities (term structure). These variations introduce nonlinearities in Vega, especially for options further out-of-the-money or near their expirations.

Impact of cross sectional Factors on Implied and realized volatility of equity options 🧵

Reference paper Link - papers.ssrn.com/sol3/papers.cf… Stock Characteristics and Implied Volatility

Beta:

Implied volatility increases with beta, suggesting that market participants view high-beta stocks as riskier.

Should hedging affect how we look at volatility? 🧵

Hedging and volatility are intrinsically linked, such that the notion of volatility becomes meaningless without considering the act of hedging itself.

Reference - Derivatives Models on Models by Espen Gaarder Haug The derivatives market is a fundamental, "primitive" concept in finance.

This contrasts with traditional views that treat probability distributions and statistical expectations as primary tools for understanding and predicting market behavior.

Discrete Dynamic Delta Hedging under Geometric Brownian Motion: A Practical Implementation (With Python code) 🧵

This thread explores the effectiveness of discrete dynamic delta hedging under GBM

Like retweet and comment "Dynamic Hedging" to receive the code Dynamic delta hedging is a fundamental strategy used by option traders to mitigate the risk associated with movements in the underlying asset's price.

Important point on Delta Hedging that every Volatility trader should consider🧵

Do read Trading Volatility by Colin Bennett, if you are want to learn more about Vol trading Delta Hedging and Matching Maturity

To hedge an option's delta effectively, use a forward with the same maturity. Using different maturities or stock introduces dividend risk, as forwards exclude interim dividend benefits.

Detailed Thread on Option Pricing with Deep Learning

Original paper link -

Like, Retweet and comment "OPDL", to receive the code for the Deep Learning for Option Pricing cs230.stanford.edu/projects_fall_… In the paper Three neural network models are discussed
1)MLP1.
2)MLP2.
3)LSTM.