Derivatives
Estimating Volatility from Data
How to Assess a Stock’s Risk Based on Historical Data We know that in finance, “risk” usually refers to volatility, i.e., uncertainty regarding changes. Here, I want to […]
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Understand stock price as a forecast
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Derivatives
The Distribution of Rate of Return
Suppose an investment fund (or a development company) promises you an average return of 14% per year. To convince you, they show that over the last 5 years […]
Derivatives
Lognormal Property – Example
აქციის ფასის მოძრაობის მოდელი აკეთებს დაშვებას, რომ დროის ძალიან მცირე მონაკვეთში ფასის %-ული ცვლილება ხასიათდება ნორმალური დისტრიბუციის ალბათობით. განვსაზღვროთ: μ – (Expected return on stock per year – continiously […]
Derivatives
Correlated Processes
Why diversification fails during market crises… Assume two variables, (x_1) and (x_2), follow generalized Wiener processes: dx1 = a1·dt + b1·dz1 and dx2 = a2·dt + b2·dz2 The […]
Derivatives
Lognormal Property of Stock Price
1. The starting point — stock price process We assume that the stock price SSS follows a geometric Brownian motion: dS = μS*dt +σS*dz where: find also: Geometric […]
Derivatives
Ito’s Lemma in Forwards
To describe the evolution of a derivative of stock price, we use Ito’s Lemma. For example, through it we can express the fair forward price of a stock […]
Derivatives
Geometric Brownian Motion of Stock Price
A common model for the price movement process of non-dividend-paying stocks is the Geometric Brownian Motion (Robert Brown (1773–1858) was a Scottish botanist). Initially, Brown observed that dust […]
Derivatives
Ito’s Lemma & Wiener Process
Without the discoveries of the Japanese mathematician Kiyoshi Itô (伊藤 清, 1915–2008), the Black–Scholes–Merton model — and therefore the modern derivatives market — could not exist. Before moving […]
Derivatives
Wiener Process
If physicists use the Wiener process to describe the movement of molecules, for finance it is interesting because it helps us understand the price behavior of stocks and […]
Derivatives
Markov Process
Why is uncertainty considered proportional to the square root of time (√t)? A Markov process (named after the Russian mathematician Andrey Markov (1856–1922)) is a special case of […]
Derivatives
How Randomness Shapes Markets
🎲 Have you ever noticed how life, business, and markets share one thing in common — uncertainty? Finance doesn’t ignore that; it embraces it.John Hull’s classic work on […]
Derivatives
Risk-Neutral vs Replication in Option Pricing
Using the Risk-Neutral Probability Method in Option Valuation is a bit counterintuitive. Why doesn’t the option price depend on the expected movement of the underlying stock — its […]
Derivatives
Valuation of Interest Rate Swaps
When an interest rate swap is contracted, its initial value is zero. However, after some time passes, its value may become positive or negative for one of the […]
Derivatives
Extending SWAP Zero Curve
To value derivatives, traders rely on LIBOR (London Interbank Offered Rate) as the risk-free interest rate. The problem is that these rates are only available for up to […]
Derivatives
Expected Spot Price
How can we assess the market’s expectation about the future change in the price of an asset? For example, if today the spot price of one ton of […]
Derivatives
Futures on Commodities
The Futures Price Formula for Commodities Requires Additional Ingredients Such commodities can be predominantly investment-oriented (e.g., gold, silver) or predominantly consumption-oriented (e.g., copper, oil). The latter requires taking […]
Derivatives
Forward vs Futures Prices
Since forward pricing formulas are used to evaluate futures, the question arises: are futures and forwards priced the same? Theoretically, in the long run, it is said that […]
Derivatives
Valuing Forward Contracts
Before we move to valuing a forward contract, it’s important to understand the connection between the spot price of an asset and its forward price. Let’s start with […]