u/askepticalbureaucrat

So, for ease and formatting, I made a quant stackexchange post here but I believe that I was incorrect here?

I wanted to output specifically the 2026 value of $\sim5.01\%$ for 2026, as seen in the above plot?

I believe that I have an error with SDE used $r = 4.65, \theta = 4.2$, and $\sigma = 0.55$, apparently in percentage points. That's internally possible, but then the reported standard deviation calculation is wrong in its interpretation:

$$\sqrt{\frac{(0.55)^2}{2(0.35)}}\approx 0.657$$

meaning $0.657$ percentage points if rates are measured in percentage points?

Also, the Hull–White equation that I derived isn't actually the standard Hull–White specification we're subsequently describing, as derivation starts from

$$dr_t=\kappa(\theta−r_t)dt+\sigma dW_t,$$

where $\theta$ is constant. That's essentially the Vasicek model, but the standard one-factor Hull–White is

$$dr_t=[\theta(t)−ar_t]dt+\sigma dW_t,$$

with a time-dependent drift chosen to fit today's initial yield curve?

My code:

import numpy as np

# Let's run a simulation for CIR (Cox-Ingersoll-Ross) and Hull-White models
np.newaxis
np.random.seed(42)

N = 216  # Monthly steps from 2008 to 2026
dt = 1 / 12

# 1. CIR Model simulation: dr = theta * (mu - r_t)dt + sigma * sqrt(r_t) * dW_t
theta_cir = 0.4
mu_cir = 3.3
sigma_cir = 0.35  # Must satisfy Feller condition: 2 * theta * mu >= sigma^2 to stay positive

r_cir = np.zeros(N)
r_cir[0] = 4.65

for i in range(1, N):
    # Ensure non-negative inside sqrt
    r_prev = max(0.0, r_cir[i-1])
    dr = theta_cir * (mu_cir - r_prev) * dt + sigma_cir * np.sqrt(r_prev) * np.sqrt(dt) * np.random.randn()
    r_cir[i] = r_prev + dr

# 2. Hull-White Model (Time-varying mean theta(t) or drift to fit term structure)
# Simplest time-varying mean formulation: theta(t) matches a shifting trend
theta_hw = 0.35
sigma_hw = 0.55

r_hw = np.zeros(N)
r_hw[0] = 4.65

for i in range(1, N):
    t_val = 2008 + i * dt
    # Let the long-term mean drift higher post-2021 to capture the inflation regime shift
    mu_t = 3.0 if t_val < 2021 else 4.2

    dr = theta_hw * (mu_t - r_hw[i-1]) * dt + sigma_hw * np.sqrt(dt) * np.random.randn()
    r_hw[i] = r_hw[i-1] + dr

print(f"CIR Model 2026 Terminal Value: {r_cir[-1]:.2f}%")
print(f"Hull-White (Regime-Shift) 2026 Terminal Value: {r_hw[-1]:.2f}%")
##CIR Model 2026 Terminal Value: 4.71%
##Hull-White (Regime-Shift) 2026 Terminal Value: 5.01%

Thanks! ❤️

For part (d) of my attempt seen attached, I believe based on the definitions and context in the document, $X_t = \int_0^t W_s \, ds$ is both, depending on how you view it:

• It can be treated pathwise as a Riemann integral: For each fixed sample path $\omega$, $W_s(\omega)$ is a continuous function of time, so the integral $\int_0^t W_s \, ds$ exists as a standard Riemann (or Lebesgue) integral.

• It is also an Itô process: As shown on the second page, its differential is given by:

$$d\left(\int_0^t W_s \, ds\right) = W_t \, dt$$

An Itô process is generally defined as a process of the form $X_t = X_0 + \int_0^t \mu_s \, ds + \int_0^t \sigma_s \, ds$. Here, it has a drift term $\mu_t = W_t$ and a diffusion term $\sigma_t = 0$ (meaning it is a special case where the stochastic differential term $dW_t$ has a coefficient of zero)

Therefore it can be viewed as both?

I made an attempt to describe this pretty artwork in Greek. Does my wording make sense? Is my handwriting legible? The η / υ / ι, ν, μ, ξ, ζ. θ, ρ, χ. ω letters especially?

I tried writing it more naturally, as I would in English.

Thanks 🧡

So, I made some notes regarding converting Stratonovich to Itô integrals, per the following:

• get the martingale property for pricing (for risk-neutral valuation): under the risk-neutral measure, discounted asset prices must be martingales (fair games with no predictable drift) to prevent arbitrage? as in \mathbb{E}\left[\int f(s) dW_s\right] = 0?

• non-anticipation: as financial markets operate in real-time, the trading strategies and pricing models can only use information known up to the present moment (mapped to the left-endpoint rule in Itô calculus, and not the future midpoint of an interval found with Stratonovich calculus)?

My question ultimately pertains to why Stratonovich integrals are used to begin with in finance? I've used them in physics, but can it be due to taking the limit of smooth physical systems driven by colored noise naturally converges to the Stratonovich interpretation via the Wong–Zakai theorem? Also, as the standard chain rule ((d(f(X)) = f'(X) \circ dX)) applies here, it helps make calculations easier (along with writing down coordinate-invariant equations or physical laws on curved spaces intuitive)?

I can't find that much information about this in my textbooks, or online, so wanted to get your industry experience.

Thanks! 🧡

​

I'm aware of Mandelbrot's 1963 paper of historical cotton price changes and where he that financial returns do not follow a light-tailed Gaussian distribution. Instead, he proposed that they follow a family of heavy-tailed models known as Stable Paretian distributions. I used a standard Cauchy here in this plot.

However, is VaR still used as it summarizes complex risk into one single number (we are 99% confident we won't lose more than $2 million tomorrow, etc.)? Also, it expresses risk in dollars (or local currency is used)?

I can imagine the heavy tails are kept in mind by many quants when running risk analysis, but was also wondering how the VaR provides risk metrics for banks nowadays? Is Expected Shortfall used more often?

Irish girl here (we're struggling with the same topic here) and I was wondering what your opinion was regarding this topic in the upcoming referendum?

Also, would one canton vote more in favour of it, vs another which would vote against it (eg., those in Romandie, etc.)

https://www.admin.ch/en/newnsb/kUHEB2euQu8MyfopZ16zE

Sending love from the sunburnt island 🔥🥀

I wanted derive the MLE for a financial maths problem, but I was unsure if my method was succinct? My assumption for the last part (to remove the summation symbol ($\sum$) and get the expression with $n$, we look closely at how each term inside the sum behaves as we sum it from $i = 1$ to $n$) and therefore

- $\-frac{1}{2}\ln(2\pi)$ does not depend on $i$ (it is a constant with respect to the index $i$) therefore becomes -$\frac{n}{2}\ln(2\pi)$

- $-\frac{1}{2}\ln(\sigma^2)$ this also does not depend on $i$ (it only depends on the variance parameter $\sigma^2$) and therefore becomes $-\frac{n}{2}\ln(\sigma^2)$

- the last part does depend on $i$ because of $R_i$ (each individual asset return changes for every index $i$)? and therefore is $-\frac{1}{2\sigma^2} \sum_{i=1}^{n} (R_i - \mu)^2$

Is this somewhat correct?

Επισκέφτηκα την Αθήνα πριν από δύο χρόνια και αγόρασα μια μικρή ελιά. Από τότε έχει μεγαλώσει πολύ! Μένω στην Ιρλανδία, οπότε πρέπει να τη διατηρώ σε μια πήλινη γλάστρα, χαχα. Έβγαλε και μικρές ελιές αυτό το καλοκαίρι! Αυτό θα εντυπωσίαζε έναν Έλληνα; 😊

Με αγάπη από την Ιρλανδία! 🇮🇪❤️🇬🇷

Barberstown Castle 📍

Not a cellphone in sight. Just living in the moment.

Στη χερσαία Αττική, φυτρώνουν άγριες ανοιξιάτικες ή χειμερινές ανεμώνες (γένος Anemone), όπως αυτές που εμφανίζονται σε ορεινές και ημιορεινές περιοχές (για παράδειγμα, στους πρόποδες ή στις πλαγιές του Υμηττού);

Θα ήθελα πολύ να δω τα κόκκινα, μπλε και μοβ λουλούδια. Επίσης, το μαύρο μουστάρδα (φυτρώνουν και κίτρινα λουλούδια στην περιοχή);

Θα νοικιάσω αυτοκίνητο από το αεροδρόμιο και θα οδηγήσω στην περιοχή τον Φεβρουάριο/Μάρτιο για να τα ψάξω. Επίσης, προτάσεις για όμορφους ελαιώνες (δηλαδή σε δημόσια γη);

Διάβασα ότι φυτρώνουν στην Πελοπόννησο, αλλά θα ήταν πολύ μακριά για μένα. 😭

Χαιρετισμούς από την Ιρλανδία! ☘️❤️

Είμαι Ιρλανδή και Μιλώ ελληνικά και λατρεύω την Αθήνα. Ήθελα να περάσω μερικά χρόνια εκεί στο μέλλον. Εργάζομαι εξ αποστάσεως και μου αρέσουν τα εξής:

- εγγύτητα στη θάλασσα

- εντελώς επίπεδο έδαφος σε αντίθεση με πολλά μέρη της Αθήνας (ή την λοφώδη Κηφισιά). Αυτό κάνει το περπάτημα στα καταστήματα, τα αρτοποιεία ή την παραλία απίστευτα εύκολο!

- πολλά καφέ!

- το απευθείας τραμ και τα λεωφορεία σας μεταφέρουν κατευθείαν στο κέντρο της Αθήνας σε 25-30 λεπτά.

- ελληνικές οικογένειες όλο το χρόνο (χωρίς τουρίστες ή κέντρα ομογενών)

ποιες απόψεις έχετε όλοι; ή συμβουλές;

Crecí en España, pero soy irlandesa y he vivido aquí la mayor parte de mi vida. Me gustaría mejorar mi español coloquial y creé un diálogo.

(Ya sé que hoy en día los pedidos se hacen por internet, jajaja)

-------------------------------------------------------------------------------

Mujer: Buenas. Mercadona. Dime.

Yo: Hola, me gustaría hacer un pedido. ¿Estáis disponibles hoy? ¿Tenéis envío a domicilio?

Mujer: Claro.

Yo: Vale. ¿Estás lista?

Mujer: Venga.

Yo: Una botella de aceite de oliva, un ajo, dos cebollas, una bolsa de patatas...

Mujer: ¿Vas a cocinar una tortilla de patatas?

Yo: Jaja, ¡claro que sí!

Mujer: Te recomiendo el aceite de girasol para esta receta, para cocinarla al horno.

Yo: Fenomenal. Gracias. Esto es todo.

Mujer: Vale. Son diez euros. ¿Te gustaría pagar en efectivo o con tarjeta?

Yo: Cualquiera. No me importa. Nos vemos.

-------------------------------------------------------------------------------

¿Qué os parece? ¿Cómo puedo hacer que esta suene más natural? ¿Todavía se usa la dime?

Είμαι Ιρλανδή (έχω ξαναρχίσει το κάπνισμα λόγω της δουλειάς μου) και θέλω να προσαρμοστώ στο περιβάλλον όταν επισκεφτώ ξανά την Αθήνα σε λίγους μήνες. 😉

Been learning for years now, but always unsure how legible it is lmaoo

PhD in maths here, looking to work in finance soon.

From what I can see, the JPY has been in a severe mess for a combination of structural, economic, and geopolitical reasons, pushing USD/JPY to historic 40-year lows near 164.

- the Bank of Japan kept its interest rates exceptionally low compared to the US and other western economies. Global investors took advantage of this by borrowing cheaply in yen to invest in higher-yielding US assets, effectively flooding the market with selling pressure against the Japanese currency. (interest rate gap)

- Japan is heavily reliant on imports for energy (oil and gas) and food. A weak yen means businesses and ordinary citizens have to pay astronomically higher prices for basic goods, creating a harsh cost-of-living squeeze. (import cost headache)

- Japan's public debt sits at over 200% of its GDP (the highest in the G20). Concerns over government spending, budget deficits, and potential policy shocks have fueled persistent market volatility. (rising public debt)

So, how do quants/traders deal with this?

My assumption is the the Garman-Kohlhagen (GK) model (which is useful as a foundational framework), suffers from severe limitations during periods of extreme yen volatility? The model assumes annualised volatility of the exchange rate is a single fixed number over the life of the option. However, the volatility clusters aggressively, and long periods of low volatility are shattered by explosive, multi-day spikes that the model cannot predict or price, and thus the σ, the interest rate differential (\(r_d - r_f\)), the drift term (μ), and the delta hedging mechanism (Δ) all collapse under severe yen volatility?

So, would volatility scaling, or dynamic hedging be useful here? By using algorithms to automatically reduce position sizes when currency fluctuations increase to keep daily risk exposure constant? Or, models continuously alter hedge ratios instead of holding static positions, buying or selling underlying assets as market prices shift?

Can you help this dumb laywoman understand?

Θα επισκεφθώ την Αθήνα τον Ιανουάριο του 2027 και ήθελα να κάνω περισσότερα τοπικά πράγματα (είδα την Ακρόπολη, το Εθνικό Αρχαιολογικό Μουσείο κ.λπ. τον περασμένο Ιανουάριο) και θα ήταν ωραία αυτά;

- να πάρω ένα τοπικό γεύμα από το πάγκο με ντελικατέσεν στο Σκλαβενίτη και να το φάω σε ένα πάρκο

- να επισκεφτώ τη Δημοτική Πινακοθήκη και να περιπλανηθώ στο Μεταξουργείο

- να παρακολουθήσω τους ψαράδες στον Πειραιά

- να κάνω πεζοπορία σε ένα τοπικό βουνό (Πάρνηθα)

- να επισκεφτώ την Αίγινα, να διαβάσω ένα βιβλίο κατά μήκος της θάλασσας και να ψωνίσω τοπικά φιστίκια

- να ψάξω για σύκα, χουρμάδες, ελιές στη μεγάλη Δημοτική αγορά

- έναν μονό ελληνικό μέτριο και μια πρασόπιτα στον κήπο με τις ελιές δίπλα στο Μουσείο της Ακρόπολης (διαβάζοντας ένα βιβλίο)

- περιηγηθείτε στο Η λαϊκή αγορά της Καλλιδρομίου

Θα μείνω εκεί για μια εβδομάδα στο Κολωνάκι και θα γράψω τη διδακτορική μου διατριβή. Οποιεσδήποτε προτάσεις θα ήταν ευπρόσδεκτες! Είμαι Ιρλανδή (συγγνώμη για την κακή γραμματική μου!) 🧡

I just don't get it.

I'm working on my PhD in stochastic wave propogation and delving into financial models as I hope to work as a quant one day. However, this fund scaled up massively to over $20–$45 billion in assets at various peaks. Then, the 439% net return in the first half of the year.

Was it ultimately down to them utilising heavy leverage (reported to be running as high as 4x or so) and heavily borrowing money from prime brokers like Bank of America, Goldman Sachs, and JPMorgan to buy concentrated baskets of AI infrastructure and memory stocks (such as SK Hynix, Micron, Nebius, and CoreWeave), alongside short bets against software companies?

I assume that when AI infrastructure tradeded violently in July, the fund suffered a brutal drawdown, wiping out massive portions of its peak value (and as they were over-leveraged, prime brokers, it forced an emergency unwind to cover margin calls)? Then, the fire sale happened?

Can someone please explain it to me?

Lastly, do some of these investors/funds bet on an aggressive P measure trend (AI is changing the world, so this stock will go up 400%, etc), but the lenders and prime brokers who control their margin accounts evaluate risk using models using the Q-measure? Where volatility \sigma dW_t is treated as an immediate threat to collateral, regardless of how brilliant somebody claims to be?