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324 votes
29 answers
241k views

What data sources are available online?

What sources of financial and economic data are available online? Which ones are free or cheap? What has your experience been like with these data sources?
96 votes
4 answers
87k views

What are the quantitative finance books that we should all have in our shelves?

Which books/papers should we all have in our shelves? There are a couple that I use regularly such as: An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation Asset ...
57 votes
5 answers
72k views

Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
Toofreak's user avatar
  • 761
21 votes
3 answers
13k views

Carr-Madan Formula

Really new to financial Maths. I am currently having problems with the Carr-Madan Formula. $$f(S_T)=f(F_t) + f'(F_t) (S_T - F_t) + \int_0^{F_t} f''(K) (K-S_T)^+ \ d K + \int_{F_t}^{\infty} f''(K)...
user22166's user avatar
  • 211
2 votes
1 answer
2k views

FX Options price vs implied vol

From the screenshot below, what is the difference between the option price by strike in the table versus the implied volatilities by delta in the chart at the bottom? https://www.investing.com/...
Student's user avatar
  • 351
6 votes
2 answers
4k views

Garman-Kohlhagen (Black-Scholes) Formula vs. Bloomberg OVML Calculator

I'm trying to price a European call option on USDJPY. We have that $S = 112.79, K = 112.24, \sigma = 6.887\%, r_d = 1.422\%, r_f = -0.519\%, T = 0.25$. My model, based on Black-Scholes, returns the ...
Vladimir Nabokov's user avatar
68 votes
9 answers
90k views

What are some useful approximations to the Black-Scholes formula?

Let the Black-Scholes formula be defined as the function $f(S, X, T, r, v)$. I'm curious about functions that are computationally simpler than the Black-Scholes that yields results that approximate $...
knorv's user avatar
  • 2,119
35 votes
4 answers
29k views

How to derive the implied probability distribution from B-S volatilities?

The general problem I have is visualization of the implied distribution of returns of a currency pair. I usually use QQplots for historical returns, so for example versus the normal distribution: ...
Thomas Browne's user avatar
3 votes
3 answers
4k views

FX Option pricing on Forward vs. Spot

In a GBM world with riskless domestic and foreign interest rates, what would be the correct model for a FX plain vanilla option given the statement that this option is priced on the forward? I guess ...
Tim's user avatar
  • 163
5 votes
1 answer
2k views

Black Scholes differential

I'm studying a BS derivation and I don't understand one part .We have a portfolio consisting of $\Delta(t)S(t)+B(t)$ where the first term is risky and the second is a riskless bond. The part i don't ...
ab94's user avatar
  • 366
7 votes
1 answer
2k views

Online sources for quantitative finance research

What are the sources one can search for or view / download research articles and other publications on quantitative finance in addition to the Internet search engines?
Alper's user avatar
  • 1,036
48 votes
6 answers
117k views

A simple formula for calculating implied volatility?

We all know if you back out of the Black Scholes option pricing model you can derive what the option is "implying" about the underlyings future expected volatility. Is there a simple, closed form, ...
jessica's user avatar
  • 2,098
1 vote
2 answers
3k views

Effect of Implied volatility on option delta

I am currently hedging a short put option where strike is 6027 and expiry is 30th Mar 2023. As per my understanding when option is ITM increase in volatility will decrease the delta and decrease in ...
Sumit's user avatar
  • 45
37 votes
6 answers
14k views

How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of ...
sets's user avatar
  • 1,461
27 votes
3 answers
15k views

Explaining the Risk Neutral Measure

What is the Risk Neutral Measure? I don't believe this has been answered on the internet well and with all the parts connecting. So: What is the risk neutral measure/pricing? Why do we need it? How ...
Trajan's user avatar
  • 2,502
15 votes
2 answers
5k views

Variance replication using options

I would like to understand the intuition behind the following question: Why a certain weighted sum of prices of put and calls is equivalent to the implied variance of an underlying? A variance swap ...
Escachator's user avatar
1 vote
1 answer
2k views

Finite Difference Method in Greeks (Options)

I need a way to approximate the analytical formula of Greeks of a generic call option using the Finite Difference Method. For example, the FD method for Delta/Gamma is the following one: Now, I am in ...
John_maddon's user avatar
47 votes
16 answers
34k views

Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
CuriousMind's user avatar
7 votes
2 answers
2k views

Is it possible to have only one volatility surface for american options (that fits both calls and puts)?

Put-Call Parity does not hold for american options. Hence, I don't see how it would be possible to have one surface that would encompass both calls and puts. For example: Let pick a call lying in the ...
Rodrigo's user avatar
  • 320
6 votes
3 answers
9k views

Why and when we should use the log variable?

Normally, I see finance papers use the real ratios but log regarding non-ratio variables. For example, some papers used log(asset) or log(1+firm age) or log GDP, but regarding the ratio, they use the ...
Phil Nguyen's user avatar
6 votes
1 answer
5k views

Extended Hull White Interest Rate Model for Zero Coupon Bond

Let's take the following three SDEs: $$dr=u(r,t)dt + w(r,t)dX$$ $$u(r,t)=a(t)-br$$ $$w(r,t)=c$$ where $b$ and $c$ are constants and $a(t)$ an arbitrary function of time $t$. If Zero Coupon Bond $Z(...
quant1's user avatar
  • 75
6 votes
4 answers
1k views

How can we estimate new stock price after a large purchase?

Suppose someone buys $4bn of a particular stock over the period of a few weeks. Depending on how much that stock is being traded, you would expect that the price goes up in a visible way compared to ...
doublefelix's user avatar
29 votes
9 answers
20k views

Any known bugs with Yahoo Finance adjusted close data ?

Yahoo Finance allows you to download tables of their daily historical stock price data. The data includes an adjusted closing price that I thought I might use to calculate daily log returns as a ...
Paul's user avatar
  • 523
29 votes
5 answers
14k views

Local Volatility vs. Stochastic Volatility

Are there any empirical observations or practices when to prefer Local Volatility Model for pricing over Stochastic Model or vice versa?
Andrey Taptunov's user avatar
28 votes
2 answers
31k views

Transformation from the Black-Scholes differential equation to the diffusion equation - and back

I know the derivation of the Black-Scholes differential equation and I understand (most of) the solution of the diffusion equation. What I am missing is the transformation from the Black-Scholes ...
vonjd's user avatar
  • 27.4k
17 votes
3 answers
32k views

Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz Portfolio Theory?

I have seen the following formula for the tangency portfolio in Markowitz portfolio theory but couldn't find a reference for derivation, and failed to derive myself. If expected excess returns of $N$ ...
Slow Learner's user avatar
  • 1,160
6 votes
2 answers
7k views

Calculating alpha and its meaning

According to wikipedia, CAPM model is described by: $E(R_{i})=R_{f}+\beta _{{i}}(E(R_{m})-R_{f})$ And according to website such as http://investexcel.net/jensens-alpha-excel/, $\alpha = E(R_{i}) - ...
codeedoc's user avatar
  • 245
4 votes
3 answers
1k views

Derivation of BS PDE problem using Delta hedging

I've always been confused with Delta hedging. It is well-known that for a (smooth enough) function of $(S,t)$ we have, due to Ito's lemma, that: \begin{eqnarray*} dC = \left(\frac{\partial C}{\partial ...
Vim's user avatar
  • 903
3 votes
1 answer
1k views

Quantlib: day-by-day evaluation of option value

I'm using Quantlib in Python to price an FX option. I'm comparing the result to Bloomberg, to make sure the code is working correct. I want to calculate the P&L of a certain option trading ...
Wynn's user avatar
  • 105
109 votes
7 answers
199k views

Where to download list of all common stocks traded on NYSE, NASDAQ and AMEX?

I have a very basic data question: how to get a list of all common stocks traded on NYSE, NASDAQ and AMEX? I would need to be able to get the approximate list of common stocks as is available in ...
Samo's user avatar
  • 1,091
64 votes
8 answers
109k views

How to annualize Sharpe Ratio?

If I know the daily returns of my portfolio, I need to multiply the Sharpe Ratio by $\sqrt{252}$ to have it annualized. I don't understand why that is.
David's user avatar
  • 701
50 votes
4 answers
8k views

How much data is needed to validate a short-horizon trading strategy?

Suppose one has an idea for a short-horizon trading strategy, which we will define as having an average holding period of under 1 week and a required latency between signal calculation and execution ...
Tal Fishman's user avatar
  • 13.4k
27 votes
17 answers
92k views

What programming languages are most commonly used in quantitative finance?

What programming languages are the most common in quantitative finance, and why are these languages used? Note: I do not mean, what languages are used to develop the accounting system at a hedge fund:...
17 votes
5 answers
48k views

Bachelier model call option pricing formula

Does anybody have the Bachelier model call option pricing formula for $r > 0$? All the references I've read assume $r = 0$. I don't speak French, so I can't read Bachelier's original paper.
Galsunja's user avatar
  • 171
13 votes
4 answers
4k views

Understanding $N(d_1)$ and how to use the stock itself as the numeraire?

Assume the stock price follows a geometric Brownian motion Then in Black-Scholes pricing model, $N(d_2)$ is the risk-neutral probability that the option expires in-the-money. However, it is said that $...
sanliusinger's user avatar
7 votes
4 answers
2k views

Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$

Develop a formula for the price of a derivative paying $$\max(S_T(S_T-K))$$ in the Black Scholes model. Apparently the trick to this question is to compute the expectation under the stock measure. So,...
Trajan's user avatar
  • 2,502
6 votes
2 answers
1k views

how to calculate vega in stochastic vol?

since vega is defined as option value changes regarding the implied vol parallel shift, how is vega defined or calculated in stochastic vol models since implied vol is not an input there? thank you.
Odyssey's user avatar
  • 131
111 votes
17 answers
17k views

What concepts are the most dangerous ones in quantitative finance work?

There are a few things that form the common canon of education in (quantitative) finance, yet everybody knows they are not exactly true, useful, well-behaved, or empirically supported. So here is the ...
49 votes
8 answers
51k views

How does the "risk-neutral pricing framework" work?

I've struggled for a long time to understand this - What is this? And how does it affect you? Yes I mean risk neutral pricing - Wilmott Forums was not clear about that.
Jack Kada's user avatar
  • 809
33 votes
5 answers
65k views

How to simulate stock prices with a Geometric Brownian Motion?

I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula: $$R_i=\frac{S_{i+1}-...
user1690846's user avatar
22 votes
1 answer
9k views

Skewness and Kurtosis under aggregation

Returns possess non-zero skewness and excess kurtosis. If these assets are temporally aggregated both will disappear due to the law of large numbers. To be exact, if we assume IID returns skewness ...
Bob Jansen's user avatar
  • 8,542
6 votes
5 answers
22k views

Carry calculation on an interest rate swap

I was hoping that I can get help on a simple yet not so straight forward topic : Looking at valuing the costs of holding an IRS in the books this would entail marketed-to-market due to price ...
user29352's user avatar
3 votes
1 answer
1k views

Option Pricing for Illiquid case

I am currently studying crypto options trading and have observed that there is often a lack of liquidity for options (such as BTC Options) on various exchanges, including Binance. In many cases, there ...
Starlord22's user avatar
1 vote
2 answers
1k views

Black Scholes Theta Finite difference

I am trying to obtain the Theta from Closed Formula by using Finite Difference methods and I observe some discrepancies. For instance, here with the following parameters: Spot:50, Strike:50, Rate: 0....
ALFRAM's user avatar
  • 135
0 votes
1 answer
899 views

Delta hedging error in B-S (hedging with implied vol) question

I have been thinking about this for a while and am at my wits end. Now assume I am pricing a call at implied vol $s$, whereas the realized volatility is $σ$. Let $C$ be the incorrect pricing function. ...
Arshdeep's user avatar
  • 1,915
59 votes
14 answers
167k views

Where to get long time historical intraday data?

I am looking for long time historical intraday day data on the S&P500 composite for a time horizon like 10 years with a - for example 10-minutes tick - or prices for call/put options on the S&...
user190080's user avatar
32 votes
7 answers
53k views

How to calculate historical intraday volatility?

Sorry for what must be a beginner question, but when I went to write code I realized I didn't understand exactly how historical volatility, or statistical volatility, is defined. Wikipedia tells me "...
Darren Cook's user avatar
  • 1,427
29 votes
5 answers
25k views

What are the advantages/disadvantages of these approaches to deal with volatility surface?

I would like to know if someone could provide a summarized view of the advantages and disadvantages of the approaches on the volatility surface issues, such as: Local vol Stochastic Vol (Heston/SVI) ...
AZhu's user avatar
  • 803
28 votes
5 answers
15k views

Is there a standard model for market impact?

Is there a standard model for market impact? I am interested in the case of high-volume equities sold in the US, during market hours, but would be interested in any pointers.
shabbychef's user avatar
  • 2,836
24 votes
2 answers
21k views

Discrete returns versus log returns of assets

There have been similar posts here already but nevertheless I find the question worth posting: why do some people claim that log returns of assets are more suitable for statistics than discrete ...
Richi Wa's user avatar
  • 13.7k

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