All Questions
2,099
questions
328
votes
29
answers
244k
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
102k
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
77k
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 ...
- 761
21
votes
3
answers
14k
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)...
- 211
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 ...
- 440
1
vote
1
answer
3k
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/...
- 361
68
votes
9
answers
93k
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 $...
- 2,119
35
votes
4
answers
30k
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:
...
- 1,180
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 ...
- 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 ...
- 366
48
votes
6
answers
119k
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, ...
- 2,108
3
votes
2
answers
4k
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 ...
- 65
38
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 ...
- 1,471
28
votes
3
answers
17k
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 ...
- 2,562
3
votes
1
answer
2k
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 ...
- 115
1
vote
1
answer
3k
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 ...
- 45
15
votes
2
answers
7k
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 ...
- 438
6
votes
2
answers
4k
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 ...
- 310
47
votes
16
answers
36k
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 ...
- 777
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?
- 1,046
6
votes
3
answers
12k
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 ...
- 307
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 ...
6
votes
1
answer
6k
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(...
- 75
29
votes
5
answers
15k
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?
- 2,239
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 ...
- 523
28
votes
2
answers
32k
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 ...
- 27.5k
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$ ...
- 1,170
6
votes
2
answers
8k
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}) - ...
- 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 ...
- 903
112
votes
18
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 ...
109
votes
7
answers
203k
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 ...
- 1,091
64
votes
8
answers
113k
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.
- 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 ...
- 13.5k
27
votes
17
answers
93k
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:...
16
votes
5
answers
51k
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.
- 161
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 $...
- 199
8
votes
4
answers
3k
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))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
- 2,562
6
votes
2
answers
2k
views
how to calculate vega in stochastic vol?
since vega is defined as option value changes regarding the implied vol parallel shift, how is vega defined or calculated in stochastic vol models since implied vol is not an input there? thank you.
- 131
3
votes
2
answers
2k
views
Black Scholes Theta Finite difference
I am trying to obtain the Theta from Closed Formula by using Finite Difference methods and I observe some discrepancies.
For instance, here with the following parameters:
Spot:50, Strike:50, Rate: 0....
- 155
2
votes
2
answers
4k
views
How to Bloomberg compute the implied Yield ? What is FX swap basis spread?
Question 1:
You can see Bloomberg EUR/USD FXFA<go> page attached below
EUR 3 months yield=3.9412
US 3 months yield= 5.6683
Spot Rate: 1.0580
How does it find FX swap rate as 1.062732?
Question ...
- 125
51
votes
8
answers
52k
views
How does the "risk-neutral pricing framework" work?
I've struggled for a long time to understand this - What is this? And how does it affect you?
Yes I mean risk neutral pricing - Wilmott Forums was not clear about that.
- 829
34
votes
5
answers
67k
views
How to simulate stock prices with a Geometric Brownian Motion?
I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula:
$$R_i=\frac{S_{i+1}-...
- 807
32
votes
7
answers
55k
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 "...
- 1,427
22
votes
1
answer
10k
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 ...
- 8,581
8
votes
1
answer
5k
views
Continuous delta hedge formula
When we buy a call and continuously delta hedge using some implied volatility $\sigma_i$, what is the formula for our aggregate profit given that the actual realized volatility is $\sigma_r$?
Say $...
- 81
6
votes
5
answers
24k
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 ...
- 61
4
votes
2
answers
3k
views
Which models do Bloomberg/Reuters use to derive implied volatility for interest rate derivatives with negative forward rates?
can anybody tell me which models Bloomberg and Reuters ares using to derive implied volatility for interest derivatives with negative forward rates?
I know that Black-76 is the standard model, and ...
- 41
2
votes
1
answer
3k
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 ...
- 67
0
votes
1
answer
1k
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.
...
- 2,521
59
votes
14
answers
171k
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&...
- 699