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**Question:** I don't understand why a Monte Carlo simulation needs correlated random variables. Isn't each simulation thread independent?
**Background:**
Specifically, I'm referring to the below example on pg 319 in the Malz text (http://ca.wiley.com/WileyCDA/WileyTitle/productCd-0470481803.html).
He describes a M... |
By definition, the payoff of a log-contract of maturity $T$ writes
$$ \phi(S_T) = \ln\left(\frac{S_T}{S_0}\right) $$
Let $\Pi_t$ denote the $t$-value of such a contingent claim. We are interested in the price at $t=0$, best known as the *option premium*. Theory tells us that the latter premium can be computed as
$$ \Pi... |
I was working on my finical math homework where I need to find the minimum variance portfolio. I need to use the following matrix expression.
[![enter image description here][1]][1]
Nowhere in the class notes does the instructor say what the u vector in the above expression is. At first I thought it was the expecte... |
A book I am reading defines a positive linear functional as a "price functional" from a set of adapted processes to the real numbers. Specifically, it defines a "consistent price functional" as one that gives the value zero to every dividend stream corresponding to any trading strategy. However, it also says that the v... |
What is the point/benefit of using a dollar-neutral strategy ***in addition to*** a Beta-neutral strategy? What exactly does a dollar-neutral strategy buy the investor? What's useful about balancing long and short positions?
Using the notation in [Qian *et al.*'s book][1], in the mean-variance optimization problem, wh... |
According to [Duffie, Pan and Singleton (2000)][1] for any real number $y$ and any $a$ and $b \in \mathbb{R}^n$, the price of a security that pays $\exp(aX_t)$ at time $T$ in the event that $bX_t \leq y$ is given by:
$$G_{a,b}(y;X_{0},T,\chi)=\frac{\psi^{\chi}(a, X_{0}, 0, T)}{2}-\frac{1}{\pi} \int_{0}^{\infty} \frac{... |
**The problem:**
>Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is standard $\mathbb P$-Brownian motion.
>
>Let $X = \{X_t\}_{t \in [0,T]}$ be a stochastic process where $X_... |
I was asked the following question at an interview:
> Suppose that you're an OTC trader for an exchange that trades bitcoin. What would you do in the following scenario?
>
> A large corporate investor wants to initiate an OTC trade for $50,000$ bitcoin (BTC) to be guaranteed at a price within $1\%$ of the mid-market ... |
Consider reading [Lorenzo Bergomi][1]'s excellent book -- or at least the **first chapter** available [here][2] for download --, it will help you clarify things.
Some remarks as to your original question:
- It is well known that, under a pure diffusion assumption, the total P&L of a delta hedged European option (i.e... |
**Background Information:**
I am not sure this is relevant:
Terminal value pricing:
If the derivative $X$ equals $f(S_T)$, for some $f$ then in the value of the derivative at time $t$ is equal to $V_t(S_t,t)$, where $V(s,t)$ is given by the formula
$$V(s,t) = \exp{(-r(T-t)E_{\mathbb{Q}}(f(S_T)|S_t = s)}$$
And the... |
> If we have a Black-Scholes model $B_t = \exp{(rt)}$ and $S_t = S_0\exp{(\sigma W_t + \mu t)}$ then is it complete?
>
> What if $W_1$ and $W_2$ are independent Brownian motions. Then the two-stage Black-Scholes model $$B_t = \exp{(rt)}$$ $$S_1(t) = \exp{(W_1(t) + W_2(t) + t)}$$ $$S_2(t) = \exp{(W_1(t) + 2W_2(t) + 2t)}... |
We should check the martingale properties.
Let $(\Omega ,\mathcal{F},\{\mathcal{F}\}_{t\ge 0},\mathbb{P}) $ be a filtered probability space. We define the class of functions ,$\mathcal {V} =\mathcal {V}(t,T)$, as follow
$$\psi(t,\omega):[0,\infty)\times\Omega\to\mathbb{R}$$
such that
- $(t,\omega)\to \psi(t,\omeg... |
Regarding your "false sense of confidence" statement: most of it has nothing to do with VaR as a framework itself, it's about how you implement it. Naive HS-VaR fails you? Sure thing, but who's to blame in the situation except yourself?
Most of the pratitioners use VaR in its simplest form, which yields predictably po... |
I use `python DataReader` for getting past stock data.
I heard that `Adj Close` data could be different each other if data range is different between datas. This is because (as I know of) `Close` data is adjusted based on the LAST DATE of period.
For example,
> APPL1_data : 2010-01-01 ~ 2016-01-01 adj close data
... |
Quote
>**Hedging with variance is not comparable to puts**
>
>Due to the lack of convexity of a variance swap hedge, we believe it is best to compare long variance hedges to hedging with futures rather than hedging with puts. Although variance hedges might be cheaper than put hedges, the lack of convexity for long vol... |
I am writing code to calculate and plot the Exponential Moving Average(**EMA**) for different chart **timeframes**(tick/1M/5M/1H etc). I found the formula for calculating EMA:
> **EMA = Price(t) * k + EMA(y) * (1 – k)**
>
>Where,
>
>*t = today, y = yesterday, N = number of days in EMA, k = 2/(N+1)*
But I can't figur... |
Trying to implement some monte carlo simulation for the first time. For the sabr model (http://www.javaquant.net/papers/managing_smile_risk.pdf), would this work?
[![enter image description here][1]][1]
Here, a = volatility of volatility, and s = volatility, and r = correlation of wiener processes.
If its ok, then w... |
Thank you all for answering my question.
I wanted to know what influence has the exponential-Lévy model on a call price (how the curve changes).
If we add Merton jumps, we get an EDPID like this one: [![enter image description here][3]][3]
with:
[![enter image description here][4]][4]
What is your financial inte... |
I'm sure [this pdf][1] will help you to understand what's lying beyond CDX.NA.IG and HY
As for the more info (from the link: @Alexander) iTraxx is product made by Deutsche Bank
[1]: https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwjl5u-zxpXTAhXjYZoKHU0yDf0QFggpMAE&url=https%3... |
I have come across two ways of measuring VaR for Fixed Income instruments thus far:
- Express the volatility in of basis points and the position in terms of sensitivity to a 1 basis point movement in yields and then multiply it by the desired largest possible movement (95% or 99%); This method is described on page 17... |
[This paper][1] gives on equation 22 an estimator for the mean reversion level of the variance process under the real world measure. The context is the Heston model, where the variance is stochastic and the paper is trying to give a proxy for the determination of the Heston model parameters under the **real word measur... |
I'm trying to understand whether there is a good way to compare forecasts for volatility from different sources i.e., implied volatility and GARCH. I'll outline a few statements that I believe and if anyone could verify if they are correct or explain why I'm wrong I would be grateful.
> $\textbf{1.}$ The 30 day implie... |
I have found a result that I find truly puzzling. Here is an extract from a GARCH-Analysis I have performed:
>Test______________Statistic_______p-Value
>
>Ljung-Box Test_____R Q(10)_____0.4047773
> Ljung-Box Test_____R Q(15)_____0.3371581
> Ljung-Box Test_____R Q(20)_____0.4098038
>Ljung-Box Test____... |
I read that since stock prices are non-stationary it does not make sense to take their covariance. So I took the log returns of stocks, computed covariance matrix, took the top few eigen vectors that explain maximum variance and instead of projecting the returns I projected the log prices on these vectors. Is there any... |
# How do you suggest I do this?
I would like to perform a statistical test to check if:
- the aggregate alpha of all funds equals 0.
- the aggregate beta of all funds equals 1.
# Data
- Sample of 1000 mutual funds over the period 1990 to 2017. I regress
funds monthly excess return over the risk free rate agains... |
In this [Investopedia article][1],
> For example, when the Federal Reserve increased interest rates in
> March 2017 by a quarter percentage point, the bond market fell. The
> yield on 30-year Treasury bonds dropped to 3.108% from 3.2%, the yield
> on ten-year Treasury notes fell to 2.509% from 2.575%, and the
> two-ye... |
I was asked this question in an interview.
> There is an option as follows. It monitors the prices of two stocks A and B, and pays the difference in their prices at time $T$, if stock A has been higher than stock B all through till $T$. There is nothing paid if stock A has fallen lesser than stock B at any time before... |
On the ["how ETF arbitrage works" page][1], investopedia says
> The arbitrage opportunity happens when demand for the ETF increases or decreases the market price, or when liquidity concerns cause investors to redeem or demand the creation of additional ETF shares. At these times, price fluctuations between the ETF and... |
**Question :** Could you please precise the $d_{1}(t)$ and $d_{0}(t)$ that you use in your formula?
That is if you are using the same $d_{1}$ and $d_{2}$ as in classical B&S Theory, your formula is incorrect as it only holds with constant deterministic rates.
I believe there are modified versions of Black Scholes ... |
I've used both approaches at different points of my career, and both are valid.
Firstly, there's a few statements that aren't quite correct.
>...AND in a file on disk in order to avoid losing all data if the program crashes.
>
>... have the TWS API receiving asynchronous data feed and append the data in a flat/binary... |
I realize that the question I ask is very related to accounting but there is no active stackexchange website for this. The question is related to Analysts' recommendation and Earnings Surprise.
Look at the chart of reported EPS and EPS consensus (from analysts' predictions) for the company Biogen Inc on Nasdaq [here][... |
Maybe this is a stupid question, i dont know, anyhow i recently read this part from an article,
[![enter image description here][1]][1]
Can someone explain the part " American put value function ...and the exercise boundary jointly solve a free boundary problem, " i though that the you had to solve for the free bou... |
From JP Morgan's Trading Credit Curves 1 and we have that:
> The MTM of a CDS contract is (for a sell of protection) therefore:
>
>$$\text{MTM} = (S_{\text{Initial}}-S_{\text{Current}}).\text{Risky Annuity}_{Current}.Notional$$
Why do we need the Risky Annuity Current? I dont see the logic behind this...
>Apparentl... |
I'm taking a financial engineering course through coursera and on a slide one of the lecturers talks about how a long hedge is used in the futures market. Here is the text:
> Today is Sept 1st. A baker needs 500,000 bushels of wheat on December 1st. So, the baker faces the risk of an uncertain price on Dec. 1st.
>
> H... |
Find expressions for the process $\psi=(\psi(t),\ 0\leq t\leq T)$ , so the portfolio $(\phi,\ \psi)$ is self-financing when:
(1) $\phi(t)= \int_{0}^{t}S_{s}ds $
(2) $\phi(t)=S_{t}$
where $\phi(t)$ is an Ito process.
Where is the error in my Ito product rule? I can't figure out what the cross-variation terms shoul... |
The following list ranges from the staples to some of the more eclectic titles:
**Factor Investing**
1. Eugene F. Fama, Kenneth R. French. *[A five-factor asset pricing model.][1]* _Journal of Financial Economics_, Volume 116, Issue 1, April 2015, Pages 1–22
> A five-factor model directed at capturing the size, val... |
I leveraged the github code (https://gist.github.com/ghalimi/4591338) to compute IRR using Newton method. When I replicated the codes step by step in excel, I'm able to find the optimized resultRate around -0.006. However when I ran the code directly, it generated a resultRate that doesn't accomplish a 0 NPV at all. Ca... |
While calculating european call option prices, using the variance-gamma model formula provided by Madan, Carr & Chang (1998), I noticed that, holding all other things constant, the value of an ITM call option in the VG-model seems to decline with $\sigma$ when the skewness parameter $\theta < 0$.
Below is depicted, fo... |
What kind of structured product has the following payout?:
- Guaranteed capital at maturity $T$
- It pays a coupon at time $t$ equal to
$$C_0\sum_{i=1}^{\infty}\max(S(t)/S(t-1)-1-C_1 i;0)$$
with $C_0,C_1>0$?
(The underlying S(t) can be some stock index for instance.)
|
The Euro swaption market is changing from cash to physical settlement quotation in July 2018 $-$ see e.g. ["Euro swaptions market prepares for pricign revamp (Risk, 2018)"](https://www.risk.net/derivatives/5399806/euro-swaptions-market-prepares-for-pricing-revamp). When describing the issues around cash settled swaptio... |
According to _Dynamic Hedging: Managing Vanilla and Exotic Options_ (Taleb, 1997), the Parkison volatility estimator has several meaningful properties. It is defined
$$P=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\frac{1}{4\log\left(2\right)}\left(\log\left(\frac{S_{H,i}}{S_{L,i}}\right)\right)^{2}}$$
where $S_{H}$ and $S_{L}$ a... |
Your estimator $\hat{s_i}$ for stock $i$ is an unbiased estimator of its latent standard deviation $\sigma_i$ (which is constant for your model). When applying your "window rolling" for calculating $\hat{s_i}$, you get a time-series $ts_{\sigma_i}$ for each stock $i$.
With an intercept-only OLS-regression for each tim... |
Using Thomson Reuters Eikon I can extract the monthly NAV and Dividen Payments of a fund. I would like to calculate the monthly returns of a fund now. Would this be the right approach?
# Fund Date NAV Div #
1 1 10 -
1 2 11 1
Return = (11+1)/10 -1 = 0.2 = 20% |
The Fama/French (1993) paper is based on the widely used CRSP database, maintained by the University of Chicago's Booth School of Business. It provides data for NYSE-, AMEX-, and NASDAQ-listed securities from December 31, 1925 through the present.
## The CRSP database ##
However, in the beginning only NYSE stocks are... |
>As there are many comments but no answer, let me just sum up some of the comments.
# Solving a PDE or doing some probability?
Usually when one wants to solve the SDE they get a distribution for $X$. In the real world though it is often more useful know some of the properties of this distribution, such as its expecte... |
> What are some indicators that a given security might be inefficiently priced?
>
> What about efficiently priced (i.e., how can we estimate the
> degree of information already baked into price)?
You would need to get lower level data than what was used in [this][1] paper that was referenced [here][2]. The MSF data ... |
**Short answer:**
Your conversion from Macaulay Duration $MacD$ to the mod. Duration $ModD$ is correct, but your statement
> [...] which is % as well.
is **incorrect**.
_______
## Continuous time ##
$MacD$ is defined as
$$MacD = \sum_{i=1}^n{t_i \frac{CF_i \cdot e^{-y t_i}}{V}}$$
where $V= \sum_{i=1}^N{PV_i}$ and t... |
I need to improve the speed of convergence of PRNG Monte Carlo. I'm opening a new thread for that purpose and I have question / need confirmation about the algorithm.
I'm pricing options with Heston, QE scheme.
- For an European Down and In Put on a basket, I was thinking about using a vanilla put as a control variat... |
As a complement to the answer by Enrico, the ISIN identifies a **financial instrument** in a unique way. Tradition is, of course a reason and many codes, such as the CUSIP code in the US, predate the ISIN.
But there are a few other solid underlying principles that are applied in financial industry. These are pieces of... |
It is possible, yes, but it requires assumptions. But, philosophically speaking, this is the case as with all pricing, of any instrument. For example, given only the price of a 6Y and 7Y IRS can you correctly price the 6.5Y IRS rate? Well, yes you can, but it depends upon your assumptions about interpolation which is a... |
In addition to getting the right transition model for the Kalman filter, the main obstacle to optimizing filter performance is to implement an optimal initialization. I use an iterative approach to initialize or "tune" the Kalman filter, known as [adaptive tuning][1]. I do this because I've found alternative methods of... |
## Preliminary ##
The empirical finding of a strong negative cross-sectional relation between idiosyncratic volatility and future stock returns is highly inconsistent with the predictions of all theoretical models and therefore known as the *idiosyncratic volatility (IVOL) puzzle*. This phenomenon is also persuasive in... |
I follow a course, and get to the point that one bond price discounted by another one is a martingale:
$$
\frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale }
$$
I can not comprehend the proof below, mainly what is measurable inside of expectations:
$$
\mathbb{E}^{T_1}_t \big[ \frac{P(T,T_0)}{... |
In particular lets consider a zero-beta asset $i$ (in the CAPM sense). Let
1. $R_f$ be the risk free rate
2. $R_i$ the return on the asset $i$
3. $R_m$ the return on the market portfolio
4. $\beta=\frac{Cov(R_i,R_m)}{Var(R_m)}$
5. $E_P (E_Q)$ the expectation under $P$ the historical probability ($Q$ the risk neut... |
According to [this link](https://quantivity.wordpress.com/2011/02/21/why-log-returns/), there are some reasons we have to use log returns.
But I can not understand the first reason provided in the link:
> First, log-normality: if we assume that prices are distributed log normally (which, in practice, may or may not be... |
I know its probably been asked bevor but i just don't get it.
I have 2 values (Oil and corn price) and i want to check if they are cointegrated. Bevor that, i have tested if they really are non stationarity (and they are assuming to ADF,KPSS and PP).
# Eigen Test
The value of the test statistic is: 3.17 10.9942
... |
Let us denote $\delta$, the Libor's tenor (e.g. 3M), $P(t, T)$ the price of a zero coupon bond price paying 1 unit of currency at $T$, and $L_t(T, T + \delta)$ the forward 3M Libor starting at $T$ and ending at $T+\delta$, seen from $t$:
$$L_0(T, T + \delta) = \frac{1}{\delta} \left(\frac{P(0, T)}{P(0, T + \delta)} - ... |
## Preliminary ##
The approach in the paper of [Davis/Fama/French (2000)](https://doi.org/10.1111/0022-1082.00209) is to give insight in four contrary explanations on the value-anomaly. [Daniel/Titman (1997)](https://www.jstor.org/stable/2329554) trace the value-premium to the value-characteristic, not to a risk-based ... |
**EXERCISE**
>We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$
>
>Let C be a financial derivative of the form $C:=h(S_1^{1})$ where $h\geq0$ is a convex function.Show that the hedging value $\overlin... |
In finite differences for the black scholes method, you move backwards in time, since of course you know the prices at time $t = T$, and then you iterate until you get to time $t = 0$.
However, why then in this code does the time move forwards? Here, *cur_t* is current time, and as you can see, he iterates and each ti... |
The theorem says if $U$ is a numeraire and let $\mathbb{Q}^U$ be the corresponding measure. Then for every tradable asset $S$, the relative price $S_t/U_t$ is a martingale under $\mathbb{Q}^U$. But I don't know the meaning of "tradable asset". For example: in Brigo's "interest rate models" p39:
[![enter image descript... |
From a paper I am reading, it is written
[![enter image description here][1]][1]
These equations do not make any sense. If $s = k$, i.e. if we are pricing ATM options, then this volatility is identically zero, hence useless.
How am I to make sense of these formulas? Even if $s \approx k$, we get a value close to ze... |
### Clarification on the regression coefficients ###
Cochrane (Asset Pricing, rev. edition, 2005) states (p. 247):
> It it easier to do this in a more standard setup, with left-hand variable $y$ and right-hand variable $x$. Consider a regression
$$y_{it} = \beta´x_{it} + \epsilon_{it}$$
$$i = 1,2,..,N$$
$$t = 1,2,...... |
Within the framework you are proposing, it would make no sense. It would be failing to distinguish noise from signal. Extreme events are rarely triggered by measures of central tendency. It is like flipping heads 20 times in a row, what caused that?
From a physicist's, magician or con man's perspective that is a va... |
[Tomas bjork- arbitrage theory in continuous time][1].
Appendix B, proposition B41 says:
[![statement][2]][2]
[![proof][3]][3]
The proof is not clear to me.
Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the part where Bjork seems to imply that
$E^Q (X/G) . E^P (L/G) = E^P[(L.E^Q(... |
I found this code on [plotly][1] site, using CVXOPT to find the efficient frontier, and then, the optimal Portfolio.
The optimal function is
def optimal_portfolio(returns):
n = len(returns)
returns = np.asmatrix(returns)
N = 100
mus = [10**(5.0 * t/N - 1.0) for t in range(N... |
## Fama-MacBeth procedure (Step 1): ##
> So if my understanding is clear, first we would use the cross sectional regression to estimate 4*5,000=20,000 betas?
That is not right, because betas (and other risk-factor loadings) are estimated by a time-series regression:
$$ R_{i,t}^e = \alpha_i + \beta_{i, MktRf} MktRf_t... |
I'm preparing to take a test on equity derivatives and have met with some difficulty, so I need some help on the following question:
----
>Please analyze the following share purchase plan. Assume client is committed to purchasing a fixed number of shares?
>
>1. Original payout : client receives the average of daily c... |
As a followup to this old question, [Private Equity: Direct Alpha vs Excess IRR](https://quant.stackexchange.com/questions/35652/private-equity-direct-alpha-vs-excess-irr), I have a new one.
In automating PME calculations, the [Direct Alpha (DA)](https://en.wikipedia.org/wiki/Public_Market_Equivalent#Direct_Alpha) app... |
I'm studying the [BARRA Predicted Beta model][1], and the common factor covariance between portfolio $p$ and the return on the market $m$ is defined as the product of the transposed vector of the factor exposures for the portfolio, the factor covariance matrix, and the vector of the factor exposures for the market:
$C... |
I'm reading a book called "**The #1 Guide to Startup Valuation: How to value your startup in 12 easy steps**" (p. 22-23) by *Joachim Blazer*.
As one of the building blocks, namely "Return", the Carry Multiple is calculated.
There it says literally:
> Assume that the investor buys shares with a payoff of 100.
>
> And... |
I'm reading Chapter 13 of Hull's book and am stuck on how he got from stock returns to continuously compounded stock returns. As a recap, he built the generalized Wiener Process, which describes a change in some variable x is a function of a drift term and stochastic term.
> dx = a * dt + b * ∈ * sqrt(dt)
>
> where a... |
In his paper ([link][1]), he has the equations:
> b1 = k + ƛ - (ρ * σ)
>
> b2 = k + ƛ
k is the rate of mean reversion, ρ is the correlation between the two Wiener processes, σ is vol of vol, what is ƛ?
I have yet to figure out what ƛ is.
Thanks!
[1]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.139.... |
You can construct some financial ratios, such as ***P/E Ratio*** or ***P/B***.
----------
## P/E Ratio ##
P/E Ratio stands for Price/Earnings. As Price you can use the market value of equity and for Earnings the Total Net Profit from firm's Income Statement. Equivalently, you can divide both measures with the outstan... |
From the [Q&A][1] on the PRIIPs KID:
> **4. The principal component analysis of Annex II, Point 23 ensures the consistent simulation of curves. Is it mandatory to use this method also for PRIIPs that depend on only one interest rate underlying, e.g. bonds with yearly coupons of max (12m EURIBOR + 0.5%, 0.3%)?**
>
> A ... |
[![This is regarding operational losses of a financial institution with insurance. How is part (c) computed? ][1]][1]
**
Referring to part (c) First is 5000. Am I supposed to replace the 8000 with 5000 while maintaining its probability? For the second part, which includes the cost of insurance, do I add it to the l... |
I am trying to solve the following exercise from Bjork's *Arbitrage Theory in Continuous Time*:
> Consider a model for the stock market where the short rate of interest
> $r$ is a deterministic constant. We focus on a particular stock with
> price process $S$. Under the objective probability measure $P$ we have
> the ... |
I would like to consider a slight generalisation of [this](https://quant.stackexchange.com/questions/36000/determine-price-of-financial-contract) question, which I recall here:
> At date of maturity $T_2$ the holder of a financial contract will
> obtain the amount: $$ \frac{1}{T_2−T_1}\int^{T_2}_{T_1}S(u)du $$ where
>... |
(My question)
-------------
Please show me how to solve from (2) to (4) with computation processes.
These are too difficult to solve.
# Thank you for your help in advance. #
----------
## (Cross-link) ##
I have posted the same question on https://math.stackexchange.com/questions/3347148/bond-option-hedging/33493... |
According to a [recent Bloomberg article][1], Michael Burry of "The Big Short" fame claims that
> In the Russell 2000 Index, for instance, the vast majority of stocks are lower volume, lower value-traded stocks. Today I counted 1,049 stocks that traded less than \$5 million in value during the day. That is over half,... |
### Preliminary ###
This answer provides evidence (and confirms your supposition!) that the HML factor in Germany is no longer rewarded with higher returns in the cross-section of German stocks. From July 1992 - December 2018, the average HML-premium is 0.49% (2.26) per month, but insignificantly -0.28% (-1.01) from J... |
Based on your comments on other answers, i would like to provide you a summary on the difference of the CAPM-Alpha and Jensen's-Alpha.
___________________
### CAPM ###
The CAPM is an **economic model** for asset pricing. It states that the equation
$$E[r_i - r_f] = \beta_i E[r_m- r_f]$$
holds for any asset $i$. $r... |
I am trying to get a better understanding of Central Limit Theorem and how it can be used in life and in finance. From what I have read, the BSM model assumes the underlying asset's simple returns are normally distributed. It also assumes the underlying asset prices are lognormally distributed.
My first question is:... |
>Let $C_K(t,T)$ be the value of a European call with strike $K$ and maturity $T$ on a stock with value $S_t$ at time $t$. Then for all $t\leq T$ we have $$C_K(t,T)\leq S_t.$$
$\textbf{Proof}$: We derive an arbitrage opportunity by contradiction.
Assume $$C_K(t,T)>S_t$$ for some $t\leq T$. Consider the following:
At ... |
I am new to the pricing of bonds:
Suppose that I would like to price a floating-rate bond with par value \$100, with maturity at $T$ years from now, paying coupons semi-annually.
Suppose that $r_{n-0.5,n} $ denotes the annual interest rate from time $n-0.5$ to time $n$, where $n \in \Pi:=\{ 0.5, 1, \ldots, T-0.5, T\... |
Following the event study paper [USING DAILY STOCK RETURNS The Case of Event Studies][1]
let us suppose that I have daily stock returns for 50 companies from the date 2012-01-01 until 2014-01-01. and I want to exmine the effect of a single day event (i.e. some announcement on 2013-02-01) and I use 250 day range as t... |
I'm struggling with what the **exact** meaning of "stock prices are lognormal" (and its use to show normality of returns). My assumption was that given ${S_t}$ are stock prices and returns are defined as $r_t = \frac{S_t-S_{t-1}}{S_{t-1}}$, then we assume $S_t$ is lognormal, and then:
$$\log(1+r_t)=\log\left(\frac{S_... |
I am working on a project related to Nonlinear ***BS*** partial differential equation, with terms for transaction costs and/or discrete hedging.
I have two questions:
1. Is there any exact solution to the **Nonlinear BS** equation?
2. I have read a paper which numerically solved a **Nonlinear BS** and compared r... |
I think the question you are asking how certain you can be that 10% is the right probability is really how accurate is your model, and how can you attest to its accuracy.
>For example: A man will toss a coin. What is the probability that it comes up heads?<br>
>
>You will say 50%, and this probability is not a random... |
I'm reading an interview book called [A Practical Guide to Quantitative Finance Interview ](http://quantfinanceinterviews.com/) by Xinfeng Zhou and I cannot make sense of the solution provided by the book, so I really appreciate your advice.
**Question:** Birthday Problem from Chapter 2:
> You and your colleague know... |
I have the following question on futures options:
There is a Black’s model, which is a variant of the Black-Scholes formula that is used to price stock options. The Black’s model prices future options.
https://en.m.wikipedia.org/wiki/Black_model
The approach in the pricing model uses Magrabe’s formula. Both the agr... |
## NYSE is the abbreviation for *New York Stock Exchange* ##
Most financial researchers like Fama/French use the CRSP database for US financial data. It is maintained by the University of Chicago's Booth School of Business and provides data for NYSE-, AMEX-, and NASDAQ-listed securities from December 31, 1925 through ... |
Let $S_t$ be a geometric brownian motion and let $M_t = \sup_{0 \le u \le t} S_t$ be the maximum process.
Define $X_t = \frac{M_t}{S_t}$.
A book I am reading states the following:
[![enter image description here][1]][1]
**How does this follow from Ito's formula?** Where does the indicator function come from?
[... |
I'm trying to find a solution to the following problem:
> Assume a portfolio of $n$ zero coupon bonds mapped in risk by their respective DV01. Assume that the ZC portfolio created cannot exceed max and min set DV01 bounds on key maturities (1Y, 2Y, etc.). Also the total DV01 (sum of key maturities DV01) cannot exceed ... |
[![enter image description here][1]][1]
Show the expected return of the portfolio and how to derive the return variance of the long-short portfolio? (see picture)
[1]: https://i.stack.imgur.com/e2mJP.png |
In the run up to the 2008 financial crisis, many securities that were risky were rated much higher, at say, AAA. Raghuram Rajan in his book "Fault Lines" argues that this could does not necessarily have to arise from maleficence on the part of rating agences:
> Roughly 60 percent of all asset-backed securities were ra... |
To give some background, I give you some pieces from
[Geman, El Karoui and Rochet (1995)][1] which use a **change of numéraire**. A fundamental observation is the change of measure formula $\mathbb{E}^\mu[X]=\mathbb{E}^\nu\left[\frac{\mathrm{d}\mu}{\mathrm{d}\nu} X\right]$.
Defintion 2 defines what a numéraire is.
... |
## Background information
You need to put some structure on this to get a sense of what is going on. **Skip down if you're acquainted with asset pricing.** Define $p_t$ as the price of a security at time $t$. Its payoff at time $T$ is given by $x_T$. In the absence of arbitrage, there exists some positive random varia... |
I am trying to convert a 90 day SOFR market rate into a fair equivalent 3 month Libor rate. I understand since SOFR is by nature an in-arrears backwards looking rate, so for SOFR instruments such as a 90 day SOFR swap we need to wait until the end of the accrual period to know the fixing value.
I'm trying to use the ... |
I am working on calculate the cumulative return and I have monthly return rate as input. So I found formula of cumulative return:
$$
\text{cumulative} = (1 + r_1) (1 + r_2)(1 + r_3) - 1
$$
so I used `(df+1).cumprod()-1` in my python code
while when I used the result to calculate maximum drawdown, it shows weird.
[... |
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