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Are there values of the strike price for which an American put and European put have the same no-arbitrage price? |
I am looking into how to measure volatility, and I am not sure if I have confused myself too much in my research. So now I really need your help. So please either confirm my understanding of volatility, or else correct me.
The thing I am struggling with is conceptualizing that volatility isn't observable.
For exa... |
Why is volatility unobservable even ex post? |
Let me start from the beginning. What do you observe in financial markets? The data, the information that is given to you in as raw a form as possible, are things like bid prices, ask prices and trading volumes. That's data. Usually, people will take the mid-point of the bid and ask spread, define this as the fair valu... |
I'm a Management with Finance student and we have recently learned about options. Because I find it easier to learn these things when I have some context to apply them to, I put $100 in my brokerage account and started trading some options that I thought had potential.
What I got really confused with was the bid/ask p... |
Trading options - real life vs. textbook? |
The following statements are taken from [the Wikipedia page for forward measure][1].
> Let
$$B(T)=\exp \left(\int _{0}^{T}r(u)\,du\right)$$
be the bank account or money market account numeraire and
$$D(T)=1/B(T)=\exp \left(-\int _{0}^{T}r(u)\,du\right)$$
be the discount factor in the market at time 0 for maturit... |
How can I use the Radon-Nikodym theorem to show that forward measure is indeed measure? |
The following is a question taken from Heard on the Street.
> Let $L$ denote the three-month US dollar LIBOR rate.
Consider an interest rate swap arrangement where Party A pays $L$ to Party B, and Party B pays $24\% - 2L$ to Party A.
Can you reverse engineer this deal and express it in simpler terms?
The answe... |
The following is a question taken from Heard on the Street.
> Let $L$ denote the three-month US dollar LIBOR rate.
Consider an interest rate swap arrangement where Party A pays $L$ to Party B, and Party B pays $24\% - 2L$ to Party A.
Can you reverse engineer this deal and express it in simpler terms?
The answe... |
So I have Time Series data for Gas Spot and Futures Prices (first 6 front quarters and first 3 front years) from 2009-2019 and I want to evaluate the performance of a 3- year static hedge vs. 3- year Rolling hedge. Say I have a yearly need of 100’000 barrels of oil. So my plan was to pick a point in 2009 say 1.1 and th... |
Are there some alternatives to the CVaR measure for portfolio optimization, which are easier to implement for ex. with a linear program? They can be just approximations of CVaR or measures representing similar concepts.
**Edit**: I would _prefer_ some exact (analytical) methods over (meta-)heuristics, as they seem t... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/3CekE.png
If the returns are log-normal, how do I find the probability of negative returns, since log can only take positive values as an argument. Also, for time = more than 1 year, how do I find the probabilities? |
Is time diversification effective? |
I am going to perform [Tag:factor] crowding analysis for my dissertation and I am struggling to build factor portfolios from the S&P 500 in [Tag:R].
I built my dataset from the S&P 500 and I am able to add a column that signal me for each stock and time if it is going to be in the long ptf, in the short ptf or neithe... |
Do Fama-French factor portfolios require optimization? |
When computing the index-delta for a swap in a multi-curve framework, only the last cash tenor seem to show sensitivity. Could anyone explain with formulas why it is the case ?
For example a 15Y swap with index curve=EUR3M and discount=OIS, the EUR3M-delta with shows zeros for all tenors except the 15Y.
Anyone co... |
I was looking through the principal-protected notes issued by Lehman Brothers. One of them was the ["100% Principal Protection Absolute Return Barrier Notes Linked to the S&P 500 Index"](https://www.sec.gov/Archives/edgar/data/806085/000119312507207369/dfwp.htm). The description says:
> If the Index never closes 24.... |
Why do some principal-protected notes reset the gains to zero? |
Is it possible to create a synthetic long single stock future using the stock and it's vanilla options with the caveat that selling naked puts is NOT allowed? That is, you can write puts, but they must be cash-secured for 100% of the potential loss. This implies that a long-call short-put combo is not allowed.
If th... |
I am trying to forecast volatility. I am on the tactical asset allocation team. No one on our team knows machine learning or any programming languages. We are fundamental equity research analysts trying to find a way to forecast volatility. We were thinking of maybe using the VIX futures? |
How would you forecast volatility without using any programming languages or machine learning or anything of that sort? |
When computing the index-delta for a swap in a multi-curve framework, only the last cash tenor seem to show sensitivity. Could anyone explain with formulas why it is the case ?
For example a 15Y swap with index curve=EUR3M and discount=OIS, the EUR3M-delta will show zeros for all tenors except the 15Y.
Anyone cou... |
From experience, the most reliable data sources are from central banks. The problem is that they often only report the exchange rate to their own currency. For example, the European Central Bank reports every currency against the EUR ([ECB](https://sdw.ecb.europa.eu/browse.do?node=9691113)), the US Fed against the USD ... |
I'm currently preparing an exam about energy markets. The knowledge of notions of quantitative risk management accounts for the 50% of the total exam. During my university education, though, I didn't follow a specific risk management course.
What are the most important topics that I should revise? Could you suggest m... |
What does it mean in asset-pricing if something is priced or not?
|
Asset pricing: Is Beta priced or not? |
**The payoff**: you are missing one point. It's not only about the upside, it's also about the _downside_. Your performance $P_T$ at maturity $T$ (the Final Valuation Date) is basically:
$$P_T=P_0\left(1+1_{\left\{\max_{0\leq t\leq T}\left|\frac{I_t}{I_0}-1\right|\leq B\right\}}\left|\frac{I_T}{I_0}-1\right|\right)$$
... |
**The payoff**: you are missing one point. It's not only about the upside, it's also about the _downside_. Your performance $P_T$ at maturity $T$ (the Final Valuation Date) is basically:
$$P_T=P_0\left(1+1_{\left\{\max_{0\leq t\leq T}\left|\frac{I_t}{I_0}-1\right|\leq B\right\}}\left|\frac{I_T}{I_0}-1\right|\right)$$
... |
**Does anybody know wheter any bank uses D-Wave quantum annealer for doing optimization?**
Quantum annealers are single purpose quatum computers used for optimization. They implement quantum simulated annealing physically and thus they use real quantum effects, not simulated as on classical computers. For this they ... |
Basically, you have to choose whether to use a forward-looking or a backward-looking method of forecasting volatility. Let's start with the VIX. The VIX is an implied volatility index. Option pricing models require the volatility of the underlying asset as an input. Volatility is not an observed quantity, so the people... |
*What does it mean in asset-pricing if something is priced or not?*
It means does the market take risk factor X into account when setting the price of an asset. If it does then on average the asset will earn a Risk Premium over time. In the case of Beta it means do High Beta stocks earn a high long term return like ... |
I was looking through the principal-protected notes issued by Lehman Brothers. One of them was the ["100% Principal Protection Absolute Return Barrier Notes Linked to the S&P 500 Index"](https://www.sec.gov/Archives/edgar/data/806085/000119312507207369/dfwp.htm). The description says:
> If the Index never closes 24.... |
I have one yield curves for EUR6M and I want to produce EUR3M using a parallel shift to EUR6M curve.
I can just add spread in 6M curve. I am facing problem that my EUR3M curve will have many more expiry nodes then 6M because my spread has many more points. I want to interpolate my spread and 3M curve during addition ... |
trying to have a back of the enveloppe way of working out generic forward starting swap rates like 2y2y or 5y3y to put in a spreadsheet without too much loss of accuracy. Whats a good way to look at it / work it out for exple 3y2y?
Thanks all! |
A large amount of literature in finance accepts the standard deviation in price as if it were an accurate measure of "risk."
What are some other financial theories for how to allocate capital between assets that are not correlated? What principles would lead to an equal, cagr, or some other weighting?
Using sigma... |
What are non-variance (non Markowitz) based theories of capital allocation between non-correlated assets? |
A large amount of literature in finance accepts the standard deviation in return as if it were an accurate measure of "risk."
What are some other financial theories for how to allocate capital between assets that are not correlated? What principles would lead to an equal, cagr, or some other weighting?
Using sigm... |
Reading this news about the valuation of airbnb makes me wonder.
https://www.reuters.com/article/us-airbnb-debt/airbnbs-new-1-billion-investment-comes-at-lower-valuation-sources-idUSKBN21P3IM
> Silver Lake and Sixth Street received warrants that can be exercised
> at an \$18 billion valuation, below the \$26 bil... |
Convertible bond strike price typically higher or lower than current stock price? |
Risk Budgeting, to which Risk Parity is closely related.
Related to that: https://quant.stackexchange.com/questions/40340/risk-parity-equal-risk-contribution-with-tail-risk-measures?rq=1 |
Basically, you have to choose whether to use a forward-looking or a backward-looking method of forecasting volatility. Let's start with the VIX. The VIX is an implied volatility index. Option pricing models require the volatility of the underlying asset as an input. Volatility is not an observed quantity, so the people... |
Some context: The actuarial curriculum offers two papers on quantitative finance (QF):
- *CT8: Financial Engineering* (Utility theory, Measures of risk, MVPT, CAPM, Binomial model, stochastic calculus, Black-Scholes)
- *ST6: Financial derivatives* (this is essentially an extension of CT8, and goes into the mo... |
How comprehensively do actuarial exams cover quantitative finance? |
I've done some active trading in my personal account at Fidelity. With surprising frequency when I enter market orders I am filled at the *bid* for buys and at the *ask* for sells! How and why does this happen?
Following is an example: I entered three market orders to sell, during which the NBBO stayed at `7.17 / ... |
How am I buying at the bid? |
I am pricing a nth to default swap with 5 reference names. I coded in R the following routine but it looks there is a bug or something that R cant read. Could anyone please help me to improve my Code ?
Many thanks
----------------
```R
library(moments)
library(psych)
library(plyr)
library(copula)
lib... |
Sorry if this question is simple, but in the place I work they want to implement a daily check to check the swap's curve steepness. What does this mean and why would this check be necessary? |
Steepness of a curve? |
someone can help me to solve this problem:
$B_t$ is a Standard Brownian Motion.
Let $Y_t=tB_t$.
Using Ito formula, find drift and volatility of $Y_t$.
The result I found is $dY_t=B_tdt+t\cdot dB_t$ but in my book the solution is $B_tdt+dB_t$.
Which result is right?
Thank you |
Reading this news about the valuation of airbnb makes me wonder.
https://www.reuters.com/article/us-airbnb-debt/airbnbs-new-1-billion-investment-comes-at-lower-valuation-sources-idUSKBN21P3IM
> Silver Lake and Sixth Street received warrants that can be exercised
> at an \$18 billion valuation, below the \$26 bil... |
Newly issued convertible bond conversion price typically higher or lower than current stock price? |
Reading this news about the valuation of airbnb makes me wonder.
https://www.reuters.com/article/us-airbnb-debt/airbnbs-new-1-billion-investment-comes-at-lower-valuation-sources-idUSKBN21P3IM
> Silver Lake and Sixth Street received warrants that can be exercised
> at an \$18 billion valuation, below the \$26 bil... |
> I assume below because if now AirBnb is at 19, investor will just convert at 18 and make $1 immediately?
Wrong, that call option (the warrant part of the deal) has intrinsic value even if the current share price is above the strike. Else all investors would exercise their convertible bonds as soon as the spot goes... |
I'm new to curve construction for swaps pricing concept and I am having hard time to understand dual curve construction and what difference it from single curve ? how do we construct it ? can someone please care to explain this to me ? thank you |
Given the interest rate, prices of the stock at time 0,1,2 where T=2 is the expiry date, and the value of the American put at time 0, how do I calculate its strike price?
|
How to calculate strike price for an American put given its value at time 0 and the binomial tree of stock prices? |
Is it possible to attribute performance of indices (monthly returns and risk measures - Sharpe ratio, etc.) to their sector weights (if I know them)?
Example: I know the monthly performance of various indices and I know the proportions they invest in each sector each month (e.g. health care 10%, utilities 5%, etc. i... |
Do methods exist to determine trading ranges around benchmark weights/strategic asset allocation weights for a tactical asset allocation from the correlation structure between the individual asset classes and their respective information ratios?
I am aware of the following approaches:
1. The approach laid out in... |
Given the interest rate, prices of the stock at time 0,1,2 where T=2 is the expiry date, and the value of the American put at time 0, how do I calculate its strike price? The question gives it the same value as the European put. Does this mean that it must have the same strike price as the European put?
|
The formula for the US Dollar Index (DXY) that every resource on the internet provides is:
$$DXY = 50.14348112 × EURUSD^{-0.576} × USDJPY^{0.136} × GBPUSD^{-0.119} × USDCAD^{0.091} × USDSEK^{0.042} × USDCHF^{0.036}$$
Some also mention that DXY is the weighted geometrical mean of these exchange rates.
So by calculat... |
Where does the 50.14348112 coefficent in the US Dollar Index formula come from? |
I am having trouble filling in a few steps in the derivation.
From Martin (2017), we get the following assumptions:
1. Constant continuously compounded rate $r$;
2. The underlying doesn't pay dividens;
3. Under the risk-neutral measure, the underlying follows $dS_t = r S_t dt + \sigma_t dZ_t$.
The fair st... |
In a scenario where no historic data exists in an option market, how would one come up with implied volatility for pricing of options, under the Black-Scholes-Merton model?
My professor has suggested using realised volatility from the stock prices and incorporating a profit margin. Is there some relationship that e... |
someone can help me to solve this problem:
$B_t$ is a Standard Brownian Motion.
Let $Y_t=tB_t$.
Using Ito formula, find drift and volatility of $Y_t$.
The result I found is $dY_t=B_tdt+t\cdot dB_t$ but in my homework paper the solution is $B_tdt+dB_t$.
Which result is right?
Thank you |
1. Can't new data be divided into those that ought be digested by humans, and those that don't?
2. If so, why aren't humans digesting the data that ought be digested by humans?
[Sell-offs could be down to machines that control 80% of US stocks, fund manager says](https://www.cnbc.com/2018/12/05/sell-offs-could-b... |
Why doesn't algorithmic trading require humans to digest new data? |
The "magic" in Cover is the rebalancing effect across all of the multiplicities of all the possible portfolios. At least, assuming, or at least allowing for, infinite time.
In its presented form, take 2 securities and construct 101 portfolios, each containing 0-100% integer proportions of either. AND REBALANCE ALL ... |
I have historic time series for spot and futures and I want to now simulate future price paths for 1 day to get the distribution and from there compute the value at risk. My question is now since i am generating correlated random numbers, what correlation am i supposed to input into the cholesky decomposition. Is it su... |
I hedge a portfolio of Global Equities (200 stocks within MSCI World universe) by shorting futures on MSCI World Net Total Return. The hedge is calculated using Beta. Beta is calculated using a risk model looking at historical volatility for the past year.
When there is no market crash, volatility is relatively low ... |
What is swimming delta as in risk attribution? |
What is "swimming delta" as a risk attribute in pnl explain? |
From Lando (2004)* I am trying to replicate the following figure (Section 2.6 Default Barriers: The Black-Cox Setup):
[![enter image description here][1]][1]
The spreads are computed as follows:
$$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$
where:
- $T$ is the time to maturity of the bond,
- $D$ is the fac... |
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**):
[![enter image description here][1]][1]
The spreads are computed as follows:
$$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$
where:
- $T$ is the time to maturity of the bond,
- $D$ is the... |
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**):
[![enter image description here][1]][1]
The spreads are computed as follows:
$$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$
where:
- $T$ is the time to maturity of the bond,
- $D$ is the... |
For Monte Carlo simulation it is necessary to suggest a distribution function $F$.
You want to simulate based on the observations $(S_{1}, \dots , S_{t}, F_{1}, \dots, F_{t})$. Here, you assume your risk factor changes $(X_{S,t+1} = \log(S_{t+1}/S_t), X_{F,t+1}= \log(F_{t+1}/F_t))$ to be bivariate normal. Now, you... |
From Investopedia:
> Essentially, CFDs are used by investors to make price bets as to
> whether the price of the underlying asset or security will rise or
> fall.
That's the same with stocks, right?
So What is an example situation where it would be better use CFDs instead of stocks? |
Why someone would prefer CFDs rather than stocks? |
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**):
[![enter image description here][1]][1]
The spreads are computed as follows:
$$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$
where:
- $T$ is the time to maturity of the bond,
- $D$ is the... |
We know that white noise isn't forecastable because of its random aspect. White noise is also stationary, and which is confusing me, is that we always try to make a serie stationary to make forecasts, hence I don't understant why white noise stationarity doesn't make it forecastable...
The only conclusion I see is t... |
white noise not forecastable ? stationarity doesn't imply forecastability? |
"Swimming delta" is another name for "floating delta" (as opposed to sticky delta). books.google.com/books?id=LnLgAgAAQBAJ&pg=PA170 Glen Swindle's excellent book "Valuation and Risk Management in Energy Markets" happens to be in Google books and explains it well on page 170.
I'm just going to quote from his book he... |
From Lando (2004)* I am trying to replicate the following figure (Section **2.6 Default Barriers: The Black-Cox Setup**):
[![enter image description here][1]][1]
The spreads are computed as follows:
$$s(T) = \frac{1}{T}\ln\frac{D}{B_0}-r$$
where:
- $T$ is the time to maturity of the bond,
- $D$ is the... |
Let $X_T,Y_T$ be the terminal values of two price processes following Continuous Gaussian Motion (I.E.) let us assume no jumps. Further assume the correct forwards/futures price is given by $F^X_{t,T} = E_t[X_T],F^Y_{t,T} = E_t[Y_T]$.
We know then the following equation is True:
$cov_t(X_T,Y_T) = E_t[X_TY_T] - F... |
I am having trouble filling in a few steps in the derivation.
From Martin (2017), we get the following assumptions:
1. Constant continuously compounded rate $r$;
2. The underlying doesn't pay dividens;
3. Under the risk-neutral measure, the underlying follows $dS_t = r S_t dt + \sigma_t dZ_t$.
The fair st... |
What's the meaning of linearity in classical statistics in Prado's book? |
Because as the number of data points grows, the sum of returns would become more and more normal. As that becomes normal, smile becomes flatter and flatter. Why does the sum of returns become normal as number of data points increase? Because some periods where the volatility was high would be offset by other periods wh... |
I have calculated the VIX implied volatility according to the CBOE Whitepaper:
\begin{equation*}
\sigma^2 = \frac{2}{T} \left(\sum_i \frac{\Delta K_i}{K_i^2} Q(K_i) e^{rT} \right) - \frac{1}{T} \left( \frac{F_0}{K_0} - 1 \right)^2
\end{equation*}
on an intraday basis (1-Minute) for two maturities around 30 da... |
The risk free rate is $0,01$ while the risky rate follows a $2$ period binomial model and the risky rate at time $0$ is 1, u= $1,5$ and d=$0,6$: how can I determine a Swap rate of IRS with maturity 2 years (whose floating leg is represented by risky rates)? My idea is to create the binomial tree for the risky rate in o... |
The risk free rate is $0,01$ while the risky rate follows a $2$ period binomial model and the risky rate at time $0$ is 1, where u= $1,5$ and d=$0,6$: how can I determine a Swap rate of IRS with maturity 2 years (whose floating leg is represented by risky rates)? My idea is to create the binomial tree for the risky rat... |
I am trying to get my head around the CAPM model and all the intricacies of portfolio management. I have written some code to help me visualise what happens to the risk-return characteristics of my portfolio as I vary the weightings amongst three stocks (classic bullet shape).
What I don't understand is what is spec... |
I don't think the accepted answer fully answers the question. In particular simulating a non-stationary VAR is not guaranteed to give you a cointegrated system, and we may be interested in simulating $n>2$ time series. To be specific, the question is asking how to simulate a vector time series $Y_t \in \mathbb{R}^n$ wi... |
I don't think the accepted answer fully answers the question. In particular simulating a non-stationary VAR is not guaranteed to give you a cointegrated system, and we may be interested in simulating $n>2$ time series. To be specific, the question is asking how to simulate a vector time series $Y_t \in \mathbb{R}^n$ wi... |
I don't think the accepted answer fully answers the question. In particular simulating a non-stationary VAR is not guaranteed to give you a cointegrated system, and we may be interested in simulating $n>2$ time series. To be specific, the question is asking how to simulate a vector time series $Y_t \in \mathbb{R}^n$ wi... |
How are you deriving $r_s$? If you use a WACC calculator like this one [WACC calculator](https://assetbrief.com/resources/wacc-calculator-9xr5y8r), the cost of equity definitely matters. Based on your derivation, let's look at an extreme case where a company is financed 100% through debt. It would say that the cost of ... |
I am looking to estimate the CVA/DVA for a portfolio of options.
For simplicity sake, let's assume there are two FX options in the portfolio, one long and one short. Both options have the same expiry and are on the same underlying.
I understand my exposure on day one is the current mtm and I can estimate the po... |
When people mention "commodity options", they almost invariably mean "options on commodity futures contracts". Why do commodity options have futures as underlying, and not the commodities themselves as underlying? In other words, why do people trade options on commodity futures rather than options on commodities? |
Why are options on commodity futures traded instead of options on spot commodities? |
Suppose I am a farmer who entered into a futures contract to deliver crops in 6 months. Suppose that a natural disaster destroyed all my crops just before the delivery date. As a result, it will be impossible for me to deliver. What happens to me, the other party, and the futures contract? Do I default on my obligation... |
What happens to farmers' futures contracts if a natural disaster destroys their crops? |
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