instruction stringlengths 15 26.1k |
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I'm dissatisfied with the usefulness of delta and would like to get your feedback on a slight tweak on it.
##Example
Consider two options for a made-up stock at \$5 with IVs around 120%.
###Option A:
- ITM call expiring in 180 days
- Purchase price: \$2.03
- Delta: 0.76
###Option B:
- ITM call expiring in... |
I am working on calculate the cumulative return, and I have monthly return rate as input.
So I found formula of cumulative return:
cumulative = (1 + return1) * (1 + return2) * (1 + return3) - 1
so I used (df+1).cumprod()-1 in my python code
while when I used the result to calculate maximum drawdown, it shows w... |
I will be glad to help, but let me first advise you away from working on this topic until you have an academic position. This topic has been poison for me, but I am slogging on anyways. Before you use anything I do, get permission from your academic advisor.
I have an unpublished article on options pricing, and I ... |
As mentioned in Carr-Madan's paper, [here][1], the European call option is:
$$
C_T(k)=\frac{e^{\alpha k}}{\pi}\int_0^\infty\mathcal{Re}\left(e^{-iuk}\psi(u)\right)du
$$
where
$$
\psi(u)=e^{-rT}\frac{\phi_T(u-(\alpha+1)i)}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u}
$$
and $\phi_T(u)$ is the characteristic function... |
Suppose we have the following subordinated stochastic differential equations:
$dR(t)=\mu dt+\sigma (Y(t))dW_{1}(t)$
$dY(t)=f(Y)dt+g(Y)dW_{2}(t)$,
where $W_i$'s are standard Wiener process such that $dW_{i}(t)=\xi _{i}(t)dt$, $\xi _{i}(t)$ being the zero-mean Gaussian White noise with $
\left< \xi _i\left( t \... |
The 13 Week Treasury Yield on [Yahoo Finance][1] is
Date Open High Low Close* Adj. close**
31 Mar 2020 0.0700 0.0700 0.0280 0.0300 0.0300
30 Mar 2020 -0.0430 0.0200 -0.0530 0.0130 0.0130
27 Mar 2020 -0.1080 -0.0580 -0.1080 -0.0580 -0.0580
The 3 month Treasury Yield on [U.S.... |
Why is the treasury yield on Yahoo finance different from that on U.S. Department of Treasury? |
### Practical Approach to Convexity & Timing Adjustments
Omitting derivations because earlier answers have already covered them, let's address how we can explicitly calculate the adjustments. Hull (pp. 765, equation (33.2), OFaOD 9th ed) gives an explicit formula for the convexity and timing adjustment for a CMS swa... |
### Practical Approach to Convexity & Timing Adjustments
Omitting derivations because earlier answers have already covered them, let's address how we can explicitly calculate the adjustments. Hull (pp. 765, equation (33.2), OFaOD 9th ed, 2015) gives an explicit formula for the convexity and timing adjustment for a C... |
A DCC-GARCH model specifies the volatility of each asset as a univariate GARCH model (this is the first stage of DCC). Therefore, I am not sure whether you can actually use DCC for modelling spillovers. How can there be spillovers when the volatility of any given asset is modeled using only the past information on that... |
Some markets support pre-market and after-market trades. These trades take place outside the opening hours of the stock exchange. My questions are:
* If the stock exchange is closed, how and where do these after-hours trades take place?
* Who records the transactions? |
Where do trades take place when the stock exchange is closed? |
The problem: you do not know the correct ticker for stocks listed on multiple exchanges.
The solution: Use the [search endpoint](https://www.alphavantage.co/documentation/#symbolsearch) to find the ticker.
Let's say we want to find the correct ticker for Tsingtao Brewery listed in Hong Kong and Shanghai:
1. Se... |
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 w... |
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
m = 10000 # number of simulations
n = 5 # number of assets
I... |
I am trying to bootstrap the 6m sterling swap curve using the depos and swap rates codes are below. I am a newbie and have been following (or at least trying to!) the python cookbook. I wanted to know how do I add FRA to the mix?i.e. how do I create helpers for FRAs
assuming I have these FRAs
```
FRAs = {(1, 7):... |
There is a new text by Antoine Savine that combines some of the more traditional information found in Meyers etc with a solid foundation of applications to writing a quant library, the table of contents can be found here:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3281877
|
Below is a problem that I am working on. I believe that my incomplete solution is correct as far as it goes. I would like to know if my solution is incorrect. I plan to solve the system of two equations by using a computer program such as SciLab. What I am wondering is, does this system of equations have a unique solut... |
I am trying to bootstrap the 6m sterling swap curve using the depos and swap rates codes are below. I am a newbie and have been following (or at least trying to!) the python cookbook. I wanted to know how do I add FRA to the mix?i.e. how do I create helpers for FRAs
assuming I have these FRAs
```
FRAs = {(1, 7):... |
Below is a problem that I am working on. I believe that my incomplete solution is correct as far as it goes. I would like to know if my solution is incorrect. I plan to solve the system of two equations by using a computer program such as SciLab. What I am wondering is, does this system of equations have a unique solut... |
Having to implement a replication strategy for European options, I encounter the following problem:
- Delta tells me how many shares to hold at time t in my replication strategy. To do so, I simply iterating through times t up to T, and there comes the last portfolio rebalancing. At t=T, in the delta formula, $\Delt... |
Black-Scholes Delta value at maturity? |
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... |
So I was trying to estimate the performance of a static hedge vs dynamic hedge in the electricity market and I came up with some weird findings. When I used the minimum variance hedge approach using regressed returns of spot to futures I was receiving an R-Square of around 0.05 (Hedge effectiveness also in this case) a... |
If you regress spots and futures prices you are likely to end up with a case of spurious correlations (it’s an actual article by Granger and easily found). Perhaps a cointegration analysis would be a better tool. This is because the time series may not be stationary. Returns are typically (more) stationary which is why... |
There is a missing link to early options pricing literature which had been overlooked. Put-call parity along with static delta hedging were understood in actionable detail well before BSM and trading and risk management were accomplished through heuristic methods which indeed continued to be used after BSM.
Would po... |
I am working on the following equation (I want to apply Ito's lemma on it):
[![enter image description here][1]][1]
and I know that:
[![enter image description here][2]][2]
and also
[![enter image description here][3]][3]
and
[![enter image description here][4]][4]
[1]: https://i.stack.img... |
I have a quick question.
Currently I study daily OHLCVs of some stocks and find that many of them have the Hurst exponent not being equal to 0.5. I know that if it is less than 0.5 then it is a mean reverting series, if more than then it is a trending series. But my question is if $H$ $\neq$ $0.5$ then it is a serie... |
If you regress spots and futures prices you are likely to end up with a case of spurious correlations. Perhaps a cointegration analysis would be a better tool. This is because the time series may not be stationary. Returns are typically (more) stationary which is why regressing them is usually more sensible. But no gua... |
I used daily log-returns to estimate the parameters in the Bates model, and I want to price a contingent claim using these estimates. I know that I have to distinguish between parameters estimated under $P$ and parameters calibrated under $Q$. But let us for a moment assume market price of risk is zero, and these to me... |
I have calculated exponentially weighted variances (and covariance) for a future and the underlying index.
Now that I have exponentially weighted variances for my 2 assets using a lookback period of 1 year, and knowing that the portfolio of 2 assets volatility depends on the correlation between these 2 assets, do I ... |
Imagine a portfolio is made of 20m USD invested in equities and -18m USD of MSCI World futures (sell 18m USD short). The annualised volatility of the 20m USD in equities (Equ.) is 15% and the annualised volatility of the -18m USD in futures (F.) is 18%. Correlation between the equities and the future is 85%.
The aim... |
Portfolio VaR of a hedge portfolio (long index, short future): What total exposure to take to calculate VaR? |
Consider $r(t)$ the spot rate of default-free interest where $B(t,T)$ represents the $T$-maturity zero-coupon bond price at time $t$. Also assume, we are taking the expectation under the risk-netural probability measure $P$. Assume $0\leq j< k.$
$E_P(e^{-\int_j^k r(t)dt})=B(0,j)$
(1) Is this expression correct?
... |
In the Black-Scholes formula, the unit of time is usually in years from what I understand. An [online calculator][1] I found allows the users to input the time in days and years.
Would the formula still be accurate if I were to plug in say 1 hour for the time variable, by first converting it to years, given that 1 h... |
Does the Black-Scholes formula work when unit of time is in hours? |
I'm not sure what you're asking here quite, it seems to me that you are inputting a shorter time to maturity (from one day to one hour) and noticing a decrease in the contract value. Theta, the derivative of the option price with regards to time, is negative for for all options so this will always be the case no matter... |
Say you have a portfolio consisting of options each having a market implied volatility. If you now use some stochastic volatility model like GARCH to calibrate the real world volatility of the underlying, and then perform simulations of the correlated stock processes using the GARCH stochastic volatility. What volatili... |
I will be glad to help, but let me first advise you away from working on this topic until you have an academic position. This topic has been poison for me, but I am slogging on anyways. Before you use anything I do, get permission from your academic advisor.
I have an unpublished article on options pricing, and I ... |
I'm currently working with the following GARCH process from Heston and Nandi (2000):
\begin{align*}
r_{t+1} - r_f &= \lambda h_{t+1} - \frac{h_{t+1}}{2} + \sqrt{h_{t+1}}z_{t+1} \\
h_{t+1} &= \omega + \beta h_t + \alpha \left( z_t - \gamma \sqrt{h_t} \right)^2
\end{align*}
given $z_{t+1} \sim N(0,1)$, w... |
Let's look at below UTX/RTN merger as an example:
https://www.fool.com/investing/2020/03/30/raytheon-united-technologies-merger-gets-green-lig.aspx
> The merged companies will from that moment forward be known as
> Raytheon Technologies Corporation, and the stock of this new, merged
> entity will trade as ... |
Let's look at below UTX/RTN merger as an example:
https://www.fool.com/investing/2020/03/30/raytheon-united-technologies-merger-gets-green-lig.aspx
> The merged companies will from that moment forward be known as
> Raytheon Technologies Corporation, and the stock of this new, merged
> entity will trade as ... |
Ideally code in SAS, R, Python or Matlab for calculations involving counterparties holding positions in energy markets on multiple price curves, with a Monte-Carlo methodology or a simpified parametric approach? |
It's not clear to me that most people understand what it means to "learn the dynamics" of a problem and momentum vs mean-reversion really just sounds like vague narrative in a world where the details matter more than anything.
Is there anything about this metaphor that is more subtle and I am missing?
Are people ... |
Is there anything about this metaphor of momentum and mean-reversion in markets that is more subtle, more general. What *factors* are amenable to the interpretation?
Are people almost always referring to this kind of setup with log prices $x_t$, latent "momentum" $v_t$, latent mean $m_t(X_t)$, $X_t \equiv \{X_s;s\le... |
What the most general but precise description one can make about mean-reversion and momentum strategies? |
The first thing you should ask concerns the amount of information contained in your estimator of the Hurst exponent.
Let's say you approach this problem from the perspective of frequentist statistics. No matter how you compute the Hurst exponent, the fact of the matter is that it is a statistic (i.e., a function of ... |
I'm currently working with the following GARCH process from Heston and Nandi (2000):
\begin{align*}
r_{t+1} - r_f &= \lambda h_{t+1} - \frac{h_{t+1}}{2} + \sqrt{h_{t+1}}z_{t+1} \\
h_{t+1} &= \omega + \beta h_t + \alpha \left( z_t - \gamma \sqrt{h_t} \right)^2
\end{align*}
given $z_{t+1} \sim N(0,1)$, w... |
I'm currently working with the following GARCH process from Heston and Nandi (2000):
\begin{align*}
r_{t+1} - r_f &= \lambda h_{t+1} - \frac{h_{t+1}}{2} + \sqrt{h_{t+1}}z_{t+1} \\
h_{t+1} &= \omega + \beta h_t + \alpha \left( z_t - \gamma \sqrt{h_t} \right)^2
\end{align*}
given $z_{t+1} \sim N(0,1)$, w... |
I've just found out about this forum while searching for answers for an InfluxDB performance "issue". I'm using it to store financial tick data (7 fields per row), query it and process it into candles. At the moment I'm querying around 1 billion ticks, about 4 years of data, for a total weight of about 28GB. The first ... |
just curious to see if anyone here has come across a book or books on options trading with the practitioner in mind? My lecture slides for instance, go through black scholes and the ins and outs of the derivation, ito’s lemma etc, but they are very light on how the BSM is actually used in practice (ie reality). Any ide... |
Books on options trading with a practical bent? |
if someone could provide some clarity on the below:
1. What is meant by 'Implied FX-OIS Basis'? For example: "ON JPY trading at parity, 1W implied OIS basis moved 70BP" and "3M Implied OIS basis moved 25 BP to imply -130 BP (3M LIBOR XCCY is +4 )". This is recent commentary, however I have seen the term/concept band... |
if someone could provide some clarity on the below:
1. What is meant by 'Implied FX-OIS Basis'? For example: "ON JPY trading at parity, 1W implied OIS basis moved 70BP" and "3M Implied OIS basis moved 25 BP to imply -130 BP (3M LIBOR XCCY is +4 )".
My attempt: We have Interest rates implied by FX Swap points compa... |
if someone could provide some clarity on the below:
1. What is meant by 'Implied FX-OIS Basis'? For example: "ON JPY trading at parity, 1W implied OIS basis moved 70BP" and "3M Implied OIS basis moved 25 BP to imply -130 BP (3M LIBOR XCCY is +4 )".
My attempt: We have Interest rates implied by FX Swap points compa... |
I am doing event study and calculated Abnormal return with the help of market model [$r_\text{actual} - (\alpha+\beta r_m)$]. Then I calculated t-stat and next CAR (cummulative abnormal return). Now I have two doubts:
1. I am confused when there is overpriced or underpriced security.
2. What more can I interpret... |
I am an doing event study and calculated abnormal return with the help of market model [$r_\text{actual} - (\alpha+\beta r_m)$]. Then I calculated a t-stat and next CAR (cumulative abnormal return). Now I have two doubts:
1. I am confused when there is overpriced or underpriced security.
2. What more can I inter... |
if someone could provide some clarity on the below:
1. What is meant by 'Implied FX-OIS Basis'? For example: "ON JPY trading at parity, 1W implied OIS basis moved 70BP" and "3M Implied OIS basis moved 25 BP to imply -130 BP (3M LIBOR XCCY is +4 )".
My attempt: We have Interest rates implied by FX Swap points compa... |
**Definitions**
Spot rate: the interest rate applied to a given spot investment to be repaid at maturity, as a single cash flow.
Par rate: the interest rate such that the PV of the cash flows (lets say semi-annually) between spot and maturity is equal to par.
**Example data**
For 6-monthly rates:
Spot ra... |
Why do increasing spot rates have to be equal to or larger than the corresponding par rates? |
I'm currently working with the following GARCH process from Heston and Nandi (2000):
\begin{align*}
r_{t+1} - r_f &= \lambda h_{t+1} - \frac{h_{t+1}}{2} + \sqrt{h_{t+1}}z_{t+1} \\
h_{t+1} &= \omega + \beta h_t + \alpha \left( z_t - \gamma \sqrt{h_t} \right)^2
\end{align*}
given $z_{t+1} \sim N(0,1)$, w... |
It is well know that VaR is not subaddtive measure which means that condition
$$
\text{VaR}(X+Y) \leq \text{VaR}(X) + \text{VaR}(Y),
$$
where $X$ and $Y$ are portfolios, is not satisfied. As a result, in some cases a simple division of portfolio to its subportfolios can lead to lower risk. Especially for this f... |
Why is VaR metric still used? |
it's tough trying to figure this out myself and I after a few hours I thought I'd ask for help:
I'm trying to do tick imbalance bars from 'Advances in Financial Machine Learning', using equations:
>>bt={bt−1,|Δpt|Δpt,if Δpt=0ifΔpt≠0 (1)
>>θT=∑Tt=1bt (2)
>>T∗=ArgminT(|θT|≥E0[T]|2P[bt=1]−1) (3)
and either... |
[enter image description here][1]it's tough trying to figure this out myself and I after a few hours I thought I'd ask for help:
I'm trying to do tick imbalance bars from 'Advances in Financial Machine Learning', using equations:
[![enter image description here][2]][2]
[![enter image description here][3]][3]
[!... |
[enter image description here][1]it's tough trying to figure this out myself and I after a few hours I thought I'd ask for help:
I'm trying to do tick imbalance bars from 'Advances in Financial Machine Learning', using equations:
[![enter image description here][2]][2]
[![enter image description here][3]][3]
[!... |
> how, if even possible, can we replicate a put option when
> short-selling the underlying is not allowed but we may trade inversely
> correlated stocks?
Ok first thing first. Your self-financing portfolio fails because you have two different risk factors (brownian motions), so you will never be able to replicate ... |
> how, if even possible, can we replicate a put option when
> short-selling the underlying is not allowed but we may trade inversely
> correlated stocks?
Ok first thing first. Your self-financing portfolio fails because you have two different risk factors (brownian motions), so you will never be able to replicate ... |
How could we use 5 year Tips minus 2 Year Tips to get insight of Inflation Expection? |
> how, if even possible, can we replicate a put option when
> short-selling the underlying is not allowed but we may trade inversely
> correlated stocks?
Ok first thing first. Your self-financing portfolio fails because you have two different risk factors (brownian motions), so you will never be able to replicate ... |
Why are we compounding continuously in Finance?
I have searched around, but i Cannot find an explination on why we actually do it.
I assume that we, in theory, do it, because every interest earned, is reinvested, but since that is not reality, then What is the answer?
Does it has something to do with the present... |
> how, if even possible, can we replicate a put option when
> short-selling the underlying is not allowed but we may trade inversely
> correlated stocks?
Ok first thing first. Your self-financing portfolio fails because you have two different risk factors (brownian motions), so you will never be able to replicate ... |
I am going to start a project with an US company. I do not really trust them. I want to know if they are a "real" company. In some European countries you can consult some (often paid) databases and there you can see their yearly revenues, profit, etc. I tried to find something similar in the USA but did not find it. An... |
I am trying to run a factor analysis on European corporate bonds and I need historical data since mid - late 90s on :
1.bonds listed
2.spread
3.firm´s total debt at the time
4.Sector of firm
5.DTS(duration times spread)
6.rating for each month
Do you know which databases provide may have these?
If... |
Why are we compounding continuously in finance?
I have searched around, but I cannot find an explination on why we actually do it.
I assume that we, in theory, do it because every interest earned is reinvested. But since that is not a reality, then what is the answer?
Does it has something to do with the present... |
I am currently trying to compute the P&L of a FX swap and to understand it's implications.
Let's say when we sell 1M EUR spot eur/usd at 1.08 and at the same time buy a one month month forward eur/usd at 1.10. Then the net cash flow would be + USD $ 16,835. I have a few questions on this.
1.) Is this a P&L gain ?... |
I am currently trying to compute the P&L of a FX swap and to understand it's implications.
Let's say when we sell 1M EUR spot eur/usd at 1.08 and at the same time buy a one month month forward eur/usd at 1.10. Then the net cash flow would be + USD $ 16,835. I have a few questions on this.
1.) Is this a P&L gain ?... |
Let the market risk be $\sigma_m=28\%$. A portfolio consists of four stocks, all with the same weight ($w_i=0.25$ for all $i$). We also know that $\sigma_a=18\%,\sigma_b=36\%,\sigma_c=22\%,\sigma_d=17\%$ where $\sigma_i$ is the standard deviation of stock $i$ and the portfolio's risk is $\sigma_p=24\%$
Calculate the... |
Let $X_t=e^{W_t}$ where $W_t$ follows the Wiener process. The answer is given as $X_t/2$.
My attempt at a solution: (which I'm afraid is poor from a mathematical standpoint)
I applied Ito's lemma as $dX_t=\frac{\partial X_t}{\partial W_t}dW_t+\frac{1}{2}\frac{\partial^2 X_t}{\partial W_t^2}(dW_t)^2$ and using the... |
Let $X_t=e^{W_t}$ where $W_t$ follows the Wiener process, calculate the drift. The answer is given as $X_t/2$.
My attempt at a solution: (which I'm afraid is poor from a mathematical standpoint)
I applied Ito's lemma as $dX_t=\frac{\partial X_t}{\partial W_t}dW_t+\frac{1}{2}\frac{\partial^2 X_t}{\partial W_t^2}(d... |
> Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift.
The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):
I applied Ito's lemma as
$$dX_t=\frac{\partial X_t}{\partial W_t}dW_t+\frac{1}{2}\frac{\partial^2 X_t}{\partial W... |
What is vega really? |
Where to retrieve financial data? |
I'm writing my thesis on the relationship between financial performance and sustainability. For that I'm comparing the returns of 2 portfolios, the first having really good ESG scoring firms, while the second portfolio consists of similar firms to the first (same peer group), but showing poorer ESG scores. I'm then usi... |
Assume for now we are working in a stohastic volatility (SV) setting,
$$
dS_r = \sqrt{v_r} S_r dW
$$
and
$$
dv_r = a(v_r,r)dr + b(v_r,r) dZ
$$
with
$$
dWdZ = \rho dr
$$
Let $C(S_t,v_t,t)$ denote the SV price of a claim today. Let's define (variance) vega as the change in the option value if time $t$ varia... |
DDM definitely used in practice. Probably the Two Stage Dividend Discount model is used more often that the 1 stage or Gordon model described in the Wikipedia article.
From time to time there are surveys of techniques used by equity analysts, for example *Equity Valuation: A Survey of Professional Practice* by G. P... |
What is vega, really? |
Since I am very new to this topic, i am struggling to connect the option pricing with Spot Trading.
Is there a way to use the Black-Scholes model to derive entry and exit Events? Any blogs or papers?
I've just started to research the Black-Scholes model of using online courses on coursera and somehow fail to fin... |
How to use black scholes for spot trading? |
This is somewhat non-technical question, but it seems like this forum is still the best place for it.
I'm reading Shleifer's *Inefficient Markets*, and he points out that "for futures and options, close substitutes are usually available, although arbitrage may still require considerable trading", where substitutes r... |
To answer this question, we must fix a bit of the vocabulary, first. I will try to stick as close as possible to your conventions:
**Spot rate**: (also called _zero rate_) is the annualised rate of return on a non-coupon-bearing bond (hence _zero_). For a given maturity $t$, let us call $r_t$ the corresponding spot ... |
This is somewhat non-technical question, but it seems like this forum is still the best place for it.
I'm reading Shleifer's *Inefficient Markets*, where he points out that
> [...] for futures and options, close substitutes are usually available, although arbitrage may still require considerable trading
where... |
Is there a reason why futures and options have more substitutes than other financial instruments? |
To answer this question, we must fix a bit of the vocabulary, first. I will try to stick as close as possible to your conventions:
**Spot rate**: (also called _zero rate_) is the annualised rate of return on a non-coupon-bearing bond (hence _zero_ coupon bond). For a given maturity $t$, let us call $r_t$ the corresp... |
Quantopian is an incredible tool for the quant community, but it is Python based only.
Just wondering if is there anything like Quantopian in R that you reccommend? |
Is there anything like Quantopian in R? |
I'd like to check how QuantLib does interpolation on rates if I use ZeroCurve constructor. As it was mentioned [here](https://quant.stackexchange.com/a/44779/15127), by using `curve.nodes()` you can get a list of rates, used for interpolation.
So, if I build a curve with spot rates with the following code
```
impo... |
I am trying to replicate the beta value that yahoo calculates but I am getting different results. According to Yahoo, its beta is calculated using 5 year returns against the SP500: [yahoo beta][1]
[1]: http://finance.yahoo.com/education/stocks/article/101033/Beta
I assume that daily values are used though it ... |
How to use Radon-Nikodym theorem to show that forward measure is indeed measure? |
Assuming the options do not pay dividends, is there a strike price that satisfies this? |
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