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Can one use the GARCH model to estimate the realized volatility, [such as done in this paper][1], rather than forecast the volatility, from (high frequency) price/tick data?
[1]: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=306002 |
Several resources I saw introduce the notion of bid/ask spread when trying to price options in incomplete market, I don't understand why the notion is introduced since we are interested on the price that will be given by the seller of the option so why considering the bid-ask spread ? To be sure that the $bid\leq ask$ ... |
Why do we worry about the bid/ask spread when pricing option in incomplete market? |
There exists a LIBOR Market Model with stochastic volatility for pricing and hedging exotic (e.g. path-dependent) interest rate options with smile. However let us consider the following approach:
1. calibrate standard SABR to vanilla options with available tenors
2. interpolate (either linearly or more sophistica... |
I try to understand the equal risk contribution (ERC) portfolio as described in [On the Properties of Equally-Weighted Risk Contributions Portfolios](https://www.researchgate.net/publication/45397778_On_the_Properties_of_Equally-Weighted_Risk_Contributions_Portfolios?enrichId=rgreq-e57c2f5c85f2f372662ff322fd9faf92-XXX&... |
Can one use the GARCH model to estimate the realized variance/volatility, [such as done in this paper][1], rather than forecast the volatility, from (high frequency) price/tick data?
[1]: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=306002 |
> Can someone provide a detailed example to prove the following statement: "When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk."
##### Proof:
The future value at investment horizon time $t$ of a set of $n$ cashflows is
$$
FV(t;y) = \sum_{i=1}^n C... |
**The TLDR; to your question**:
- **How can one use realized volatility as a volatility model to do out-of-sample prediction?** You extended known models to incorporate additional information procured from high-frequency data. Going from the vanilla GARCH to a realized GARCH model can be done by adding an auxiliar... |
Let us consider a stochastic market model with a fixed short (risk-free) rate $r\in\mathbb{R}$. A trader can obtain unsecured funding at a rate $f_t:=r+s_t$ where $s_t$ is its stochastic funding spread. We assume this spread follows geometric Brownian Motion dynamics:
$$\label{model:fnd-sprd}\tag{1}
\text{d}s_t=\sigm... |
I'm considering an extension of the binomial model where the risky asset can take three values at each node, that is $
S_{t+1}=\left\{
\begin{array}{ll}
S_t\cdot u\\\nonumber
S_t\cdot c\\
S_t\cdot d
\end{array}
\right.$
with $0<d<c... |
I'm considering an extension of the binomial model where the risky asset can take three values at each node, that is $
S_{t+1}=\left\{
\begin{array}{ll}
S_t\cdot u\\\nonumber
S_t\cdot c\\
S_t\cdot d
\end{array}
\right.$
with $0<d<c... |
Let us consider a stochastic market model with a fixed short (risk-free) rate $r\in\mathbb{R}$. A trader can obtain unsecured funding at a rate $f_t:=r+s_t$ where $s_t$ is its stochastic funding spread. We assume this spread follows geometric Brownian Motion dynamics:
$$\label{model:fnd-sprd}\tag{1}
\text{d}s_t=\sigm... |
It allows a bit more flexibility when fitting then requiring that the prices match the mid price? Quoted prices might be preferred over last traded price as the latter could be stale for less liquid contracts. As pointed out by @nbbo2 this makes all the prices between the bid and the ask no-arbitrage prices. |
A perpetual option solves the Cauchy-Euler ODE
$$(r-q)SV_S+\frac{1}{2}\sigma^2S^2V_{SS}-rV=0$$
The general solution is $$V(S)=aS^{n_1}+bS^{n_2}$$ where $n_1>1$ and $n_2<0$.
----------
For an **American put** option with payoff $K-S$, we find $a=0$ because we require $V(S)=0$ as $S\to\infty$. We find the fre... |
I am guessing that the first model you are referring to is the one from Rebonato: Linking caplets and swaptions prices in the LMM-SABR model (2009)? If yes, then I would say that your approach is a simplification of his model. Assuming that you are still able to calibrate to a set of swaptions that are of interests wit... |
I don't see anything inherently wrong at first sight, although adding a mean-reverting drift might be more realistic (e.g. a CIR process with $\sqrt(
{s_t}$), especially for long maturities? Also I don't know what you mean by instantaneous risk factors exactly but a log-normal process is used for the instantaneous vol... |
----
**Prelude**
A valuation tree introduces a lattice structure for derivatives valuation. To not overburden notation, let us define the tree by $K$, the fixed number of nodes exiting each state with corresponding probabilities $p_1,p_2,\ldots,p_K$ and relative jump sizes $J_1,J_2,\ldots,J_K$. Forthermore, $N$ is ... |
Coherent risk measures are defined by number of axioms (see e.g. [Coherent Risk Measure][1]) but a question that does not seem well studied is how to use them.
Let's take a coherent risk measure $\rho$ and a position X in the book. The risk is $\rho(X)$ and suppose that you are extend with another asset Y. So, in a se... |
How to use coherent risk measure for evaluating price? |
There exists a LIBOR Market Model with stochastic volatility for pricing and hedging exotic (e.g. path-dependent) interest rate options with smile. However let us consider the following approach:
1. calibrate standard SABR to vanilla options with available tenors
2. interpolate (either linearly or more sophistica... |
I don't see anything inherently wrong at first sight, although adding a mean-reverting drift might be more realistic, e.g. a CIR process with $\sqrt{s_t}$, especially for long maturities? Also I don't know what you mean by instantaneous risk factors exactly but a log-normal process is used for the instantaneous volatil... |
Derive vega for Black-Scholes call from this formula? |
Are there any advantages of pricing and hedging plain vanilla interest rate options with more complex SABR LMM instead of simplier SABR model? Should one always go with the SABR LMM as a universal model when managing a book of vanillas and exotics in order to eliminate the possibility of cross-model arbitrage? |
I can hardly think of any advantage of using the SABR LMM for a vanilla book. The SABR LMM should be anyway calibrated to the smile (usually marked in the SABR model)? If the calibration is perfect (usually it is not), the price from the SABR LMM would be the same. |
> Can someone provide a detailed example to prove the following statement: "When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk."
##### Proof:
The future value at investment horizon time $t$ of a set of $n$ cashflows is
$$
FV(t;y) = \sum_{i=1}^n C... |
> Can someone provide a detailed example to prove the following statement: "When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk."
##### Proof:
The future value at investment horizon time $t$ of a set of $n$ cashflows is
$$
FV(t;y) = \sum_{i=1}^n C... |
A perpetual option solves the ODE
$$rSV_S+\frac{1}{2}\sigma^2S^2V_{SS}-rV=0$$
The general solution is $$V(S)=aS+bS^{\gamma}$$ where $\gamma=-\frac{2r}{\sigma^2}<0$.
----------
For an **American put** option with payoff $K-S$, we find $a=0$ because we require $V(S)=0$ as $S\to\infty$. We find the free bounda... |
Are you able to provide some suggestions for resources to get started with non-flat volatility modelling? The models I am interested in are the likes of CEV, Heston, SABR etc.
I have tried looking into Gatheral's *The volatility surface. A practitioner's guide* but it is a little too heavy so looking for something i... |
You may find [A Short Note on Volatility Models][1] an interesting summary providing bird's-eye overview of general ideas in volatility modeling.
I would higly recommed [SABR and SABR LIBOR Market Models in Practice][2] as a great reference to SABR and SABR LMM accessible without prior knowledge of these models.
... |
I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $F_{i}(t)$ fixing at $T_{i-1}$ and paying at $T_i$:
\begin{align}
dF_{i}(t) = (F_i (t) +\gamma) \sigma_i^I dt \ ,
\end{align}
where $\gamma$ is a constant shift and $\sigma_i^I$ is a random variable for ... |
The bid ask bounce is a phenomenon that you observe when you look at transaction prices in a quote driven market. Because buys and sells occur randomly, the price of the last trade changes randomly between the quoted ask and the quoted bid, this constant up and down movement is called the bid-ask bounce. For a simple e... |
The bid ask bounce is a phenomenon that you observe when you look at transaction prices in a quote driven market. Because buys and sells occur randomly, the price of the last trade changes randomly between the quoted ask and the quoted bid, this constant up and down movement is called the bid-ask bounce. For a simple e... |
Are there any advantages of pricing and hedging plain vanilla interest rate options with more complex SABR LMM instead of simpler SABR model? Should one always go with the SABR LMM as a universal model when managing a book of vanillas and exotics in order to eliminate the possibility of cross-model arbitrage? |
You may find [A Short Note on Volatility Models][1] an interesting summary providing bird's-eye overview of general ideas in volatility modeling.
I would highly recommend [SABR and SABR LIBOR Market Models in Practice][2] as a great reference to SABR and SABR LMM accessible without prior knowledge of these models.
... |
I am trying to imply BS volatilities using Yahoo Finance! API (MSFT options) and Reuters zero curves. I am ignoring dividends for the moment. Note that I've also cleant the options' data (e.g., I make sure that last trade day of the option was the day on which I am valuing, and to get a nice smile I've only kept OTM ca... |
What newsletters do you read daily that help you learn new methods, keep up to date on news, new research, etc in quant finance? Looking for something similar to those morning newsletters sent out by The Economist, Reuters, etc. |
What is a good daily newsletter? |
I am learning about using the OIS fixed rate to value a plain vanilla LIBOR swap. I'm using *Bond Math* by Smith, and the accompanying online addendum.
To bootstrap the discount factors, the author treats the OIS fixed rate as a par yield. If $d_i$ are the OIS discount factors, $R_N$ is the OIS fixed rate, $m$ is t... |
Is a swap fixed rate always a par yield? |
I think you get it wrong your definition of $d_1$. Shouldn't it be $d_1 = 1 / (\text{vol} * \text{dt}) * (\log(S/K) + (r - q + \text{vol}^2 / 2) * dt)$ and not $d_1 = 1 / \text{vol} * \text{dt} * (\log(S/K) + (r - q + \text{vol}^2 / 2) * dt)$ ? |
Apologies for the rather broad question! Essentially, I wanted to ask how to build an FX forward curve from scratch. I have some basic understanding of interest rates and discounting but I am lacking the theoretical and practical knowledge to figure out how it fits together to result in an EUR/USD SP to 2Y curve, for e... |
How to build an FX curve? |
From the perspective of an XVA desk, what does the NPV actually tell you in terms of Counter-party Credit Risk? I understand that CVA + DVA gives you the NPV of any given trade (at the simplest level), But what does that mean? As an example, if I have an underlying trade of an interest-rate swap and the XVA desk decide... |
I found this [repository][1] (options pricing in Python) where they adjust IV for asian options and they use it under regular BS76 model.
I could not find any proof of this result on the web, do you know any paper where a similar result is proved or do you have any idea from where to start?
[![formula][2]][2]
... |
**Q: How could one feed alternative bars into a machine learning model?**
It's the same as how you would normally feed time bars. You are only changing how you sample information, but the bar will have the same structure. So for each dollar bar you still get, lets say, "High/Low/Open/Close" as four features, which c... |
With the imminent transition from LIBOR to SOFR next year, what are the data points practitioners are using to construct a yield curve? Also, since LIBOR implicitly took into account credit risk of the counterparty due to the fact that this is an interbank rate, how will derivatives transactions incorporate credit ris... |
So rather than a call option C(S_t,t) we have some type of asset with asset price is given by S(x,t) where x is any type of variable that the asset price depends on. I.e Price of wooden desks, W(x,t) where x is price of wood. Or price of bitcoin, B(x,t) where x is price of electricity.
Could the FK equation be used... |
Can the Feynman-Kac formula be used for asset classes that don’t have options? |
I am trying to imply BS volatilities using Yahoo Finance! API (MSFT options) and Reuters zero curves. I am ignoring dividends for the moment. Note that I've also cleant the options' data (e.g., I make sure that last trade day of the option was the day on which I am valuing, and to get a nice smile I've only kept OTM ca... |
It seems to have vanished off the web but a few years back Dupire published two tradable realized volatility estimators in 2015, I even asked a question around it here. https://quant.stackexchange.com/questions/34333/what-makes-a-realized-vol-estimate-tradeable
However, the only presentation I could find on the web ... |
I am reading about ABS avoiding esoteric instruments with complex structures and I want to learn about pricing methods and trading of these instruments thinking about futures applications for decentralized finance (DeFi), do you think these instruments can be a good alternative for retail investor or is it something us... |
How to get started with ABS? |
I found this [repository][1] (options pricing in Python) where they adjust IV for Asian options and they use it under the regular BS76 model.
I could not find any proof of this result on the web, do you know any paper where a similar result is proved or do you have any idea from where to start?
[![formula][2]][2... |
I am doing the following exercise from a math finance textbook but I got stuck at the end of the part 2. I found nothing on the internet concerning solutions of exercises from this textbook (called Introduction to stochastic calculus applied to finance for those interested in). At the end of the exercise (that I have c... |
I have a 2 year maturity AA rated US corporate bonds, and I found that it has negative Z-spread, -0.00053.
Does it make sense or it's wrong? |
Is it possible to have negative Z-spread for a corporate bond? |
I'm looking for a way to measure the trend of two curves. For example, the figure below is my forecast for the S&P 500. It can be seen that although I haven't predicted the exact index price, the trends of the two curves are quite the same, including the rising and falling time and the turning point time. I have tried ... |
I am looking at the latest 10-Q SEC Filing of ATVI stock (March 31, 2022). (RefLink: https://investor.activision.com/sec-filings/sec-filing/10-q/0001628280-22-011987)
The Balance Sheet states the following:
[![ATVI Balance Sheet March 31, 2022][1]][1]
I would like to know how to calculate the Price Per Share. Ba... |
How to calculate Share Price based on Outstanding Shares of $ATVI? |
I am looking at the latest 10-Q SEC Filing of ATVI stock (March 31, 2022). (RefLink: https://investor.activision.com/sec-filings/sec-filing/10-q/0001628280-22-011987)
The Balance Sheet states the following:
[![ATVI Balance Sheet March 31, 2022][1]][1]
I would like to know how to calculate the Price Per Share (Bo... |
I am doing the following exercise from a math finance textbook but I got stuck at the end of the part 2. I found nothing on the internet concerning solutions of exercises from this textbook (called *Introduction to stochastic calculus applied to finance* by Lamberton & Lapeyre for those interested). At the end of the e... |
If Asset A is having 35% higher Sharpe Ratio than Asset B is that significant? Or should I consider that only if Asset A is having 50% higher Sharpe Ratio than Asset B. |
I am comparing two assets Rolling 10 Year Sharpe Ratios. I want to know what percentage increase in Sharpe Ratio is meaningful? |
If we look at the yield spreads on Bloomberg of some TIPS bonds, we see they have credit spread < 0 (i.e. their yield is even lower than their benchmark treasury bonds)
Why is that the case. Shouldn't treasury bonds typically have the lowest yield? How does TIPS work to have lower yield than treasury?
|
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\partial^2 f}{\partial x^2}(S(t),t)\cdot\sigma^2\cdot S(t)^2\right) \,dt
$$
From then, they argue that this process ... |
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\partial^2 f}{\partial x^2}(S(t),t)\cdot\sigma^2\cdot S(t)^2\right) \,dt
$$
From then, they argue that this process ... |
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\partial^2 f}{\partial x^2}(S(t),t)\cdot\sigma^2\cdot S(t)^2\right) \,dt
$$
From then, they argue that this process ... |
Statistically Significant Four Factor Alpha for Monthly Returns Regression but non stat sig alpha for Daily Returns. Which one should I trust? |
**How do you find the corresponding market index of a stock?**
So for example, if I have a stock, we say MSFT, and I want to find its corresponding market index - how do I do this?
I thought I needed to start by finding out which index the stock, in this case MSFT, is included in and from there I know which is it... |
here is a screenshot of the FWCM screen on Bloomberg:
https://i.stack.imgur.com/2BT3y.jpg
I'm trying to check that I understand this by calculating the 1Y1Y which according to this matrix is 3.6877%.
So the inputs are 1Yr rate at 2.7737% and 2Yr rate is 3.2756%.
((2*.032756)-(1*.027737))/(2-1) =... |
In a live example: Today is June 14, 1 hour before market close, and \$SPY (S&P 500 ETF) is currently at \$372.28 and the June 15 \$350 strike Put is being quoted for \$0.13 on the bid and \$0.14 cents on the ask. The IV is 39.00%.
The $350 strike is currently 5.98% away from the spot/ATM price.
An implied volat... |
How do market-makers profit & manage inventory when customers sell a lot of deep OTM options? |
In a live example: Today is June 14, 1 hour before market close, and \$SPY (S&P 500 ETF) is currently at \$372.28 and the June 15 \$350 strike Put is being quoted for \$0.13 on the bid and \$0.14 cents on the ask. The IV is 39.00%.
The $350 strike is currently 5.98% away from the spot/ATM price.
An implied volat... |
In the situation that you describe, the MM will indeed probably lose a large part of the premium, but also has a small chance of making a large amount of money if the market completely falls apart. Thus, the book is not balanced and the MM has to live with that profile. If you think it’s a bad situation, would you pre... |
Let $r_{s, t}$ and $r_{f, t}$ be the return rates of the spot and futures of a commodity at time $t$. The hedging ratio based on variance minimization is calculated by finding the minimum of the variance of the combined returns:
$$\beta_t = \rho_{sf, t} \frac{\sigma_{s, t}}{\sigma_{f, t}},$$
where $\rho_{sf, t}$ ... |
Let $r_{s, t}$ and $r_{f, t}$ be the return rates of the spot and futures of a commodity at time $t$. The hedging ratio based on variance minimization is calculated by finding the minimum of the variance of the combined returns:
$$\beta_t = \rho_{sf, t} \frac{\sigma_{s, t}}{\sigma_{f, t}},$$
where $\rho_{sf, t}$ ... |
I think the answer to this question is that it's not as simple as you're making it out to be.
If i'm market making options, there is no requirement that i hedge that option with the exact same strike, perhaps i am happy to sell a different (portfolio of) option(s) such that i am net recieving premium and am comforta... |
I think the answer to this question is that it's not as simple as you're making it out to be.
If i'm market making options, there is no requirement that i hedge that option with the exact same strike, perhaps i am happy to sell a different (portfolio of) option(s) such that i am net recieving premium and am comforta... |
I am trying to reperform a vanilla swap. The problem is that the vanilla swap object does not seem to be using the exact fixings of the underlying index.
```
import pandas as pd
import QuantLib as ql
calendar = ql.SouthAfrica()
notional = 481000000
fixed_rate = 0.0565
day_count = ql.Actual365Fixed()
sett... |
I am trying to reperform a vanilla swap. The problem is that the vanilla swap object does not seem to be using the exact fixings of the underlying index.
```
import pandas as pd
import QuantLib as ql
calendar = ql.SouthAfrica()
notional = 481000000
fixed_rate = 0.0565
day_count = ql.Actual365Fixed()
sett... |
I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) and stumbled across something similar; iterated Brownian motions. Iterated Brownian motions have been defined as follows:... |
That's because the default of the library is to estimate floating-rate coupons at par on the curve. This can cause the end dates of the underlying forward rates to differ from the ones implied by the index.
It's possible to change the default behavior by executing the line:
ql.IborCoupon.createIndexedCoupon... |
I'm reading the Barra risk model handbook (2004) available online and trying to understand the methodology. I've read a few materials on portfolio theory, so I can get at least the theoretical ideas behind a multi-factor risk model. But when you try to design one in reality, how do you test its predictions of the covar... |
How to test a risk model? |
If I have a stock, we say MSFT, and I want to find its corresponding market index - how do I do this?
I thought I needed to start by finding out which index the stock, in this case MSFT, is included in and from there I know which is its market index. However, I have discovered that it is very difficult to find a ans... |
How do you find the corresponding market index of a stock? |
I'm reading the Barra risk model handbook (2004) available online and trying to understand the methodology. I've read a few materials on portfolio theory, so I can get at least the theoretical ideas behind a multi-factor risk model. But when you try to design one in reality, how do you test its predictions of the covar... |
#### 1. Basic model assumptions
Let $X_k^{(i)}$ denote the return at time $k=1,2,...,n$ for stock $i=1,2,...,m$.
Assume that each stock return series is a stationary series and that $X_k^{(i)}\sim \mathcal{N}(\mu_i, \sigma_i^2)$.
Now assume that you have a prediction $Y_k^{(i)}\sim \mathcal{N}(\nu_i, \eta_i^2)$ and... |
#### 1. Basic model assumptions
Let $X_k^{(i)}$ denote the return at time $k=1,2,...,n$ for stock $i=1,2,...,m$.
Assume that each stock return series is a stationary series and that $X_k^{(i)}\sim \mathcal{N}(\mu_i, \sigma_i^2)$.
Now assume that you have a prediction $Y_k^{(i)}\sim \mathcal{N}(\nu_i, \eta_i^2)$ and... |
I am trying to understand from a mathematical and financial point of view the mechanism behind the so-called gamma squeeze. Is there a good source to read about this/ My questions are:
* what are the conditions that lead to gamma squeeze?
* what long/short positions favour the creating of a gamma-squeeze situation?
... |
#### 1. Basic model assumptions
Let $X_t^{(i)}$ denote the return at time $t=1,2,...,T$ for stock $i=1,2,...,m$.
Assume that each stock return series is a stationary series and that $X_t^{(i)}\sim \mathcal{N}(\mu_i, \sigma_i^2)$.
Now assume that you have a prediction $Y_t^{(i)}\sim \mathcal{N}(\nu_i, \eta_i^2)$ and... |
If I understand this correctly, we could simulate this process as follows. Let $N(0,t)$ denote the the Normal distribution with variance $t$.
Given some fixed level $t$, simulate
* $B_1(t)\sim N(0,t)$, i.e. the first iterate is a typical Brownian motion for time horizon $t$.
* Then, $B_2(t)\sim N(0,|B_1(t)|)$ is a... |
I'm using Montecarlo to estimate the value of an option,
$$\overline V(S_T, r, \sigma, T;N)=\mathbb{E} \left[V(S_T, r, \sigma, T)\right]$$
which comes with a standard error $SE$.
I'm using "bump-and-reval" (finite differences) to compute the greeks. So for example for Delta:
$$
\Delta \approx \frac{\overline... |
I am struggling a bit with some basic stuff lately:
Consider a SV model
\begin{align}
dS_t &= \sigma_t S_t dW_t \\
d\sigma_t &= b(\sigma_t,t) dZ_t
\end{align}
with $dW_t dZ_t = 0$.
I know that zero correlation does not imply independence, and in fact $S_t$ is clearly not independent of $\sigma_t$.
Howeve... |
I am struggling a bit with some basic stuff lately:
Consider a SV model
\begin{align}
dS_t &= \sigma_t S_t dW_t \\
d\sigma_t &= b(\sigma_t,t) dZ_t
\end{align}
with $dW_t dZ_t = 0$.
I know that zero correlation does not imply independence, and in fact $S_t$ is clearly not independent of $\sigma_t$.
Howeve... |
why the sign of RHO in BSM and Black76 Model is oppsite? |
Why the sign of RHO in BSM and Black76 Model is opposite? |
Can someone explain to me why the 3M Zero Rate is not equal to the 3M Cash Rate?
Thanks.
[![enter image description here][1]][1][![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/7u3ic.png
[2]: https://i.stack.imgur.com/X2fjh.png |
Why the 3M Zero Rate is not equal to the 3M Cash Rate? On Bloomberg yield curve bootstrapping |
Does settlement method of an instrument affect its payoff? |
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