Questions tagged [derivatives]
A financial contract whose payoff is linked to the evolution of an underlying security.
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Innovative ways of visualizing financial data
Finance is drowning in a deluge of data. Humans are not very good at comprehending large amounts of data. One way out may be visualization.
Traditional ways of visualizing patterns, complexities and ...
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How to estimate real-world probabilities
In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings.
However, the behavior of ...
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Stochastic modelling of derivatives on dividends
I consider pricing and risk analysis of derivatives on dividends of the members of equity indices (such as Dow Jones EuroStoxx). There are options but I focus on futures.
What are common stochastic ...
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Probability distribution of maximum value of binary option?
A binary option with payout \$0/\$100 is trading at \$30 with 12 hours to
expiration.
Assuming the underlying follows a geometric Brownian motion (hence volatility remains constant), what ...
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Derivation of VIX Formula
I've read a lot of derivations about VIX formula. I can say it is -adjusted- fair strike of variance swap. But I can't see how it goes from variance swap rate to VIX formula. In particular I can't see ...
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Why is the SABR volatility model not good at pricing a constant maturity swap (CMS)?
I have heard that the SABR volatility model was not good at pricing a constant maturity swap (CMS). How is that?
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Quantitative Derivatives Trading vs. Time
Most quantitative investment strategies focus on the changing prices of a commodity or equity over time. Derivatives, however, make this more complicated. How can I apply quantitative strategies to ...
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How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?
I have $\frac{dS_t}{S_t} = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs
$$(S_T f(S_T))^+$$
How do I express the stock dynamics using the ...
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Why a self-financing replicating portfolio should always exist?
According to my understanding the derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that it should be possible to ...
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Differences between main classes of interest pricing derivatives models
There seems to be 3 main classes of interest rate pricing models: 1) Short rate models, 2) Heath Jarrow models and 3) Libor Market Model. My book doesnt seem to explain why we need all these different ...
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What are the major models for energy derivatives, particularly electricity derivatives?
Aside from Black-Scholes with crazy skews, what major models are used for energy derivatives? I'm thinking particularly of electricity derivatives, but I'm also interested in natural gas and other ...
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Why Must Dividends Be Reinvested to Use Risk-Neutral Pricing?
Assume the price of a stock $S_t$ paying continuous dividend $a$ satisfies
$$
dS_t = S_t\left((\mu - a)dt + \sigma dW_t\right).
$$
The risk-neutral pricing formula states that if $\mathbb{Q}$ is any ...
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What is a self-financing and replicating portfolio?
I try to understand the derivation of the Black-Scholes equation based on the "constructing a replicating portfolio".
From mathematical point of view it looks simple. We assume that:
Stock prices is ...
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Why discounted derivative price is a martingale?
Usually after showing that discounted stock price process is martingale under the risk-neutral measure, most authors say that this implies that the discounted derivative price process is a martingale ...
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What is the replicating portfolio of swaptions for a constant maturity swap (CMS)?
How do you replicate the payoff of a constant maturity swap rate?
That is, if the payoff of a contract pays the 5-year swap rate every year for 10 years, how would you replicate this payoff using ...
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option chain data visualization, sunburst
I think option chains are not represented in the best way. With more and more options products coming out and trading on the various exchanges, I see vendors struggling to keep up with a good way to ...
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Present and future role of pricing quants
While looking up on quants, I came across many sources that cited 'pricing quants' as one of the biggest chunks among all quant positions. But then I also came across many software companies providing ...
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What is the connection between default probabilities calculated using the credit rating and the price of a CDS?
I'm working on a tool to price Credit Default Swaps. I've already done the standard pricing tools. I'm working on a pricing tool which uses the credit rating for the default probabilities used in the ...
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How to think about pricing this weather call option
So as opposed to the normal structure using a reference temperature and HDD/CDD, I'm looking at pricing a call option with a structure similar to the following:
Daily option on maximum daily ...
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Why does the price of a derivative not depend on the derivative with which you hedge volatility risk?
I'm trying to derive the valuation equation under a general stochastic volatility model.
What one can read in the literature is the following reasoning:
One considers a replicating self-financing ...
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Heston Model Integration Oscillations
Is there a way to reduce oscillations for the numerical integration when evaluating the Heston model. I am pricing a series of 5000 options scattered over the Heston model parameter space and I find ...
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Theoretical limits for contango and backwardation
What do you think would be the theoretical limit for contango? What about backwardation?
This was asked in an interview. I am still not so sure about the answer.
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Refer some most recent books of derivatives pricing by C++
Could you refer some most recent books of derivatives pricing by C++ including Tree method, Finite difference method, Monte Carlo etc.
Once I read a series of <...
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Are there financial instruments that make a bet on traded volume instead of price or its derivatives?
For most financial instruments we can go long or short and make a bet on the price. In the case of options we can bet on derivatives of price and other factors (e.g., interest rates).
Is there an ...
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Replicating a portfolio with a certain payoff function
Assume there are two stocks $S_1$ with price $p_1(t)$ and $S_2$ with price $p_2(t)$ where $t$ indicates time. Assume, there is a hypothetical derivative $D$, which is such that, price of $D$ at a time ...
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New ways of communicating risk
One of the scapegoats of the financial crisis was value at risk. Still communicating risks effectively to clients is a big challenge and hugely important (also to keep your job as a quant!) In this ...
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Calibration of nested pricing models consistently on two different classes of derivatives
Hi everyone, I'm programming in MATLAB and I have the following optimization problem in calibrating several nested specifications of pricing models.
Summary: I have two pricing models ($1$ and $2$, $...
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Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
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What is meant by the funding cost of a derivative?
Numerous sources refer to the 'funding cost' of a derivative.
I'm confused as to exactly what cost is being referred to here. To illustrate my confusion, consider purchasing an uncollateralised OTC ...
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Radon Nikodym derivative when changing numeraires
I note from Wikipedia that if $Q$ and $Q^N$ are two measures corresponding to numeraires $M$ and $N$, then the Radon Nikodym derivative is given by: $$\frac{dQ^N}{dQ} = \frac{M(0)}{M(T)}\frac{N(T)}{N(...
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Is it true that pricing an IR swap doesn't require any stochastic model but calculation of the PFE of an IR swap would?
Pricing an IR swap doesn't require any stochastic model but calculation of the PFE for an IR swap would require the Hull White Model or any other stochastic short rate or forward rate model.
Is this ...
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CMS Pricing - Convexity Adjustment by Replication [closed]
I'm trying to learn CMS pricing, but didn't get the logic of this method. Previously cited articles about this method is pretty complex.
I'd be glad if you can provide me with simpler articles or ...
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Black-Scholes formula for Poisson jumps
For underlying asset
$$d S = r S dt + \sigma S d W + (J-1)Sd N$$
here $W$ is a Brownian motion, $N(t)$ is Poisson process with intensity $\lambda.$
Suppose $J$ is log-normal with standard deviation $\...
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On short-rate-models: Black-Karasinski (with constant parameters) compared to Vasicek
When modelling the term structure of interest rates, one widespread possibility is using the Black-Karasinski model, which is given by the following stochastic process
$$d\ln{r}=[\theta(t)-a(t)\ln{r}]...
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About Option Adjusted Spread, rate curves and bonds comparison
I have few questions about using OAS as a measure of risk:
does OAS allow for comparison between bonds with and without embedded options (e.g. a callable bond against a plain vanilla one against a ...
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Seeking criticism of model assumptions
I have been trying to publish a new calculus and options model for seven years. I have been consistently desk rejected, so what I am trying to do is get criticism of my assumptions because they ...
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Non attainable claim - Incomplete market
I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market.
As an example, let us have the following market ($B = (B^1, B^2, B^3)$...
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What is a Constant Maturity Swap (CMS) rate?
I have been searching in books and on the internet for a basic definition and explanation of CMS rates, but I cannot find anything clear and simple. Can you explain (maybe with an example) what a CMS ...
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When would open interest equal trading volume?
I know the difference between open interest and trading volume. Open interest is the number of contracts, long or short, outstanding. Trading volume is the number of contracts traded in a day.
...
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What are the books in which to study the basics of the derivative financial instruments?
Books similar to Options, Futures, and Other Derivatives by John C. Hull. I need another academic book that explains the basics of quantitative finance derivatives (forward, futures, options)
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Price of an asian option with squared of average payoff
Is there a closed form solution of the following price formula? Assuming $dS_t=rSdt+\sigma S_t dW_t$ under the Q dynamics
$e^{-r(T-t)}\mathbb{E}_t^\mathcal{Q}[(\frac{(\int_0^T S_u du)}{T})^2]$
I ...
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Derive vega for Black-Scholes call from this formula?
Is it possible to get the right formula for vega of a call option under the black scholes model from this formula?
$$\frac{\partial{C}}{\partial{\sigma}}=\frac{S_0}{\sqrt{2\pi}}{e^\frac{-d_+^2}{2}}(\...
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What is a standard credit default swap contract and where can I find spread data? What alternatives exist to judge creditworthiness?
I'm doing some work for a company and one of my tasks is to research credit default swaps on banks and to write a page about them explaining what they are and how they're used to evaluate the banks' ...
user2706
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Use of interest rate swaps in liability-driven investing
You probably have home across recent events in the UK bond markets. The Financial Times article "The reason the BoE is buying long gilts: an LDI blow-up" from Sep. 28, 2022 goes through why ...
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Relationship between Vega and Gamma in Black-Scholes model
my question is the following one: I don't manage to prove that, in Black-Scholes model, single-signed Gamma options have values that are monotonic in the volatility. I am looking for an exhaustive and ...
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How to hedge the fixed leg of a swap contract?
I happened to get this question for Fixed Income Swap contract. (let's assume it's it's not cross currency).
If the fixed leg is paying 10% interest rate in this contract, but in the market the ...
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Market data for options
Looking for recommendations on places to get market data for options. I'm looking at NYSE and NASDAQ only.
My current solution is my broker, Tradeking. I can request realtime data for 700 option ...
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The positivity of the market price of risk
Does the market price of risk, be it of stochastic volatility, interest rate or equity return, have to be positive? What is the rationale if it does?
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Replicating a square derivative with calls and puts
I have a derivative that pays off $S_T^2$ at time $T > 0$ with $S_T$ denoting the price of a non dividend-paying stock at $T$. I came across a question about how one can statically replicate this ...
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the cash flows behind closing out futures positions
I always get confused about the cashflows occurring when a futures position is closed out. For example, say it is January and I enter into a long December Futures position with a futures price F(jan). ...
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