Questions tagged [option-pricing]
Questions about models for the valuation of option contracts.
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What are some useful approximations to the Black-Scholes formula?
Let the Black-Scholes formula be defined as the function $f(S, X, T, r, v)$.
I'm curious about functions that are computationally simpler than the Black-Scholes that yields results that approximate $...
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Are there any new Option pricing models?
Back in the mid 90's I used the Black-Scholes Model and the Cox-Ross-Rubenstein (Binomial) Model's to price Options. That was nearly 15 years ago and I was wondering if there are any new models being ...
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Why Drifts are not in the Black Scholes Formula
This question has puzzled me for a while.
We all know geometric brownian motions have drifts $\mu$:
$dS / S = \mu dt + \sigma dW$
and different stocks have different drifts of $\mu$. Why would ...
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How should I calculate the implied volatility of an American option in a real-time production environment?
There are many models available for calculating the implied volatility of an American option. The most popular method, employed by OptionMetrics and others, is probably the Cox-Ross-Rubinstein model. ...
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How to show that this weak scheme is a cubature scheme?
Weak schemes, such as Ninomiya-Victoir or Ninomiya-Ninomiya, are typically used for discretization of stochastic volatility models such as the Heston Model.
Can anyone familiar with Cubature on ...
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How do we use option price models (like Black-Scholes Model) to make money in practice?
In quantitative finance, we know we have a lot of option price models such as geometric Brownian motion model (Black-Scholes models), stochastic volatility model (Heston), jump diffusion models and so ...
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Probability of touching
For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the ...
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Is there a relationship between Risk Neutral Pricing framework and Nash Equilibria?
Based on the Fundamental Theorem of Asset Pricing, the risk neutral price of a contingent claim on an asset in a liquid, arbitrage free market can be determined by switching to an equivalent $Q-$ ...
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Explaining the Risk Neutral Measure
What is the Risk Neutral Measure?
I don't believe this has been answered on the internet well and with all the parts connecting.
So:
What is the risk neutral measure/pricing?
Why do we need it?
How ...
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What is the implied volatility skew?
I often hear people talking about the skew of the volatility surface, model, etc... but it appears to me that there isn't a clear standard definition unanimously used by practitioners.
So here is my ...
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When do Finite Element method provide considerable advantage over Finite Differences for option pricing?
I'm looking for concrete examples where a Finite Element method (FEM) provides a considerable advantages (e.g. in convergence rate, accuracy, stability, etc.) over the Finite Difference method (FDM) ...
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What is the trickiest thing to get right in Rates Quant recently (2019)?
What are the biggest challenges for Rates Quants in 2019? Most quants have been through a lot over the past years-shifting their SABR models in JPY swaptions, fixing the FVA models for negative rates, ...
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Why does implied volatility show an inverse relation with strike price when examining option chains?
When looking at option chains, I often notice that the (broker calculated) implied volatility has an inverse relation to the strike price. This seems true both for calls and puts.
As a current ...
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Risk Neutral Probability
I read that an option prices is the expected value of the payout under the risk neutral probability. Intuitively why is the expectation taken with respect to risk neutral as opposed to the actual ...
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What causes the call and put volatility surface to differ?
I currently have a local volatility model that uses the standard Black Scholes assumptions.
When calculating the volatility surface, what causes the difference between the call volatility surface, ...
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From Fourier Transforms to Option Values
I am trying to understand how Fourier transforms & Characteristics functions can be used to calculate option values.
However, I am having difficulty following the process that is used in several ...
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How to solve for the implied stock lending rate given equity options prices?
When market makers price options on hard-to-borrow equities, they include the cost to borrow the underlying equity that their broker is going to charge them to sell the security short to hedge. I'm ...
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Hedging Covid-19 and other low probability high loss risks
Covid-19 and similar risks are low probability, high loss events. Does it make sense to utilize options to provide hedges for such events? For example, should one utilize long positions in deep out-...
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Formal proof for risk-neutral pricing formula
As you know, the key equation of risk neutral pricing is the following:
$$\exp^{-rt} S_t = E_Q[\exp^{-rT} S_T | \mathcal{F}_t]$$
That is, discounted prices are Q-martingales.
It makes real-sense ...
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Duality between constant rebalanced portfolio (CRP) and corresponding derivative
One of the greatest achievements of modern option pricing theory is finding corresponding dynamical trading strategies in linear instruments with which you can replicate and by that price derivative ...
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Bachelier model call option pricing formula
Does anybody have the Bachelier model call option pricing formula for $r > 0$?
All the references I've read assume $r = 0$. I don't speak French, so I can't read Bachelier's original paper.
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How to get greeks using Monte-Carlo for arbitrary option?
Let's assume I have an arbitrary option that I can price using Monte-Carlo simulation. What is the general approach (i.e. without relying on specific option type) to calculating the greeks in this ...
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How does volatility affect the price of binary options?
In theory, how should volatility affect the price of a binary option? A typical out the money option has more extrinsic value and therefore volatility plays a much more noticeable factor. Now let's ...
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How can one compute the Greeks on VIX Futures
I am guessing the short answer to this question is "use the chain rule and linearity of the derivative," but I am looking for more specific advice on how to compute the derivatives of a VIX futures ...
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Max option leverage strike
Since options represent leveraged stock investments, at which strike $K$ does a European option provide maximum leverage?
Hereby define leverage $L$ as ratio of Delta/Optionprice:
$$L(K)=\frac{\...
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How do I price OANDA box options?
How do I price OANDA box options without using their slow and
machine-unfriendly user interface?:
http://fxtrade.oanda.com (free demo account) sells "box options":
If you already know what a box ...
user59
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Methods for pricing options
I'm looking at doing some research drawing comparisons between various methods of approaching option pricing. I'm aware of the Monte Carlo simulation for option pricing, Black-Scholes, and that ...
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Are there comprehensive analyses of theta decay in weekly options?
Are there comprehensive analyses of how much theta a weekly options loses in a day, per day?
I know what the shape of theta decay looks like, in theory, where the decay towards zero happens more ...
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How to price a volatility-index option?
There exist several volatility indices, such as the CBOE Volatility Index (VIX). There are also options on such indicies.
What is the best way to price a volatility-index option? Is there a simple ...
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What are important model and assumption-free no-arbitrage conditions in options trading?
In the paper "Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula" (Espen Gaarder Haug, Nassim Nicholas Taleb) a couple of model-free arbitrage conditions are mentioned which limits ...
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How do different models impact option Greeks?
If I trade an option using delta, vega, Prob OTM, etc. these are derived from a model. How do leading models impact valuations in terms of the Greeks?
I suppose to form a baseline it would have to be ...
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Consensus on Cauchy distribution for stock prices
What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded.
My motivation is to find ...
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How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?
I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
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Historical Volatility vs Implied Volatility Performance in Pricing Options
I consistently read on academic papers, when pricing options, using implied volatility is better than using historical volatility. Because, market is more "forward-looking" and historical data is "...
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Ways of treating time in the BS formula
The Black-scholes formula typically has time as $\sqrt{T-t}$ or some such. My questions:
What is the granularity of this? If we treat $t$ as the number of days, then logically on the day of expiry, ...
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What are the main flaws behind Ross Recovery Theorem?
Stephen Ross’ new paper claims that it is possible to separate risk aversions and historical probabilities if the Stochastic Discount Factor is transition independent using Perron-Frobenius Theorem.
...
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How to numerically obtain delta?
The delta in option pricing, also called the hedge ratio, is expressed as the sensitivity of the option price to the underlying price change.
The analytical solution for the most common option ...
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How to price very short dated options?
I was wondering if there is any industry standard in pricing very short dated options, from say 6h options down to 5 minute options.
My thinking is that as time to expiry gets shorter and shorter, the ...
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Transformation of Volatility - BS
I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev
\begin{equation}
\sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}}
\end{...
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Black-Scholes under stochastic interest rates
I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
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What tools are used to numerically solve differential equations in Quantitative Finance?
There are a lot of Quantitative Finance models (e.g. Black-Scholes) which are formulated in terms of partial differential equations. What is a standard approach in Quantitative Finance to solve these ...
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How to transform process to risk-neutral measure for Monte Carlo option pricing?
I am trying to price an option using the Monte Carlo method, and I have the price process simulations as an inputs. The underlying is a forward contract, so at all times the mean of the simulations is ...
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How should FX options be priced when a currency is artificially capped?
The question is inspired by yesterday's (06/09/11) historic announcement by the Swiss National Bank that it would impose a ceiling on the franc of 1.20 against the euro.
I would like to know if there ...
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How to price a Swing Option?
I'm working in the commodity market and I've to price Swing Options with MATLAB, preferably with finite element.
Has anyone already priced these kind of derivatives?
I'm thinking about using the ...
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Jim Gatheral's ansatz
In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes $$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$
where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total ...
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How to choose a risk-neutral measure when the market is incomplete?
I am more of a probabilist than a financial mathematician. I am currently working on the features of American put options under a particular stochastic volatility model.
Like most stochastic ...
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Option Pricing Model Calibration In Practice
I'm curious how an option pricing model like the Heston model is calibrated in practice.
Here's how I imagine it happens:
Let's say I have access to the most recent option prices on a given stock ...
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Reference on Markov chain Monte Carlo method for option pricing?
I have to implement option pricing in c++ using Markov chain Monte Carlo. Is there some paper which describes this in detail so that I can learn from there and implement?
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Pricing of a Foreign Exchange Vanilla Option
To understand how Bloomberg prices foreign exchange vanilla options , I extract the following screenshot from its OVML function.
The Black-Scholes formua for vanilla options are
\begin{split}
& P=...
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Can the Heston model be shown to reduce to the original Black Scholes model if appropriate parameters are chosen?
Summary
For Heston model parameters that render the variance process constant, the solution should revert to plain Black-Scholes. Closed from solutions to the Heston model don't seem to do this, even ...
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