Questions tagged [options]

A contract that gives the owner the right, but not the obligation, to buy or sell a security at a fixed price in the future.

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63 votes
9 answers
78k views

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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49 votes
9 answers
8k views

Option pricing before Black-Scholes

According to the Wikipedia article, Contracts similar to options are believed to have been used since ancient times. In London, puts and "refusals" (calls) first became well-known trading ...
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48 votes
6 answers
111k views

A simple formula for calculating implied volatility?

We all know if you back out of the Black Scholes option pricing model you can derive what the option is "implying" about the underlyings future expected volatility. Is there a simple, closed form, ...
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46 votes
16 answers
30k views

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 ...
38 votes
6 answers
44k views

What type of investor is willing to be short gamma?

As far as I understand, most investors are willing to buy options (puts and calls) in order to limit their exposure to the market in case it moves against them. This is due to the fact that they are ...
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34 votes
5 answers
16k views

Skew arbitrage: How can you realize the skewness of the underlying?

It's not clear to me how to realize skewness. In other words, how do you implement skew arbitrage? There seems to be no well-known recipe like in volatility arbitrage. Volatility arbitrage (or vol ...
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32 votes
4 answers
26k views

How to derive the implied probability distribution from B-S volatilities?

The general problem I have is visualization of the implied distribution of returns of a currency pair. I usually use QQplots for historical returns, so for example versus the normal distribution: ...
27 votes
4 answers
2k views

Who has introduced the term 'vega' and why?

The sensitivity of the option value $V$ to volatility $\sigma$ (a.k.a. vega) is different from the other greeks. It is a derivative with respect to a parameter and not a variable. To quote from Paul ...
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26 votes
3 answers
8k views

Papers about backtesting option trading strategies

I am looking for all kinds of research concerning option trading strategies. With that I mean papers that publish results on different option trading strategies properly backtested with real-world ...
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26 votes
7 answers
10k views

When does delta hedging result in more risk?

A question from an interview book: When can hedging an options position make you take on more risk? The answer provided is the following: Hedging can increase your risk if you are forced to ...
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25 votes
3 answers
4k views

What does the VIX formula measure and how does it work?

I have read the CBOE's white paper on the VIX and a lot of other things, but I need to honestly say, I don't really get it, or I am missing something important. In semi-layman's terms, is the VIX ...
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25 votes
5 answers
8k views

Why hold options when you can dynamically replicate their payoff?

When holding vanilla options, you can cancel out, theoretically, all risk with dynamic (delta) hedging. Then you earn the "risk free rate of return". Why would you make such a portfolio when you can ...
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24 votes
6 answers
3k views

Setting the r in put-call parity?

Put-call parity is given by $C + Ke^{-r(T-t)} = P + S$. The variables $C$, $P$ and $S$ are directly observable in the market place. $T-t$ follows by the contract specification. The variable $r$ is ...
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23 votes
6 answers
29k views

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 a clear standard definition is not unanimously in place among practitioners. So here is ...
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23 votes
6 answers
4k views

Why are options trades supposed to be delta-neutral?

I'm reading Natenberg's book, and he says that all options trades should be delta neutral. I understand that this prevents small changes in the underlying price from changing the price of the option, ...
21 votes
7 answers
19k views

What is the "delta" option quoting convention about?

At my work I often see option prices or vols quoted against deltas rather than against strikes. For example for March 2013 Zinc options I might see 5 quotes available for deltas as follows: ...
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21 votes
3 answers
8k views

Is there an all Java options-pricing library (preferably open source) besides jquantlib?

I am looking for an all-java implementation of black scholes, preferably open source. I found jquantlib and quantlib (C++). Any other recommendations? The jquantlib site seems to be down. I'd prefer ...
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21 votes
3 answers
6k views

Is there a popular curve fitting formula of options skew vs strike price or vs Delta?

I was trying to build a options trading/optimization system. But it often gets more inaccurate as it scans through the far from ATM options because, you know, options skews. That is because I did ...
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20 votes
8 answers
15k views

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 ...
20 votes
3 answers
12k views

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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20 votes
4 answers
21k views

What is a Heat Rate Option?

I tried a search with google but I can't find a clear definition of what a Heat Rate Option is. I would appreciate if someone could explain to me what this type of option is. My understanding is ...
19 votes
4 answers
5k views

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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19 votes
3 answers
3k views

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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18 votes
3 answers
10k views

Carr-Madan Formula

Really new to financial Maths. I am currently having problems with the Carr-Madan Formula. $$f(S_T)=f(F_t) + f'(F_t) (S_T - F_t) + \int_0^{F_t} f''(K) (K-S_T)^+ \ d K + \int_{F_t}^{\infty} f''(K)...
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18 votes
4 answers
7k views

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 ...
18 votes
1 answer
2k views

Bergomi: Skew arbitrage

In his paper "Smile Dynamics IV" (https://www.fields.utoronto.ca/programs/scientific/09-10/finance/derivatives/bergomi.pdf) as well as in his book "Stochastic Volatility Modeling" (...
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18 votes
1 answer
14k views

What is the best live options data API?

What is the best/cheapest service to get real-time (as real-time as you can get) on stock options? I'm looking for the fastest update on the ENTIRE market, with a few stocks prioritized, so I need ...
17 votes
7 answers
6k views

Why do institutional Traders prefer Short Selling instead of Buying Puts?

Why is it more common for Institutional Traders to short sell stocks when they have a bearish stance instead of Buying Puts? The limited loss potential of Buying Puts seems like a better choice.
17 votes
6 answers
41k views

How to calculate stock move probability based on option implied volatility and time to expiration? (Monte Carlo simulation)

I am looking for one line formula ideally in Excel to calculate stock move probability based on option implied volatility and time to expiration? I have already found a few complex samples which took ...
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17 votes
9 answers
9k views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
17 votes
9 answers
8k views

Why are options called what they are called?

This may be a very obvious question, but can someone tell me where and when the names call and put originated? And similarly, where do the terms American and European option come from? Aside from the ...
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17 votes
1 answer
1k views

How much can be said about the Greeks without picking a model?

Let $C(S, K, \sigma, r, T)$ be the price of a call option. How much can be said about the Greeks without picking a model? Or at least without full Black-Scholes? Below, I write down everything I know ...
17 votes
3 answers
559 views

How do you characterize dividends for equity options?

While many systems like to treat dividends as a continuous yield when pricing equity options, it works quite poorly for short-dated options. In the short run, deterministic dividends are clearly the ...
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16 votes
2 answers
20k views

Gamma Pnl vs Vega Pnl

Why does Gamma Pnl have exposure to realised volatility, but Vega Pnl only has exposure to implied volatility? I am confused as to why gamma pnl is affected (more) by IV and why vega pnl isnt affected ...
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16 votes
2 answers
2k views

Can you replicate an option on an arbitrary basket of stocks?

Since a market index is nothing more than a basket of stocks, you can create your own index by putting together stocks of your choice. The only difference is that you can trade options on major ...
16 votes
1 answer
2k views

Is QuantLib more trouble than it's worth?

I'm just starting to work with QuantLib and wonder if I'm going down a very wrong path. I'm working on a site that presents the visitor with a table of streamed real-time options data, including ...
16 votes
1 answer
608 views

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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15 votes
3 answers
28k views

Does implied volatility vary for calls vs puts?

Volatility skew tells us that options with the same maturity at different strikes can have different implied vol. However, can a corresponding call and put for the same strike and maturity have ...
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15 votes
4 answers
4k views

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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15 votes
5 answers
9k views

Best way to store hourly/daily options data for research purposes

There are quite a few discussions here about storage, but I can't find quite what I'm looking for. I'm in need to design a database to store (mostly) option data (strikes, premiums bid / ask, etc.). ...
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15 votes
3 answers
2k views

What benchmark/index to use for backtesting a portfolio of stock options?

What benchmark should I use for backtesting a model for when I should buy an option of a particular stock? For equities, one could say their portfolio outperformed the S&P 500. I would like to ...
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15 votes
2 answers
1k views

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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14 votes
4 answers
14k views

Why is realized volatility typically lower than implied volatility?

A number of quantitative finance textbooks mention something along the following lines, without further explanation: A typical feature of implied volatility from stock index options is that it is ...
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14 votes
2 answers
7k views

How to extrapolate implied volatility for out of the money options?

Estimation of model-free implied volatility is highly dependent upon the extrapolation procedure for non-traded options at extreme out-of-the-money points. Jiang and Tian (2007) propose that the ...
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14 votes
3 answers
7k views

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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14 votes
3 answers
17k views

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 ...
  • 1,822
14 votes
2 answers
1k views

Can you fully hedge an option in the presence of counterparty risk?

The derivation of the Black-Scholes model assumes no counterparty risk. Does the presence of counterparty risk invalidate the argument behind the model? EDIT: The question is about options in general,...
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14 votes
4 answers
3k views

Implementing a Fast Fourier Transform for Option Pricing

So, I'm in need of some tips regarding a small project I'm doing. My goal is an implementation of a Fast Fourier Transform algorithm (FFT) which can be applied to the pricing of options. First ...
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14 votes
1 answer
3k views

How should I estimate the implied volatility skew term when calculating the skew-adjusted delta?

I'm trying to come up with the implied volatility skew adjusted delta for SPY options. I'm working with the following formula: Skew Adjusted Delta = Black Scholes Delta + Vega * Vol Skew Slope. I ...
13 votes
5 answers
42k views

Why do some people claim the delta of an ATM call option is 0.5?

I am looking for a mathematical proof in terms of differentiating the BS equation to calculate Delta and then prove it that ATM delta is equal to 0.5. I have seen many books quoting delta of ATM call ...
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