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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2
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1answer
5k views

Implied state price density (Question 1 - derivation of the formula)

I came upon the term "implied state price density" in a couple of papers. As far as I understand the concept one basically tries to extract the "pricing density" from the market data. For the sake ...
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2answers
2k views

Lower bound of ITM Calls when computing Implied Volatility

Assuming the Black Scholes model and pricing formula of a European call option. Then, if the call is ITM, i.e. if $ln(\frac{S}{K})>0$, the $d_1$-term will go towards infinity as $\sigma$ goes to ...
5
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2answers
3k views

call vs put open interest

I have been observing the data for US stock options. In general, it seem like there are more open interest for call rather than for put, is there a reason people like to write more call? at the ...
5
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3answers
830 views

Pricing when arbitrage is possible through Negative Probabilities or something else

Also now asked about here: Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated? Assume that we have a general one-...
5
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1answer
367 views

Other numerraire choices when applying Feynman Kac

all of the books and notes I have seen on the Feynman Kac formula mostly applied to Risk neutral measure, i.e. different interest rate models, stochastic volatility, etc. I think risk neutral measure ...
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3answers
7k views

How does an option's time value depend on moneyness?

How does an option's time value (also known as extrinsic or instrumental value) depend on how far it is in the money or out of the money? In other words, how does the time value change as the ...
6
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3answers
6k views

Greeks for binary option?

How to derive an analytic formula of greeks for binary option? We know a vanilla option can be constructed by an asset-or-nothing call and a cash-or-nothing call, does that help us? Wikipedia states ...
5
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2answers
3k views

pricing using dupire local volatility model

I am reading about Dupire local volatility model and have a rough idea of the derivation. But I can't reconcile the local volatility surface to pricing using geometric brownian motion process. If I'm ...
4
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1answer
476 views

European Call price for an asset with mean reverting (Vasicek model) dynamics

Let's look at a stock with a mean reverting price dynamics: $$dS_t = a(S-S_0)dt + \sigma dW_t$$ If we let $\sigma=0.25$ and $a=-0.5$ then the variance of this process is: $$Var(S_t) = 0.199\sim0.2$$ ...
3
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1answer
511 views

Variance of a straddle (Black Scholes)

I am trying to determine the variance of the payout of a straddle. For puts and calls individually: Var[P] = E[P^2] - E[P]^2 Var[C] = E[C^2] - E[C]^2 where: $$ E[...
3
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3answers
200 views

Is “at the money” referent to the spot or forward price?

This may be a trivial question, but one I wasn’t sure about. Imagine I want to buy a 1 year ATM straddle. Does “at the money” imply buying closest to the current spot price, or does it mean to buy at ...
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4answers
4k views

List of ISIN for Options, Swaps, Derivatives?

In pages like isin.org or openfigi you can search by an ISIN and you will get information about the share, bond, fund... However, for options , derivatives the search returns 0 results. Is there a ...
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1answer
1k views

Paradox in option expiry as volatility goes to infinity

As volatility goes to infinity, the delta of a call option goes to 1. The delta approximates the probability that the option expires in the money. So it seems that the probability of expiring in the ...
4
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3answers
10k views

Negative time value european options

I have a basic question for which I feel like I should have found the answer by googling it, but I didn't get a definitive answer, so here I am: Can the time value for a plain vanilla (European) ...
3
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1answer
269 views

Cap option on Libor

We denote discount factor $D(t),$ zero coupon bond $B(t,T),$ $E_t[X] = E[X|\mathcal{F}(t)]$ and $T$-forward measure $E_t^{T}[\ ]....
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3answers
7k 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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9answers
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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4answers
4k 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 ...
14
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2answers
15k 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 ...
6
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6answers
14k views

Option trading API other than Interactive Brokers

I'm looking for an options broker that provides an execution API. I'd like to ideally test on a papertrading version of it before connecting to a real execution engine. I know IB offers that, but they ...
16
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6answers
40k 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 ...
17
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1answer
13k 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 ...
23
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7answers
9k 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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4answers
6k 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 ...
14
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5answers
39k 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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2answers
2k views

Interpertation of delta hedge error in Black Scholes

I have spent some time to prove the delta hedge error as described in this paper paper page 16-17 by Davis. The proof is discussed here Deriving Delta Hedge error in the B-S setup (part 2) (a post by ...
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6answers
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, ...
13
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4answers
12k 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 ...
8
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1answer
1k views

Risk management tools for long term Gamma/Vega sellers subject to margin calls

TL;DR: if you're a retail investor and you systematically sell long-term vertical spreads while staying Delta-neutral, your main risk comes from Vega and the Gamma of opening gaps that can throw you ...
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5answers
12k views

What is the intuitive reason why the Gamma and the Theta tend to have the opposite sign?

Quoting Hull's book: When gamma is positive, theta tends to be negative. The portfolio declines in value if there is no change in S, but increases in value if there is a large positive or ...
14
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3answers
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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2answers
3k views

Beta vs. Implied Volatility statistical arbitrage using options

Let two underlyings, $S_{1}$ and $S_{2}$, are correlated and $\beta$ is the slope of their returns linear regression, that is, it says how much $S_{1}$ co-variates with $S_{2}$ variance. For instance,...
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5answers
625 views

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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2answers
2k views

Delta Hedging with fixed Implied Volatility to get rid of vega?

I'm wondering if i should use a floating IV or a fixed IV to delta hedge my options every day. I've read this post but would like different information : Delta Hedging with fixed Implied Volatility ...
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2answers
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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1answer
1k views

How to use a change of numeraire to price this option?

I recently asked this question regarding how to price an option with payoff: $$\text{Payoff}_T = (A_TR_T - A_T \lambda)^+ $$ Let's assume for generality that $A_t$ and $R_t$ are GMB's: $$dA_t = \...
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2answers
10k views

Why FX Vanilla Options are quoted in volatility

I've been curious why vanilla options are quoted (and traded) in terms of volatility. Considering that every financial institution has its own options pricing model, volatility as an input would cause ...
9
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1answer
10k views

Link between Vega and Gamma

"The vega is the integral of the gamma profits ( ie expected gamma rebalancing P/L) over the duration of the option at one volatility minus the same integral at a different volatility......
3
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1answer
1k views

Pre-trade evaluation and risk assessment of option trading strategies (in market practice)

When a trader gets conclusion of the volatility is being underestimated (via volatility cone or some other technology), actually there are multiple ways for his trading. (Let's assume the underlying ...
17
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10answers
7k 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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2answers
3k views

Constructing an approximation of the S&P 500 volatility smile with publicly available data

Besides of the VIX there is another vol datum publicly available for the S&P 500: the SKEW. Do you know a procedure with which one can extrapolate other implied vols of the S&P 500 smile with ...
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3answers
1k views

Which is riskier: a call option or the underlying?

From Joshi's Quant Interview Questions and Answers: What is riskier: a call option or the underlying? (Consider a one day time horizon and compute which has bigger Delta as a fraction of value). I ...
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6answers
9k views

Call vs. Put Option

I have two interrelated questions that have been bothering me for some time. I have read all the stuff online and it still doesn't make sense to me: Let us assume: 0% interest rate (both hedge ...
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2answers
325 views

Can the concept of negative probabilities be used to price a call option?

Edit: I'm a dumbass. The thing below is supposed to be just the motivation of asking. I want to ask for below and in general, hehe. Assume that we have a general one-period market model consisting of ...
5
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1answer
10k views

Where can I find API access to historical options data? Paid or free?

I'm looking for a company or website that provides API access to historical options data. I would prefer a provider that has a python module to access the API. Any leads would be appreciated.
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2answers
3k views

Why is the vega of an at the money option so insensitive to movements in volatility? I.e, why do ATM options have such little Vomma?

I've been trying to understand why at the money options have very little vomma. I was reading and came across a graph that showed vega as volatility changes and I couldn't grasp how the relationships ...
11
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4answers
7k views

What is the importance of alpha, beta, rho in the SABR volatility model?

I just read that SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the ...
10
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4answers
977 views

How do you check your option calculations?

I'm implementing a bunch of different algorithms to price options/find Greeks: finite difference, Monte Carlo, binomial... I'm not really sure how to check my calculations. I tried using QuantLib to ...
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3answers
1k views

How to calculate Implied Volatility for out-of-the-money options?

I'm trying to calculate the implied volatility for out-of-the-money options, and to a lesser extent, in-the-money options. Most of the literature estimations I could find for implied volatility were ...
8
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2answers
817 views

Vega of exotic options

I'am wondering if there is a standard definition to the Vega of an exotic product when the underlying model is not Black-Scholes. Let me give some examples : What is the Vega if the price is ...