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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21 votes
3 answers
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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)...
user22166's user avatar
  • 211
2 votes
1 answer
2k views

FX Options price vs implied vol

From the screenshot below, what is the difference between the option price by strike in the table versus the implied volatilities by delta in the chart at the bottom? https://www.investing.com/...
Student's user avatar
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6 votes
2 answers
3k views

Garman-Kohlhagen (Black-Scholes) Formula vs. Bloomberg OVML Calculator

I'm trying to price a European call option on USDJPY. We have that $S = 112.79, K = 112.24, \sigma = 6.887\%, r_d = 1.422\%, r_f = -0.519\%, T = 0.25$. My model, based on Black-Scholes, returns the ...
Vladimir Nabokov's user avatar
68 votes
9 answers
89k 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 $...
knorv's user avatar
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35 votes
4 answers
28k 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: ...
Thomas Browne's user avatar
48 votes
6 answers
116k 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, ...
jessica's user avatar
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1 vote
2 answers
3k views

Effect of Implied volatility on option delta

I am currently hedging a short put option where strike is 6027 and expiry is 30th Mar 2023. As per my understanding when option is ITM increase in volatility will decrease the delta and decrease in ...
Sumit's user avatar
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15 votes
2 answers
5k views

Variance replication using options

I would like to understand the intuition behind the following question: Why a certain weighted sum of prices of put and calls is equivalent to the implied variance of an underlying? A variance swap ...
Escachator's user avatar
1 vote
1 answer
2k views

Finite Difference Method in Greeks (Options)

I need a way to approximate the analytical formula of Greeks of a generic call option using the Finite Difference Method. For example, the FD method for Delta/Gamma is the following one: Now, I am in ...
John_maddon's user avatar
47 votes
16 answers
34k 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 ...
CuriousMind's user avatar
7 votes
4 answers
2k views

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,...
Trajan's user avatar
  • 2,502
5 votes
3 answers
2k views

Probability of an Option maturing In-the-money vs. Volatility

How will the probability of an option ending up in the money change if the volatility of the underlying stock increases? Intuitively, I think the answer to this is that if volatility goes up the ...
Trajan's user avatar
  • 2,502
5 votes
4 answers
15k views

Calculate strike from Black Scholes delta

I have a list of deltas and their corresponding volatilities in an FX market but I want to go from delta to strike price. In this Question similar problem is being discussed How can I calculate the ...
Sanjay's user avatar
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4 votes
3 answers
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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 ...
FrancescoD's user avatar
3 votes
1 answer
1k views

Quantlib: day-by-day evaluation of option value

I'm using Quantlib in Python to price an FX option. I'm comparing the result to Bloomberg, to make sure the code is working correct. I want to calculate the P&L of a certain option trading ...
Wynn's user avatar
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20 votes
3 answers
13k 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, ...
Jeffrey's user avatar
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11 votes
2 answers
5k views

How to derive the price of a square-or-nothing call option?

At maturity $T$, the holder of a "square-or-nothing" call option written on an underlying $S_t$ receives a payoff of the form $$ \phi(S_T) = \frac{S_T^2}{K} \pmb{1}_{\{S_T \geq K\}} = \begin{cases}\...
pkpkPPkafa's user avatar
8 votes
1 answer
15k views

Breeden-Litzenberger formula for risk-neutral densities

Based on this topic: How to derive the implied probability distribution from B-S volatilities? I am trying to implement the Breeden-Litzenberger formula to compute the market implied risk-neutral ...
user39039's user avatar
  • 441
4 votes
5 answers
10k views

Value of Call Option as Volatility goes to Infinity

Why would the value of a call option go infinity as volatility goes to infinity? I understand how you could solve this question by taking $\sigma \rightarrow \infty$ in the solution to the black ...
Trajan's user avatar
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4 votes
4 answers
7k views

Prove that the butterfly condition is always greater than zero

I need to prove that the butterfly condition is always positive under no arbitrage theorem. We are constructing a long butterfly using European call options ...
stud91's user avatar
  • 137
2 votes
1 answer
1k views

Can european call option on stock have positive theta? (assume positive interest rate)

I believe the answer is no, as minimum value of call option is S - PV(K), which can never be below S-K. The reason for the question is this paragraph in Natenberg, pg 109: Is it ever possible for an ...
Shreyans's user avatar
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17 votes
4 answers
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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 ...
user330060's user avatar
17 votes
8 answers
7k 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.
Anirban Saha's user avatar
12 votes
7 answers
5k views

What is the fair price of this option?

Without having to use Black-Scholes, how do I price this option using a basic no-arbitrage argument? Question Assume zero interest rate and a stock with current price at \$$1$ that pays no dividend. ...
Antonius Gavin's user avatar
7 votes
1 answer
2k views

Modeling Call Price w.r.t. Strike w Models that Capture Vol Smile

I am trying to model $C(K)$, the price of the call $C$ as a function of strike $K$. Because this is tied to Prob ITM - and in fact the probability density function of that particular expiration (https:...
Jared's user avatar
  • 735
6 votes
1 answer
9k views

Barrier option (autocallable) Vega profile

I have a question about the Vega profile(graph) on an autocallable option. Generally for a regular option, the vega graph looks like a normal (kinda normal) distribution with the vega highest at-the-...
Paul's user avatar
  • 63
6 votes
2 answers
5k views

Dynamic Hedge of Quanto Options

Can anybody explain to me step-by-step how can I dynamically hedge and/or replicate a quanto option with the foreign underlying asset, the foreign cash account and the domestic cash account as ...
Nikola's user avatar
  • 63
2 votes
3 answers
6k views

FX Delta Conventions

I'm currently reading Iain Clark's book Foreign Exchange Option Pricing and I got stuck at one sentence in the beginning of Section 3.3 that I feel is important to understand. He writes: FX ...
DoubleTrouble's user avatar
1 vote
1 answer
540 views

How do trading platforms estimate options pricing P&L graphs?

Using the profit/loss calculator for equity option strategies of a trading platform, it displays estimated P&L curves for some date in the future and across the prices of the underlying with a ...
mentics's user avatar
  • 113
20 votes
8 answers
16k 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 ...
Joseph Tanenbaum's user avatar
18 votes
1 answer
2k 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 ...
user357269's user avatar
14 votes
3 answers
11k views

How to exploit calendar arbitrage?

Say we are looking at European Call options in a toy environment with zero deterministic interest rates, a stock paying no dividends, no repo rates etc... Let $C(T,K)$ be the price of a call with ...
Alexander Chertov's user avatar
12 votes
1 answer
10k views

Implied dividend estimation

I am looking at two different ways of estimating the expected / implied dividends from market data. 1. Dividend futures I know that this asset class is not very liquid and might not be ...
sets's user avatar
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9 votes
4 answers
2k views

European Call Option Delta Upper Bound

For a pure equity process (with interest rate, dividend, etc., being zero) not necessarily the geometric Brownian motion, is the delta of a European call option always no higher than $1$? I am NOT ...
Hans's user avatar
  • 2,776
6 votes
5 answers
8k views

Writing an Options Strategy Backtester

I've been doing some digging, and this question has been asked many times in various forms over the years - Backtesting Options Strategies in R Are there any good tools for backtesting options ...
user avatar
5 votes
1 answer
5k views

Option prices in Bates SVJ model?

In this [post] discussed the European put and call price formulas under the Heston Stochastic Volatility model. There exists an important extension of Heston model to include diffusion jumps, known ...
emcor's user avatar
  • 5,775
4 votes
1 answer
3k views

Numerical example of how to calculate local vol surface from IV surface

I'm looking for an excel example (not a copy of Dupire's eqn) of how to convert an IV surface to a local vol surface. If unsuccessful I'll work through Dupire's eqn but would be helpful to look at an ...
Zeus's user avatar
  • 219
3 votes
1 answer
506 views

Intuition behind prices modeled by Geometric Brownian Motion

Suppose that we model a price $P_t$ to evolve per $$\frac{dP_t}{P_t}=\mu dt+\sigma dW_t$$ for $\mu\in\mathbb{R}$ and $\sigma>0$. The unique strong solution to this diffusion is $$P_t=P_0e^{(\mu-\...
Heatconomics's user avatar
2 votes
3 answers
1k views

Do basket options have a closed form valuation formula?

Suppose I'm simulating a European call option on a basket consisting of N stocks with slightly varying volatilities but all other parameters remain the same. From the perspective of an estimate, it ...
John1942's user avatar
2 votes
1 answer
2k 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 ...
user9081230912390's user avatar
0 votes
1 answer
2k views

Hedging in the Heston Model

I have simulated an underlying stock price, $S_t$ and a stochastic variance process, $v_t$ with the following stochastic differential equations from the Heston Universe: $$ dS_t = \mu S_tdt + \sqrt{...
Modvinden's user avatar
  • 137
0 votes
0 answers
302 views

Exotic option arbitrage

Suppose an exotic European option has a sub hedging (price being lower than the target) portfolio of vanilla European options all with the same expiry as the exotic option. The sub hedging portfolio ...
Hans's user avatar
  • 2,776
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 ...
knorv's user avatar
  • 2,119
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 ...
LeonTan's user avatar
  • 311
21 votes
3 answers
9k 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 ...
colin's user avatar
  • 249
20 votes
2 answers
28k 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 ...
Trajan's user avatar
  • 2,502
20 votes
4 answers
6k 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 ...
sets's user avatar
  • 1,461
17 votes
9 answers
10k 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 ...
Allanqunzi's user avatar
17 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 ...
Tal Fishman's user avatar
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17 votes
4 answers
16k 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 ...
arni's user avatar
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