Questions tagged [european-options]

An option that can be exercised only at expiration.

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why exactly does delta go to the Value 1 for atm calls with a volatility converging to zero if a positive risk free rate is assumed. (bs-modell)

Im basically looking for a further/non mathematical explanation for following answer ATM call option delta with low volatility what is meant with a positive drift? does that mean we assume the stock ...
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Understanding American option payoff at T+0

The above picture shows the payoff at expiry(in gold) and at current time T+0(in blue) for a bull call spread. I am trying to understand American options and to know if it has any significant ...
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Shout option payoff replication

I have not seen much talk about exotic options, and if they are actually traded. Is it possible to replicate the payoff of a ‘Shout option’ using standard European/American call and put options?
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Pricing look-back option

I have the monthly price data of a stock starting from December 2020 and I am considering a EU style look-back option issued in December 2020. The payoff at maturity of the look-back option is given ...
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Replication of the payoff of a chooser option

With numerical examples, how can the payoff of a chooser option be replicated with European call and put options?
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In the paper "By Implication" by Jaeckel, he says that put-call parity should never be used in practic

In this paper by Jackel (2006), on page 2, he writes: The normalised option price $b$ is a positively monotic function in $\sigma \in[0, \infty)$ with the limits $$ h(\theta x) \cdot \theta \cdot\left(...
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Confusion about payoff for an option [closed]

My teacher said that the payoff of a put is $\mathrm{max}(K-S_T, 0)$, where $K$ is the strike price and $S_T$ is the spot price at maturity. Why isn't it $K$ if $K-S_T > 0$ and $0$ otherwise (i.e. $...
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Practical use of Dual Delta?

I am wondering what the practical use of the Black-Scholes Dual-Delta is? I know it is the first derivative wrt the strike price: $$ \frac{\partial V}{\partial K} = -\omega e^{-r T} \Phi(\omega d_2) $$...
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At what threshold on delta percentage should I hedge my option portfolio?

I am able to identify and build an option portfolio with long/short call/put options across different strikes and expiries such that the gamma is positive and cost is negative. Upon inception I hedge ...
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Clarification on a Claim in Black-Scholes-Merton Derivation

In these notes: https://johnthickstun.com/docs/bscrr.pdf, towards the end of the proof of Proposition 5.2 on page 6, the author claims: $$ \log \lim_{n \to \infty} \Bbb{E}_\pi \left[\frac {S^*_n} S \...
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Reconstructing the CRR model knowing put and call prices

In an arbitrage-free single-period CRR model, the following options on a share are offered: [They are all European] (i) Call option at strike price $100$, price: $C_{0,1}=7.44$ (ii) Call option at ...
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Modelling the Implied volatility of an asset

I want to estimate the implied volatility of an asset which has not historical implied volatility data. I do have the historical realized volatility ( I have the historical prices). What would be some ...
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Can the risk neutral pdf derived from Breeden-Litzenberger Method be used to calculate vega and theta?

I have been researching volatility smoothing techniques and risk-neutral pdf. I noticed one interesting post in Does the risk neutral pdf that is derived using Litzenberger-Breeden Method correspond ...
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Pricing European Call Closed Form Spread Options in Python

I am currently trying to correctly price European Call Closed Form Spread Options using Python. The main problem I am currently running into is that I have nothing to compare the option price so that ...
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American Contingent Claim vs European Option pricing

Suppose $Y$ is an American Contingent Claim (ACC) defined as $Y = \{Y_t, t \in 0,1,...,T\}$ and asssume $U_t$ is its fair price. Also suppose $C_t$ is the arbitrage-free price at time $t$ of a ...
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Check for arbitrage - European calls with same strike price, different duration and price

I tried a lot of different things to check for arbitrage on the following calls but didn't succeed. Let's suppose we have a stock that is currently valued at 40. The interest rate is 0.05 and the ...
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BS price as the first term of option price expansion

I recently saw someone write, on a generally non-technical platform, that the Black-Merton-Scholes vanilla option price is the first term of an expansion of the price of a vanilla option. I get that ...
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Option time value is Nd1-Nd2

I can't find the below statement anywhere (rearrangement of Black-Scholes formula) : $C(0, S) = e^{-rT}N_2[F-K] + [N_1-N_2]S$ $F$ being the forward, it reads as a straightforward decomposition to ...
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Decompose Option price into greeks

I am trying to decompose option prices into various greeks and trying to see if I can recover option prices from various of its greeks. At the start of certain time ...
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In the CRR model, describe the strategy replicating the payoff $X=(S_T-K)^{ +} +a(K-S_{T-2})^{+ }$ for $a \neq 0$ [closed]

In the CRR model, describe the strategy replicating the payoff $X=(S_T-K)^{ +} +a(K-S_{T-2})^{+ }$ for $a \neq 0$ $X$ consists of two parts: European call option with strike price $K$ and expiration ...
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What is the Time Value of European Options if r=0? [closed]

As I understand it, time value for European options is as follows: What if r=0? Then puts should behave the same as calls, right? Would the time value always be nonnegative or could it be negative?
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How to hedge a dual digital option

Let us assume we have two FX rates: $ 1 EUR = S_t^{(1)} USD$ and $ 1 GBP=S_t^{(2)} USD $. Let $K_1>0, K_2>0$ be strictly positive values and a payoff at some time $ T>0 $ (called maturity) ...
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Call option on forward [closed]

What is the trade description behind a call option on a forward? How can it be described with words and not with mathematical formulas? So what is the intuition behind the following payoff: $$Payoff_{...
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Right risk free rate to price an Option using BS formula

I understand this is very basic question but I still scramble to determine what would be right risk free rate to price a simple European call option using Black-scholes formula, with maturity is 5 ...
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Calibration period

I want to calibrate some model to market data. This could fx be Bates, Kou, Black-Scholes, etc. So, for each model we have a set of parameters which need to be estimated through calibration. Now, my ...
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Derivation of Call Theta from Black Scholes Model [closed]

How is call theta mathematically derived from Black Scholes Model (without approximation) ? Please help me understand each step mathematically.
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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 ...
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Contradictory arguments for ATM/ITM/OTM option demand

I am trying to understand which of the options have the most demand, and found this discussion here. The arguments presented are as follows: ATM is more liquidly traded than ITM/OTM because they are ...
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Why is this inequality strict for arbitrage argument for European call?

in the notes about arbitrage arguments I am reading, I notice the statement We can also see that $$C^E_t>(S_t-K\mathrm{e}^{-r(T-t)})^+$$ Notice that the inequality holds STRICTLY! I don't ...
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Extension of CRR model

I'm considering an extension of the binomial model where the risky asset can take three values at each node, that is $ S_{t+1}=\left\{ \begin{array}{ll} S_t\cdot u\\\nonumber ...
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Why do we worry about the bid/ask spread when pricing option in incomplete market?

Several resources I saw introduce the notion of bid/ask spread when trying to price options in incomplete market, I don't understand why the notion is introduced since we are interested on the price ...
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Are European call and put option useful ? [Cox-Ross-Rubinstein model]

I'm new to the world of option market, but after having studied CRR model I'm wondering if European call and put option are very useful since a talk with my professor that piqued ma curiosity. In the ...
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Model-Free Implied Volatility: Data of Expired Options and Bond Price

I am attempting to calculate Model-Free Implied Volatility for several equity indices (S&P500, NASDAQ100, CAC40, FTSE100, DJIA, EUROSTOXX50, NIKKEI225, NIFTY50). I wish to get historical data of ...
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Expectation of Product of two European Option when vol smile exist

Currently I'm thinking about how to calculate the expectation of the product of two euro option, which is $E[(S_T-K_1)^+(S_T-K_2)^+]$ I can fit some parametric vol model from the market listed option ...
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Caplet stripping in the bwd-looking RFR world with/without maturity adjustment

Since the beginning of this year, LIBOR rates have ceased in some markets like GBP, CHF, and JPY and rates pricing has moved into the RFR space, using compounded overnight rates as the underlying for ...
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What is the meaning of an implied volatility of an Asian option?

Suppose that an Asian option is quoted OTC in terms of its implied volatility. What is the meaning of an implied volatility in this case? Is it an implied volatility of a vanilla European option with ...
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Proof of Calendar-Spread-Inequality

The Calendar-Spread-Inequality compares the prices of two European Call Options on the same underlying non-dividend-paying stock, but with different maturities $T_1<T_2$. Denote the value of a call ...
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how does margin affect the Option Price when Selling an Option

Currently I'm thinking the effect of margin. When selling an option, you need to pay margin everyday and mark to market. In most exchanges, margin is overcollateralized. But when buying a option, you ...
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Assymptotic behaviors of European options

In some of the numerical works on Black-Scholes generalized models, the boundary conditions on the truncated domain taken from the asymptotic behaviors of European call options, which is given by $$\...
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Volatility of American vs European Stock option return

Let's say that I hold an American Call Option (ACO) and an European Call Option (ECO) in my portfolio on the same underlying, with same strike price and same maturity date. Given that I hold both ...
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Find the value of put option using a two-period binomial model

I've been asked to find the price of a two-month European Put Option with strike price $£40$. The price at $S_0=£30$, this can move up to $£40$ or down to $£25$ ($1/3$ chance to go up, $2/3$ chance to ...
Charlie P's user avatar
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Why is call option value same as portfolio value at all times in Black Scholes model?

Following is a part of the text from Steven Shreve Stochastic Calculus for Finance II, for pricing the European Option in Black Scholes model. The argument is that today I start by selling a European ...
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What is the intuition behind a positive theta for European long puts?

I've googled extensively for an answer to this question. Very similar (if not identical) questions have popped up in this same website (example) but I never find the answers to be clear and/or precise....
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Valuation of non-deliverable option

What is the difference between valuation of deliverable and non-deliverable European options? I am not asking settlement-wise, but daily valuation. Will Black-Scholes be used for both?
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Relationship between VIX and Vega

Assuming that all other factors (such as underlying price, strike price, etc.) remain unchanged, I want to see how a spike in VIX would affect the price of the average call option? Assume Vega is ...
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how can properties of transition matrix be applied in the transcation cost of option

I am currently reading the PP BOYLE's article ' Option Replication in Discrete Time with Transaction Costs' written in 1992. Here is one place i couldn't figure out: Where does that $\widehat{p}$ ...
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Is there some reason for volatility smile minima to be displaced from ATM?

I am analyzing some options data and I see that the volatility smile has its minima a few strikes higher than the current traded price (about 2.5 % higher than spot). I have checked my data thoroughly....
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Why is implied volatility often higher for OTM/ITM european call options than ATM? [closed]

I am working on some Black-Scholes stuff and currently investigating implied volatility (IV). I understand that the typical volatility smile can be viewed as a criticism of the assumption about ...
quant_son's user avatar
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HNGARCHFIT in R (No standard deviations or P values printed)

When I estimate an HN-GARCH model using the hngarchfit() from the fOptions package in R, only the coefficient estimates are printed. There are no standard deviations or P-values printed. Does anyone ...
August's user avatar
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European call option lower bound derivation by Black-Scholes formula [closed]

Derive the lower bound of european call options: $$C(S, t)\geq[S-e^{-r(T-t)}K]^+$$ I know how to derive it using put-call parity, but is there any way to derive from Black-Scholes formula?
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