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Questions tagged [european-options]

An option that can be exercised only at expiration.

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Potential arbitrage opportunity or fallacy?

Suppose we have two European options with the same expiration: a call priced at $c$ with strike price $K_1$ and a put priced at $p$ with $K_2 (>K_1)$. Further, suppose the zero-points of the two ...
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Understanding basic options arbitrage in Hull

I’m reading Hull’s book, Options, Futures and Other Derivatives. In Chapter 11 he discusses put-call parity and the arbitrage opportunities that can result from its violation. I’m having a basic issue,...
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Implied Vol under CEV model

Consider the following steps: Suppose the underlying equity follows a CEV model $dS_t = rS_t dt + \sigma S^{0.5} dW_t$. Use the above CEV model to simulate Monte Carlo paths and price a large set (...
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How do linear wings reconcile with volatility frowns

I was recently looking at a paper that brought up that under certain market conditions the risk neutral density can exhibit thin tails, yielding a volatility frown as opposed to the more usual ...
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Volatility Surface Modelling in Python

For my master thesis, I try to create a Volatility Surface for S&P500 Index options. Every time I run my code, the surface I get is full of spikes. I'm just not sure if these are outliers which ...
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Pricing a custom option in terms of simpler instruments

I have the following custom European Option $F$ on the underlying $S$ whose pay-off at expiry $T$ follows: $$ F(T) = \min{[B, \max{[K_1-S(T), S(T)-K_2,0]}]} $$ where $B$ is a cash position and $0<...
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Describing the volatility skew with a set of options

Say you have a set of options data, and you filter the dataset based on certain criteria such as the bid-ask spread, open volume etc. and you end up with a set of liquid options based on said criteria....
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Change in price of underlying impact on delta gamma and vega

I am working my way through Natenberg's book as well as the accompanying workbook, and there is a question I cannot figure out (p86). Futures price = 149.65 time to August expiration = 8 weeks annual ...
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Barrier Puts Pricing (down-and-in put)

I am trying to price the down-and-in put option with European Style (when barrier level < strike price) by using Black Scholes Option Pricing model. but after checking the formula several times, I ...
Wannapat P.'s user avatar
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How to access the Black Sholes Formula through the Distributive Law?

Recently I read a comment on how to interpret the Black Sholes Formula and more specifically how to wrap your head around the d1/d2. Although there were many good comments, this one stood out when one ...
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Why do ATM options intuitively have higher Time Value (Extrinsic Value) than Out- and In-The-Money options?

I'm trying to get some intuition concerning the Black Sholes Formula and in doing so I've come across these graphs: Trying to understand the intrinsic value relationship with Options Value was ...
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Pricing a zero coupon callable bond

Suppose I have a 20-year zero bond with a call date in 10 years and a zero interest rate of 2%, which is currently valued at a Z-spread of 100. Now I would like to evaluate the right of termination ...
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path dependency and dollar gamma

On a previous question on this website, a user derived the following PnL of a delta-hedged option: $$P\&L_{[0,T]} = \int_0^T \frac{1}{2} \underbrace{\Gamma(t,S_t,\sigma^2_{t,\text{impl.}})S_t^2}_{\...
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Tradeability of Option (not underlying) necessary assumption in BSM?

Working with the Black Scholes Model to value european Call and Put Options I encountered a question that came up during the valuation of a (european Call) Option, which itself cannot be traded (e.g. ...
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The partial derivative of a call option with respect to $t$ [closed]

In Black-Scholes related computations, why do we not treat the stock price $S$ as a function of $t$ when taking partial derivatives with respect to $t$? For example, if $$c(t,T)=SN(d_1)-Ke^{-r(T-t)}N(...
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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(...
THATS MY QUANT MY QUANTITATIVE's user avatar
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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. $...
Cyclopropane's user avatar
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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 ...
Analysis's user avatar
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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 ...
Kaustubh Shamshery's user avatar
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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 ...
Jennifer's user avatar
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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 ...
LunaStorm's user avatar
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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 ...
Frido's user avatar
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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 ...
bigInner's user avatar
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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 ...
nimbus3000's user avatar
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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 ...
timofiej8384's user avatar
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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?
Alec's user avatar
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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_{...
Kapes Mate's user avatar
3 votes
1 answer
567 views

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 ...
Brian Smith's user avatar
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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 ...
CasMath's user avatar
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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.
Alan's user avatar
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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 ...
Shreyans's user avatar
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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 ...
Ice Tea's user avatar
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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 ...
Ice Tea's user avatar
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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 ...
G2MWF's user avatar
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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 ...
G2MWF's user avatar
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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 ...
Vishal Nagwani's user avatar
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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 ...
OneDayMemo's user avatar
6 votes
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655 views

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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