Questions tagged [european-options]
An option that can be exercised only at expiration.
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Vega of forward volatility
Imagine I have two European-style options that are the same except for their times to expiry: $T_1 \lt T_2$. And each option has its own vega $\mathcal{V}_1$ and $\mathcal{V}_2$. My question is: what ...
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Quantlib Heston MC Discrepancy between methods
I am a newbie at Quantlib (not finance) and am trying to price with the Heston model. I have implemented two different ways to verify the correctness of the Heston path generation to use in a custom ...
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Why does the lower arbitrage boundary of a European call on stock rise with time if F > S but fall if F < S?
I am reading "Option Volatility & Pricing", 2nd edition, by S. Natenberg.
On page 297, he explains the lower arbitrage boundary of European options. I don't understand the logic for what ...
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3
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Monte Carlo simulations with extremely high volatility
I am using monte Carlo simulations to price a forex option. This is a standard model and works very well with less than 1 % error from black scholes price for 10000 simulations. But, as I increase ...
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1
answer
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Upper Bound on European/American Call Option (Hull)
I recently began reading Hull's derivatives textbook, and found a line that he didn't expand on much. Let $c$ be the price of a European call, $C$ be the price of an American call, and $S_0$ be the ...
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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 ...
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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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2
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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
$$
h(\theta x) \cdot \theta \cdot\left(...
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1
answer
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Confusion about payoff for an option [closed]
My teacher said that the payoff of a put is $\mathrm{max}(K-S_T, 0)$, where $K$ is the strike price and $S_T$ is the spot price at maturity. Why isn't it $K$ if $K-S_T > 0$ and $0$ otherwise (i.e. $...
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Practical use of Dual Delta?
I am wondering what the practical use of the Black-Scholes Dual-Delta is?
I know it is the first derivative wrt the strike price:
$$
\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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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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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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answer
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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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answer
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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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141
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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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