Questions tagged [american-options]
An option that may be exercised at any time before the expiration date.
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Black Scholes/American Put/Martingale Condition
Consider a Black Scholes model with $r \geq 0$. Show that the price of an American Put Option with maturity $T > 0$ is bounded by $\frac{K}{1 + \alpha} {(\frac{\alpha K}{1 + \alpha})}^{\alpha}{S_{0}...
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American option pricing using path integrals
I am writing a brute force code in python that implements the path integral formalism for the American put option, the goal being to obtain its price at given a price $S_0$ of the underlying asset.
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Binomial option pricing model for American options on assets paying a continuous dividend yield
Let's say an asset has a continuous dividend yield of 5% (and assume interest rate is 0%). If I want to price an American call option on such an asset, I take each time step individually and construct ...
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Prove that there exists a critical price for a American call option with continuous dividends
For a American call option on a stock with continuous dividend yield, show that there exists a critical price, that is a price $S^*_t$ such that if the stock price is above this at time $t$, then it ...
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implied-information in american option
I have recently been researching European options versus American options implied information. For European options, an overview article is Christoffersen(2012). But for American options, I only found ...
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Delta-hedge experiment of American Put option
I am trying to run a delta-hedge experiment for an American Put option but there's a (systematic) hedge error which I cannot seem to understand or fix.
My implementation is found in the bottom of this ...
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Implied Volatility Discrepancy in American Options - Mathematical Reasoning?
I've been analyzing Tesla stock American options data and have observed an interesting pattern that I'd appreciate some help understanding.
For this analysis, I obtained the Implied Volatilities (IVs) ...
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Optimal exercise time in Binomial model
Let (B, S) a multi period binomial model that is arbitrage free.
I would like to prove that the unique optimal exercise time for an American call option is the maturity time T. My idea is to prove ...
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LSMC for Out of The Money paths
In the Longstaff & Schawartz article they condition on using In-The-Money (ITM) paths only for the regression. The reason for this is to obtain more accurate results and also reduce the ...
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Is American put Gamma always greater than the European one in the non-early-exercise domain?
Consider a pair of American and European puts with the same specifications except the former has the continuous early exercise right. Has anyone plotted the Gamma's of both as functions of the ...
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Pathwise sensitivities of American options - Derivative of the American payoff function
How can I compute the derivative of the payoff function for an American put option?
In the paper "Smoking adjoints: fast Monte Carlo Greeks" by Giles and Glasserman (2006) they compare two ...
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Is this an optimal stopping problem?
I am trying to work out how to approach a machine learning problem of 'learning' an optimal liquidation time/threshold, under some conditions, from historic data. The idea is a trader armed with this ...
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Continuation value in Longstaff-Schwartz: Why the expected value?
In the paper by Longstaff and Schwartz on American option pricing, the continuation value at time $t_k$ is given by:
\begin{align}
F(\omega;t_k) = \mathbb{E}_Q\Big[\sum_{j=k+1}^Kexp\Big(-\int_{t_k}^{...
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American option pricing formulation
Assuming the usual setup of:
$\left(\Omega, \mathcal{S}, \mathbb{P}\right)$ our probability space endowed with a filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\in[0,T]}$,
$T>0$ denoting the ...
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Longstaff-Schwarz LS Monte Carlo - which approach is correct? [closed]
I'm trying to understand Least-Square Monte Carlo approach for pricing american options. I'm familiar with Tsitsiklis and van Roy (2001) approach where we are going backwards with:
$V_T = h(S_T)$, ...
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If American Options always have positive time value, how can it be optimal to exercise an American Put early? [duplicate]
r > 0. I understand that money today is worth more than money tomorrow. So if volatility is 0, it's better to take the money today. But I don't understand how to square away the following:
Time ...
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Comparison of the American and European call deltas
Suppose the interest rate is zero. A stock with price $S(t)$ at time $t$ pays only one dividend at time $t_1$ such that $S(s_+)=S(t_1^-)q$ where $q\in[0,1]$ is a constant. Consider a European call and ...
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How to improve fit in American options vol surface?
I am trying to model the volatility surface of index etfs (spy, iwm and qqq). I am using the CRR model with discrete dividends and the spot model. I find that for some cases there is a noticeable ...
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How to price an american put option on a dividend-paying stock? [duplicate]
There is no Black Scholes formula for the value of an American put option on dividend paying stock eithe has been produced ? Should I use the binomial model ?
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Properties of the American derivative security price process
$$
\newcommand{\cbkt}[1]{\left\{{#1}\right\}}
\newcommand{\rbkt}[1]{\left({#1}\right)}
\newcommand{\sqbkt}[1]{\left[{#1}\right]}
$$
Shreve volume I, defines an American derivative security as follows:
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Black-Scholes PDE for American options (inequality)
I am currently working on American options.
I saw that we can derive a PDE for American style options in the same way as with BS for European options.
In a textbook, I found that the PDE leads to an ...
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Using the SABR Model to Calibrate the Implied Volatility Smile/Surface of an American Option
If I already know the implied volatility smile/surface of an American option, can I directly use the SABR model to calibrate the smile/surface, or do I need to make certain adjustments first?
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Convergence rate of Bermudan to American option
When trying to value an American option we often use grid-based methods (e.g. Monte Carlo in combination with Longstaff Schwartz; or Finite Difference Methods). As such, we are in fact estimating the ...
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Time steps in CRR Binomial Option Pricing for American Options
how do you determine the time steps required as inputs to the Cox Rubinstein Binomial Option Pricing model when trying to determine the fair price of an American option? Most textbooks and literature ...
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How can I derive the price of american options given the european options prices? [closed]
I have the european volatility surface of a given asset. What is the correct procedure to compute the price of the options with american exercise type?
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Is it possible to have only one volatility surface for american options (that fits both calls and puts)?
Put-Call Parity does not hold for american options. Hence, I don't see how it would be possible to have one surface that would encompass both calls and puts.
For example:
Let pick a call lying in the ...
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Does the risk neutral pdf that is derived using Litzenberger-Breeden Method correspond to gamma and it's integral correspond to delta?
I derived the pdf using the butterfly prices and the curve looks like gamma of an option at every strike. Is that the case or am I missing something to get the pricing of an option?
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Exotic options with lookback features [closed]
I am trying to value an american call option with a lookback feature. So the holder can choose to exercise either based on a fixed strike (K) or a floating strike equal to 10-day moving average (MA). ...
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Put price characterisation
I am reading Shreve's Stochastic Calculus for Finance II: Continuous-Time Models.
I am trying to understand the below two concepts:
Topic 8.3.3 Analytical Characterization of the Put price on Page ...
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Binomial Tree for CDF
I'm tasked with solving an optimal stopping problem relating to stochastic process representing a firms profit namely $X_t = X_0 + \mu t + \sigma Wt$ where $X_0, \mu$ and $\sigma$ are constants.
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What are some good books to get started with option theory? [duplicate]
Recently graduated in econometrics but starting to realize my knowledge is limited. Any and all tips are welcome!
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Perpetual Option Paying Chooser Option
A perpetual option solves the ODE
$$rSV_S+\frac{1}{2}\sigma^2S^2V_{SS}-rV=0$$
The general solution is $$V(S)=aS+bS^{\gamma}$$ where $\gamma=-\frac{2r}{\sigma^2}<0$.
For an American put option with ...
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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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Is there a closed form solution to calculate Fugit for stock options?
I am trying to find a quick and dirty way to estimate fugit for a basket of American options real time. Is there a quick way to provide a proxy? I know one can do binomial tree or monte carlos ...
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What is the optimal time for exercising American call and put option?
A 9 month American option (underlying) is known to pay dividend of USD 1 and USD 0.75 at the end of the ...
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American option under Ornstein-Uhlenbeck stock price
I came across with the following problem:
For the Ornstein-Uhlenbeck process $(X_t, 0\leq t\leq T)$ with initial
condition $X_0 = x$, find the stopping time $\tau$ that maximizes
$\mathbb{E}[e^{-r\...
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Early exercising American put options
I have found a proof that an American put option without dividend will never be exercised early. However, I suspect that that is not true, so there should be a mistake in the proof. The proof is as ...
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Under put call parity shouldnt the implied volatility for call and put for same strike and maturity be the same?
If all of the other inputs into black scholes (divs/rates/time to maturity/strick/current price/etc) are all the same between two pairs of calls/put contracts on the same security, shouldn't the ...
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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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American Option Valuation - Induction algorithm
The price of an American put option is given by
$$V_k = \sup_{\tau\in\mathcal{T}, \tau\ge t_K} E\{e^{-\int_{t_k}^\tau r_sds} (K-S_{\tau})^+|\mathcal{F}_{t_k}\}$$
I found in one book the following:
$$\...
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Suboptimality bias in least squares Monte Carlo for American options
In Monte Carlo pricing of American options we form two estimators:
A high estimator that is biased upward because of "look-ahead" bias (i.e., at any given time we uses future information to ...
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Generating Greeks with American Options
Investor and Software Engineer but very new to quant finance here...
I have the below code (which I'm sure will be helpful for some) and have some questions regarding the function parameters!
Is RF ...
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What is the greatest theoretical delta value?
In a few options positions I'm currently holding I noticed delta values of ~0.6 while gamma is ~1.0 which surprised me as I thought delta can never be greater than 1 - meaning for every 1\$ move in ...
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Selling American calls before ex date vs. exercising
From reading Hull OFOD (among other references), I understand that early exercise makes sense for an American call option at time $t_n$ when $$D_n > K\Big[1-e^{-r\big(T-t_n\big)}\Big]$$
for a call ...
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Bermudan pricing in Black-Scholes
Is there an "analytical" method to price American options (approximated as daily Bermudans) in the Black-Scholes model using backward induction?
$$V_T(S) = \max(K-S, 0)$$
$$V_{T-\Delta t}(S) ...
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Difference in pricing of American call and put
In Paul Wilmotts quantitative finance books he says that the the value of an American option satisfies the following
$$
\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2 \frac{\partial^2V}{\partial ...
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Early exercise American Options with Dividend
this is a basic question but I have not fully understood it.
Let's say we have dividend paying stock (continuous dividend yield), when would we exercise the Option early? Since the Dividend yield is ...
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Yield to call on American style callable bond
(Assuming current bond price is quoted and maturity, par value, strike price all known..)
I was wondering how do we calculate yield to call on American style callable bonds after the call date has ...
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Can a decrease in call open interest drive the stock price down?
I am analysing the price movement of a U.S. stock in conjunction with its open interest on calls vs puts. If within a month, the call open interest drastically declines (even relative to put open ...
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Why Vasicek model on a tree is a bad choice for pricing American option on credit prepayment?
I have an American option on a credit prepayment, i.e. the holder of the option can prepay the remaining credit if the interest rate falls below the initial strike. The pricing of this option was done ...
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