# Questions tagged [american-options]

An option that may be exercised at any time before the expiration date.

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### JDOI variance reduction method python

Has anyone read the paper 'JDOI variance reduction method and the pricing of American-style options' by Johan. I want to implement the simulation. But For Monte carlo I got different results. andI ...
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### Longstaff & Schwartz algorithm - Python: American option cheaper than European option

I have implemented the Longstaff & Schwartz algorithm for pricing American Option in Python, but I ran into an issue while doing some experiments: sometimes, for the same option, I get a higher ...
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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 ...
1 vote
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### How to calculate return on investment for an adjustment to a complex options position?

Say I currently hold a set of options positions with the same symbol/expiry that collectively have a net present value based on the estimated value at expiration of +10. I could also liquidate the ...
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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 ...
608 views

### American put option. Exercise time is a random variable, calculation of expected payoff

I got an American put option, where the payoff is $V_\tau = \max(K - X_{\tau}, 0)$ and $X_{\tau}$ is the price of an underlying at the stopping time $\tau < T$. The underlying follows a standard ...
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### Far OTM calculation issue on Bjerksund-Stensland

Has anyone come across and fixed calculation issues on boundaries using Bjerksund-Stensland 2002 (Hull, Haug or Rouah implementations) ? Thanks in advance
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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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### How to price lookback american option when its payment is distributed during its life

I would like to price a floating strike american lookback with a particular feature: I don't want to charge upfront the client, rather I would like to insert a "running fee", some sort of a dividend. ...
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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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### 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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### 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) ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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### Monte Carlo American Options Discrete Dividends

Built some tree methods to price american options with discrete dividends. But I have no way to really verify my work. Questions below: Does it make sense to build a Monte Carlo pricer to use as a ...
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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 ...
1 vote
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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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### 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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### 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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### 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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### 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: ...
1 vote
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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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One knows that the price of a bermudan claim exercisable at times $T_1, T_2,\ldots, T_N$ is $$V_0 = \sup_{\tau\in\Gamma} \mathbf{E} \left[ e^{\int_0^{\tau} r_s ds} \varphi_{\tau}\left( x_{\tau} \... 1 vote 0 answers 145 views ### 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 ... 0 votes 3 answers 261 views ### Are "American" option strategies traded OTC? Is there such a thing as an American butterfly spread? For a European butterfly spread simply buying 1 put with strike price X+a, 1 put with strike price X-a and shorting 2 calls with strike price X, ... 1 vote 0 answers 82 views ### 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? 7 votes 2 answers 948 views ### What's the connection between implied vol curve of SPX and SPY? I think there should be an obvious connection of the two implied vol curves from the SPX and SPY markets since the underlying of SPX is SP500, while the underlying of SPY is a ETF which tracks sp500 ... 2 votes 1 answer 297 views ### 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? -3 votes 2 answers 246 views ### Why is accuracy important in pricing American Options? [closed] I see a lot of academic papers talking about accuracy in pricing American Options (and finding analytic solutions). Why is there so much interest in this topic? Isn't the option price set by the ... 1 vote 0 answers 35 views ### 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). ... -1 votes 2 answers 1k views ### Bermudan Swaptions [closed] Can someone explain, in layman's terms, the mechanics behind Bermudan Swapttions ( without having recourse to pricing models )? Why are they popular? when are they used ? How are they hedged i.e ... 0 votes 0 answers 58 views ### 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. ... 9 votes 1 answer 563 views ### Why were Laguerre polynomials a good choice of basis functions for American Monte Carlo? I am implementing LSMC to price American options based on a custom model. I now need to make a choice of basis functions, so I am looking for the theoretical justification for using Laguerre ... 2 votes 0 answers 93 views ### 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! 1 vote 1 answer 107 views ### Significant digits in numerical derivative pricing I am looking for examples for the number of significant digits commonly required to find numerically the price different types of derivatives. For instance, if we have to price an American put option, ... 2 votes 0 answers 97 views ### 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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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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### Convexity of an American put option

Is the price of an American put on an underlying without dividend convex with respect to the strike?