28 votes

Exercising an American call option early

Cases where exercising a call early makes sense: There is a dividend payment the next day that is >= the interest on the strike price + a put with the same strike and expiration. Exercise the ...
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16 votes
Accepted

American Options relation between greeks

No, you should not expect such a relationship to hold in general. The reason is that American options have an "exercise barrier" which European options don't, and this results in different prices and ...
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  • 5,628
13 votes

Convexity of an American put option

It is indeed. The price of an American option is the Bermuda option in the limit that the exercising interval approaches zero. The Bermuda option at any exercising time can be evaluated inductively ...
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  • 2,471
12 votes
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Least Squares Monte Carlo

To compute the price of an American option or a callable instrument in general, at each potential exercise date, one is required to compare its continuation value (discounted risk-neutral expectation ...
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10 votes
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Convexity of an American put option

Here is a much more straightforward proof of the convexity of the American option with respect to a parameter, if it is independent of time and deterministic, than my previous one, though I am happy ...
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  • 2,471
9 votes

Exercising an American call option early

At the time of exercise, you don't know what the final expiry stock value is. Consider the portfolio consisting of the option and $K$ zero coupon bonds worth $B_t \leq 1.$ At expiry its value is $$...
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  • 6,743
9 votes
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American Call: when it's European?

it's a model-free result. The conditions are $d\leq 0, r\geq 0.$ The proof is that for a european $$ C_t > S_t - Ke^{-r(T-t)} \geq S_T - K $$ and the American is worth at least as much so you ...
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  • 6,743
9 votes

What is the industry standard pricing model for CME-traded Eurodollar future (American) options?

Having traded these options for a number of years I have some insight. It’s my belief that those that make a living specifically out of these options do have tree-style models that take into account ...
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  • 13.9k
8 votes

Least Square Monte Carlo - american Call Option

If implemented properly, least-squares Monte Carlo as originally suggested by Longstaff-Schwartz should allow you to identify sub-optimal exercise dates and a lower bound of the true option price. ...
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  • 13.8k
8 votes
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Implied Dividend from American Options (in practice)

There are 2 ways to do it. The good-enough way, and the complete and complex way. The Good-Enough Way Here you will convert to a situation where you can apply put-call parity. Begin by finding the ...
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  • 14.4k
8 votes

Why aren't american put options martingales?

European Contracts It's a really important question and as @noob2 commented, the FTAP is normally applied to European-style derivatives, even if they are (strongly) path-dependent, including barrier ...
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  • 13.8k
7 votes

American Call: when it's European?

It can also be proved by Jenson's inequality. It can only be optimal to exercise the American option if the option is below its intrinsic value; but since the "max" function is convex, the European ...
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  • 1,329
7 votes
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Yahoo Finance Implied Volatility Calculation

Probably because your risk-free rate is 0.3070664 (30%) Try 0.3%
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7 votes

A paradox about the American Put option price

So, from this simple no-arbitrage argument, we see that the price of the option must always be at least its intrisic value. Yes indeed However, at this point I realized something strange: if this ...
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7 votes
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Understanding early exercise of options - The implicit put in an American call

Let’s forget about dividends (actually assume there are no dividends). By Put Call parity $C^E(K)= P^E(K) + S - Ke^{-rt}$. Suppose that $S>K$ [otherwise you don’t even think about exercising!], if ...
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  • 1,856
7 votes
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Implied Volatility Surface - log forward moneyness

The reason is that, as shown in Proposition 2.1 of that paper, in order to exclude static calendar arbitrage, the total variance has to be strictly increasing in forward moneyness. See also the below ...
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7 votes
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Issue Using QuantLib and Python to Calculate Price and Greeks for American Option With Discrete Dividends

You're not setting the global evaluation date. If you don't, you're in December 2017 and your option has expired a good while ago. Adding ...
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7 votes
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Do Perpetual American Options have closed form functions to compute the Greeks?

The Black-Scholes differential equation is a second-order PDE in two dimensions and reads as \begin{align*} \frac{\partial f}{\partial t} + rx\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2 x^2 \...
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7 votes
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Implied vol and model calibration for an american option on a dividend paying stock - is there a market standard pricing model?

Theoretically, this is a more difficult problem than it looks like at first glance. Unfortunately, existing literature taking into account a proper dividend consideration is rare (at least from a ...
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  • 721
6 votes

American put for negative interest rates

The classic result is never early exercise an American call if $r \geq 0, d \leq 0.$ If we think in terms of FX, calls and puts are really the same thing and by switching currency, we get never early ...
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  • 6,743
6 votes
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The Upper Bound of an American Put Option

Dividends do not matter for the determination of the upper bound. Indeed, the maximum profit which the holder of a put option can make (be it through a European or an American exercise feature) is ...
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  • 13.8k
6 votes
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Figure of Stopping and Continuation Region

The exercise boundary $B_t$ for a finite maturity American put option is not a constant function of time as in your plot. As mentioned in the excerpt, $B_T = K$ at maturity. But for $t < T$, we ...
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6 votes
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Brennan-Schwartz algorithm for pricing American options

Ikonen and Toivanen don't say that the LCP is solved exactly, they simply say that the modified back-substitution is a valid algorithm to solve the LCP. A numerical error may arise around the ...
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  • 1,213
6 votes

Convexity of an American put option

Let $\mathscr{T}$ be the set of stopping times with values in $[0, T]$. Note that, for any $\tau \in \mathscr{T}$, $\lambda_1\ge 0$, $\lambda_2 \ge 0$, and $\lambda_1+\lambda_2 =1$, \begin{align*} &...
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  • 20.4k
5 votes

Pricing American with floating strike

These options can be priced by adding an early exercise premium value to the intrinsic value: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.542.3141&rep=rep1&type=pdf
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  • 5,629
5 votes

Intuition behind American Option pricing

The model here is the binomial option pricing model, so the second term in the brackets represents the expected future value of the option (under riskneutral probabilities). The aim of the option ...
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  • 5,629
5 votes

Can call options be priced with Least-Squares Monte Carlo?

American calls on a non-dividend paying stock are worth the same as European ones so there is no point to using least-squares.
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  • 6,743
5 votes

Soft American Options

It is easier to understand Taleb's distinction between 'soft' and 'hard' American options if we understand from the beginning that he is talking about FX options or behaving similarly options on ...
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  • 1,644
5 votes

Do we need to derive the PDE for the option price when applying Least Squares Monte Carlo?

You do not need the PDE to implement the LSM algorithm. The $T$ maturity American call price on time $t$ is $$v_t = \max_{\tau} E_t\left[e^{-\int_t^\tau r(u) du} (S_\tau - K)^+\right]$$ where the max ...
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5 votes
Accepted

Early exercise of American options

It is easiest to just think about volatility dropping to near zero in each of these cases, and also to assume that you will immediately trade out of the stock position. Note the following principles ...
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  • 14.4k

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