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33 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 ...
Kalev Maricq's user avatar
20 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 ...
Chris Taylor's user avatar
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13 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 ...
Quantuple's user avatar
  • 14.7k
12 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 ...
Hans's user avatar
  • 2,806
9 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 ...
Brian B's user avatar
  • 14.9k
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 ...
dm63's user avatar
  • 17.1k
9 votes
Accepted

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 ...
Hans's user avatar
  • 2,806
9 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 ...
Kevin's user avatar
  • 16k
9 votes

Is it possible to have only one volatility surface for american options (that fits both calls and puts)?

Usually, there is only one vol surface (I have never seen or heard of anyone using two). Almost certainly the most advanced commercially available vol surfaces are built by voladynamics. They also ...
AKdemy's user avatar
  • 8,924
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. ...
Quantuple's user avatar
  • 14.7k
8 votes
Accepted

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 ...
Luigi Ballabio's user avatar
7 votes
Accepted

Yahoo Finance Implied Volatility Calculation

Probably because your risk-free rate is 0.3070664 (30%) Try 0.3%
onlyvix.blogspot.com's user avatar
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 ...
Quantuple's user avatar
  • 14.7k
7 votes
Accepted

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 ...
fni's user avatar
  • 1,896
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 ...
LocalVolatility's user avatar
7 votes
Accepted

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 \...
Kevin's user avatar
  • 16k
7 votes
Accepted

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 ...
SI7's user avatar
  • 843
7 votes
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American options and stopping times

You could but there are difficulties associated with this approach. The main one is that $\tau$ is stochastic, ie it is different for different paths of $S$, so the standard Black-Scholes formula does ...
piterbarg's user avatar
  • 940
6 votes
Accepted

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 ...
Quantuple's user avatar
  • 14.7k
6 votes
Accepted

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 ...
LocalVolatility's user avatar
6 votes
Accepted

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 ...
jherek's user avatar
  • 1,394
6 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 ...
Brian B's user avatar
  • 14.9k
6 votes
Accepted

Why the focus on American put options in literature?

That’s because in the case of a non dividend paying asset (the usual studied case), an American call is worth the same as a European call. Conversely for a non dividend paying asset the American put ...
siou0107's user avatar
  • 2,680
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.
Mark Joshi's user avatar
  • 6,973
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 ...
zer0hedge's user avatar
  • 1,704
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 ...
Antoine Conze's user avatar
5 votes

What is this function in the Longstaff-Schwartz paper?

From the arguments in the lines following the proposition in this paper where the proposition is made; it really looks as if $F_X$ is a typo which should actually read $F_M$, which stands for an ...
Mats Lind's user avatar
  • 1,412
5 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*} &...
Gordon's user avatar
  • 21.1k
5 votes

Estimating optimal exercise boundary for an American call by LSM method

With $V$ American option value, $H$ holding (aka continuation) value, and $B$ bank account value, we have: $$V_N(S_N) = (K-S_N)^+$$ and for $i$ backwards from $N-1$ down to 0, we have: $$ H_i(S_i) = \...
ir7's user avatar
  • 5,043
4 votes

Perpetual American options

Use Dynkin's formula to write the expectation: $\mathbb{E}[e^{-r\tau} \phi(S_\tau)]= g(S_0)+\mathbb{E}[\int_ 0 ^ \tau (A g -rg) dt]$ where $\phi$ is the payoff. Use the infinitismal generator $A$ to ...
user9403's user avatar
  • 1,429

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