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 ...
- 596
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 ...
- 5,931
13
votes
Accepted
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 ...
- 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 ...
- 2,806
10
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 ...
- 16.2k
9
votes
Accepted
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 ...
- 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 ...
- 17.5k
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 ...
- 2,806
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 ...
- 9,419
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. ...
- 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
...
- 7,853
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 ...
- 1,896
7
votes
Accepted
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 ...
- 6,084
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 \...
- 16.2k
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 ...
- 863
7
votes
Accepted
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 ...
- 940
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 ...
- 6,084
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 ...
- 1,444
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 ...
- 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 ...
- 2,680
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 ...
- 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 ...
- 5,702
5
votes
What is the point of the regression in Longstaff Schwartz method?
You are mixing up the realization of a random variable with its expected value at a certain stage.
Let's say you are at path $i$ and time step $t_j$, what you want is not the realization of the ...
- 2,240
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 ...
- 1,422
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*}
&...
- 21.2k
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) = \...
- 5,133
5
votes
implied-information in american option
I observe that Christoffersen et al. (2012) consider the implied volatility from European options, as calculated under the BS model and other extensions of it. Therefore, implied volatilities from ...
- 1,426
4
votes
Accepted
why we drop the last term in the Barone-Adesi Whaley formula
Below is a hand-wavy way to reach the above result. I suspect there is a more elegant way to show it though.
The early exercise premium is defined as the difference between the American and European ...
- 14.7k
4
votes
Is there any useful links for option pricing (american + asian + european) using R
Below is an example of how you could plot a "call" option value with RQuantLib:
...
- 1,704
4
votes
Accepted
Intuitively understand boundaries of American Call and Put
The lower bounds are obvious since American options can be exercised at any time while European options can only be exercised at maturity.
The upper bounds are obtained from the property that an ...
- 5,702
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