19

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 call, buy the put, and sell a zero-coupon bond for (strike price + put price - dividend). The transaction has a net 0 cashflow at time 0 and at expiry you will be ...


15

If you think the stock is going to continue going up, just wait. If you think the stock has reached its peak, then short it in the open market. If the shorted stock continues to climb, you can always cover with your call option. If however the stock falls below your strike price, then let the option expire and cover at the market price. It's this very ...


9

A stochastic volatility model for a single risky asset can't be complete because you have two sources of randomness. But you can easily make it complete by adding a derivative whose value depends on the volatility. For example, if you add a variance swap in the Heston model then it becomes complete. This allows you to calibrate the model. But your ...


9

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 never early exercise. So it's worth the same as European. To prove the inequality, observe if $B_T =1,$ take $K$ units of $B_t$ and one of $C_t$ to get something ...


8

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 $$ \max(S,K) \geq S $$ since can you exercise and get the stock if $S>K$ and have $K$ otherwise. So at all times previously, $$ C_t + K B_t > S_t $$ since ...


8

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 greeks. In the case of put options (with interest rate $r>0$) as the spot price falls, at some point it becomes optimal to exercise early and take the cash. ...


7

I guess if your American-style option is in no-exercise region, you can use exactly the same bisection method as for European option.The implied volatility will be different, but the method is still the same. See for example, here, chapter 9.3.3. The applicability of bisection method for American-style options is discussed in the book "Binomial Models in ...


7

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 price satisfies the following inequality: $$c(S_t, t)=e^{-rT}\mathbb{E}[(S_T-K)^+]>=e^{-rT}\left(\mathbb{E}[S_T]-\mathbb{E}[K]\right)^+=S_t-Ke^{-rT} $$ The ...


7

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. There are many articles out there discussing this non trivial topic. @MarkJoshi can probably shed some more light, see this nice paper. You claim that your LSM ...


7

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 you exercise the American Call now you get $S - K$ that for sure is less than the intrinsic value of the European call, i.e. when the American Call is still ...


7

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 to links for details on this result. The intuition is that for European options, only the distribution of the terminal spot price is relevant. Furthermore, $...


6

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 exactly equal to the strike price $X$. This can be seen by simply looking at the payout function: the maximum profit is finite and located on the downside when the ...


6

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


6

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 strike $K$ where put and call prices are closest to each other. This might not end up being the closest-to-the-money strike, but it will do. Now run the ...


6

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 early exercise. On the other hand , those that have occasional use of these options (such as interest rate derivatives dealers who might use them to hedge otc ...


5

Standard Options on CBOE expire on the Saturday following the third Friday of a month. Additionally to that there exist weekly options. That's why you see these two series of options.


5

There are several ways to choose a particular EMM. I believe that the most popular approach is to use a "distance" between $\mathbb{P}$ and $\mathbb{Q}$. Most papers use a minimal entropy approach(for example, Fujiwara and Miyahara, Esche and Schweizer, or Hubalek and Sgarra) or a relative q-entropy approach (for example, Jeanblanc, Klöppel, & Miyahara) ...


5

For a standard American exercise option expiring at $T>0$, price is still monotically increasing in volatility under the Black-Scholes model (though obviously it is not strictly monotonic, due to early exercise rendering price insensitive to volatility in some regions of parameter space). To see this, you can use one of three techniques: Investigate the ...


5

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 holder is always to maximize the value of his option. He can at any point sell the option at the fair market price $E(V_{n+1})$ or exercise it to get $G_n$. So if ...


5

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


5

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 is true, why in the world should I exercise my put option before expiry?? The inequality seems to indicate that it would be an unwise decision to ever exercise ...


5

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 have $B_t < K$ as you would never pre-maturely exercise to receive a zero payoff. Below is a plot of the early exercise boundary that I once produced for a ...


5

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 ql.Settings.instance().evaluationDate = valuation_date before the calculations will give you the expected results.


5

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 approximation of $F$ using the first $M$ basis functions. It is argued that: The key to this result is that the convergence of $F_M(w, t)$ to $F(w;t)$ is ...


4

With respect to your first question: Yes. The regression has to determine the conditional expectation of the continuation value, i.e., the (discounted) value of the future cash flows including the exercise criteria(s) you have determined for the remaining future exercise times, conditional to the assumption that you did not exercise at or prior the current ...


4

Hum, that's one of the most important questions in financial engineering, that why no answer is proposed. If you have available data as option prices, you may calibrate a parametric EMM but nothing can tell that it's the best EMM (cause there is no best EMM). So make a choice and defend your choice by saying 'it's simple and allows beautiful result' like ...


4

For a vanilla option, this is a very slow way to get the boundary, and it's somewhat unreliable for any option. In either a more standard grid scheme or in a LS solver, you obtain the boundary by finding two nodes such that one of them has option value equal to early exercise value, and its neighbor has option value above early exercise value. This gives ...


4

Because you would make a higher profit if you sold the option on the open market at that point in time, rather than exercising it at that point in time due to the time value of money.


4

If you can dynamically hedge then you can monetize the value of your option without prematurely exercising it. Before writing about Randomness and Black Swans, Taleb wrote a book on the topic. The short version of the story is find the DV01 of your position, and take an opposite position with the same DV01. (& if you want, line up the rest of the ...


4

Because if you sell, you will get a higher value than 20USD per share. You can think of the reason behind this added value is that having a deep ITM option is better than having a stock: your downside is limited. Therefore your option is worth more on the market than it's exercise value. This is why you are better off by selling it in your case (if you know ...


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