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Kamil90
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Joined: February 15th, 2012, 2:02 pm

### American call on a dividend paying stock

I would like to show that American call can be exercised early if the stock pays dividend. So, I have a code to value an American call, what I need to know is how to incorporate the dividend. Most FDM books say dividends are set through V(t+,S)=V(t-,S-d), so basically if I implement it, I do:
1. Assume I have a numerical solution V at the grid node t_i,S_j from solving BS PDE, then I set V_updated(t_i,S_j)=V(t_i,S_j-d_i) at time node t_i if the dividend is paid and set V back to V_updated. If this time point the dividend is not paid, do nothing.
2. And because it is American, after I take V(t_i,S_j)=max(V(t_I,S_j),(S_j-K)^+).
3. Solve the BS pde again
4. Repeat
First question: is that a correct iteration algorithm?
Second question: I was not able to find this algorithm online, all the papers that come up on google consider just changing the drift from r to r-d in BS PDE, but I guess this is for the case of a continuous dividend? Why would they consider that because in reality most stocks pay discrete dividends, am I wrong?

Cuchulainn
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### Re: American call on a dividend paying stock

One approach is this
http://www.nccr-finrisk.uzh.ch/media/pdf/ODD.pdf

and the update in Alan's new book (chapter 9 AFAIR).

Kamil90
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Joined: February 15th, 2012, 2:02 pm

### Re: American call on a dividend paying stock

One approach is this
http://www.nccr-finrisk.uzh.ch/media/pdf/ODD.pdf

and the update in Alan's new book (chapter 9 AFAIR).
ok, I looked through the paper and it confirms that my scheme is correct. However, it only considers the call option in details. And clearly If I want to exercise, I better do that right before the dividend. On the other hand, would I use the same algorithm for American puts? The paper just mentions that and the fact that it could be optimal at any point in the lifetime of an option regardless of the dividends. What puts me in doubt is the fact that if I want to exercise, I want to do that right after the dividend is paid, so I get a cheaper stock. In that case, when I roll back my solver, should not I first check for max(intrinsic, value to hold) and then adjust for dividend by the continuity of an option price?

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### Re: American call on a dividend paying stock

early exercise decisions always get back to "what do i give up" versus "what do i get" - in the case of a put, ask yourself in the current interest rate environment what you get

Kamil90
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Posts: 178
Joined: February 15th, 2012, 2:02 pm

### Re: American call on a dividend paying stock

One approach is this
http://www.nccr-finrisk.uzh.ch/media/pdf/ODD.pdf

and the update in Alan's new book (chapter 9 AFAIR).
Alan,
can you please comment on puts? In the paper, only calls are considered in details. Puts are only mentioned.

Kamil90
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Posts: 178
Joined: February 15th, 2012, 2:02 pm

### Re: American call on a dividend paying stock

early exercise decisions always get back to "what do i give up" versus "what do i get" - in the case of a put, ask yourself in the current interest rate environment what you get
I am using a numerical algorithm to solve this problem and it has to work in any environment.

Alan
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### Re: American call on a dividend paying stock

One approach is this
http://www.nccr-finrisk.uzh.ch/media/pdf/ODD.pdf

and the update in Alan's new book (chapter 9 AFAIR).
Alan,
can you please comment on puts? In the paper, only calls are considered in details. Puts are only mentioned.
As Cuchulainn noted, puts are treated comprehensively in Ch. 9 of "Option Valuation under Stochastic Volatility II" (using the same HHL approach as in the linked paper). Give me a few minutes and I'll post a nice chart.

Alan
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### Re: American call on a dividend paying stock

Here are some nice early exercise boundaries in the $(t,S)$ plane for an American put with a discrete dividend at $t_D = 0.8$, option expiration at $t = 1$, and strike price $K=100$. (The other parameters are $r = 0.08$ and $\sigma = 0.40$).

The dividend amount paid is the policy $\mathcal{D}(S)$ amount shown above each figure, where $S = S(t_D-)$ is the random stock price (under GBM) immediately before going ex-dividend. The first two rows show the boundary under the Haug, Haug, & Lewis piecewise GBM approach (the linked paper above) with various dividend policy functions. The third row shows the boundary under the escrowed dividend model in the $(t,\tilde{S})$ plane, where $\tilde{S}$ is the risky part of the stock price. The chart is from Ch. 9 of the just-mentioned book, where the various chart features are explained.

Kamil90
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Joined: February 15th, 2012, 2:02 pm

### Re: American call on a dividend paying stock

I don't own the book so I can't just lookup that chapter. I referred to the paper because this is something accessible to everyone as of today. And I still did not get my answer, just shown the results. At least from the paper I could see from formula 5 how the model is implemented for calls. All I am asking if the same equation applies to puts but with different payoff function(max(K-S,0)). At first I thought this is the case, but then I started thinking on the fundamental difference between calls and puts on a stock with dividends and if I need to take this into account in my numerical algorithm.

Alan
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### Re: American call on a dividend paying stock

The modification of your algorithm for calls to make it work for puts is pretty simple.  (I  assume $r > 0$ and no other carry costs besides the discrete dividends).

For calls, it suffices to check for early exercise at $t = t_D-$, instantaneously prior to the ex-date. For puts, you need to check at every date except $t = t_D-$. Both option types have the same jump condition in the value function as you pass through the ex-date. If you do those things and reproduce the first two rows of figures, you're probably OK. (The book has a good numerical example for a detailed check).

Kamil90
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Joined: February 15th, 2012, 2:02 pm

### Re: American call on a dividend paying stock

The modification of your algorithm for calls to make it work for puts is pretty simple.  (I  assume $r > 0$ and no other carry costs besides the discrete dividends).

For calls, it suffices to check for early exercise at $t = t_D-$, instantaneously prior to the ex-date. For puts, you need to check at every date except $t = t_D-$. Both option types have the same jump condition in the value function as you pass through the ex-date. If you do those things and reproduce the first two rows of figures, you're probably OK. (The book has a good numerical example for a detailed check).
Why would I remove the date when the dividend is paid for a put? Can't I exercise immediately after the dividend is paid at the same point in time?

Alan
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### Re: American call on a dividend paying stock

I didn't say "remove the date". At amazon, search inside the book for "Tree approaches" and look at the last result, which will show Table 9.1. If you reproduce that 24.468 value and reproduce the charts I posted here, your code is probably ok and the answer to your last question is 'yes'.