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BOPM FOR PUTS AND THE DIVIDEND-ADJUSTED BOPM

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BOPM FOR PUTSAND THE DIVIDEND-ADJUSTED BOPM

Chapter 6

Binomial Stock Movement

- Assume there is a European put option on the stock that expires at the end of the period.
- Example: X = $100.

Binomial Put Movement

- The replicating portfolio consist of buying Ho shares of the stock at So and investing Io dollars.

- The RP is formed by solving for the Ho and Bo values (Ho* and Io*) which make the two possible values of the replicating portfolio equal to the two possible values of the put (Pu and Pd).
- Mathematically, this requires solving simultaneously for the Ho and Io which satisfy the following two equations.

- Equations:

- Equations:

- Equations:

- Law of One Price:

- The equilibrium price of the put is governed by arbitrage.
- If the market price of the put is above (below) the equilibrium price, then an arbitrage can be exploited by going short (long) in the put and long (short) in the replicating portfolio. For an example, see JG: 185-186.

- By substituting the expressions for Ho* and Io* into the equation for Po*, the equation for the equilibrium put price can be alternatively expressed as:

- The value obtained using the binomial put model is consistent with put-call parity.

- From Previous example for call:
- u = 1.0488, d = .9747,
- So = $100, n = 2,
- Rf = .025,

- To price a European put with X = 100, use the same recursive procedure we used for a call option.

- The value obtained using the binomial put model is consistent with put-call parity.

- Note: At Sd = 97.47, Pd = 1.56. If the put were American and priced at this value, then the put holder would find it advantageous to exercise the put: IV = 100-97.47 = 2.53.
- The BOPM for a put can be adjusted to value American puts by constraining the price of the put at each node to be the maximum of either its BOPM value or its IV:

- The arbitrage strategies underlying the multi-period put model are similar to the multiple-period call model, requiring possible readjustments in subsequent periods.
- For a discussion of multiple-period arbitrage strategies, see JG, p 191.

- If a dividend is paid and the ex-dividend date occurs at the end of any of the periods, then the price of the stock will fall. The price decrease will cause the call price to fall and the put price to increase.
- The dividend may also make the early exercise of a call profitable, making an American call more valuable than a European.

- The BOPM can be adjusted for dividends by using a dividend-adjusted stock price (stock price just before ex-dividend date minus dividend) on the ex-dividend dates in calculating the option prices.
- The BOPM can be adapted to price an American call by constraining the price at each node to be the maximum of the binomial value or the IV.

- Assume there is an ex-dividend date at the end of the period, with the value of the dividend being D.
- Let uSo and dSo be the stock prices just before the ex-dividend date.
- As shown in the Figure, the dividend does not affect the form of the replicating portfolio, but it does lower the two possible call prices which changes H, B, and Co.

- Equations:

- Assuming D = 1

- Let uSo, dSo, uuSo, udSo, and ddSo be the stock prices just before the ex-dividend date.
- Assume the dividend at end of Period 2 is $1 and the dividend at the end of Period 1 is $1.
- Assume: n = 2, u = 1.0487, d = .9747, Rf =.025.
- As shown in the Figure, H and B values reflect the ex-dividend call values.
- Note: In the formulas for H and B, the stock prices just before the ex-dividend date are used.

- For American call options on dividend-paying stocks, early exercise can be incorporated into the BOPM by constraining each possible call value to be the maximum of either its European value as determine by the BOPM or the IV just prior to the ex-dividend date.
- In our previous case, the $1 dividend in Period 1 would not make early exercise profitable if the call were American.
- As shown in the next Figure, if the dividend in Period 2 were $2 instead of $1, then there is an early exercise advantage at the top node in Period 1. The price of an American call in this case is 3.38, compared to a European value of $3.02.

- The dividend adjustments required for European puts are similar to those for European calls.
- When applicable, the dividend is subtracted from the stock price to determine the ex-dividend put price. This price is then used to determine H, I, and Po.
- The dividend adjustment for an American put does differ from the adjustment for the American call. For the put, the price at each node is the maximum of its European value as determined by the BOPM or its IV on the ex-dividend date (see JG:210):

- The value obtained using the binomial model with dividends is consistent with put-call parity.