Question:
15.5. For a stock, you are given:
(i) The prices of the prepaid forward on the stock follow a lognormal model.
(ii) The stock’s current price is 100.
(iii) The stock’s continuously compounded annual rate of return is α
(iv) The volatility of a prepaid forward on the stock is 0.2.
(v) The stock pays quarterly dividends of 1. The next dividend will be paid 3 months from now.
(vi) The continuously compounded risk-free interest rate is 0.04.
A 1-year European call option on the stock has strike price 100. The payoff on the option will be based on
the ex-dividend price.
Simulate the stock’s price using the following standard normal random numbers in the order given to simulate the change in price for each quarter.
−1.2 1.8 0 −0.3
Calculate the simulated present value of the option in this run.
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My problem is with "The prices of the prepaid forward on the stock follow a lognormal model".
I think since the prepaid foward price is actually ex-dividend stock price for discrete dividends paying stocks. Therefore, "prepaid forward follows lognormal model" can be translated into the change of "the ex-dividend" stock price follow the lognormal model.
Assume the ex-dividend stock price is St at time t and St+1 at time t+1. My understanding is St+1 / St ~ lognormal ( m, v^2) .
Therefore, this monte carlo problem should be solved as St+1/St = e^n
(where n = m + v * z, and z is the std. normal random number given)
The solution of this problem also set St and St+1 as ex-dividend prices. However, on the solution it shows:
St+1 + Dividend / St = e^n ( which is cum-dividend stock price / ex-dividend stock price = e^n ).
I am always confused with those discrete dividends problems. Could somebody explain why the solution is doing like that ?
and what is the right way to understand the " prepaid foward price follows lognormal ?
Thanks
15.5. For a stock, you are given:
(i) The prices of the prepaid forward on the stock follow a lognormal model.
(ii) The stock’s current price is 100.
(iii) The stock’s continuously compounded annual rate of return is α
(iv) The volatility of a prepaid forward on the stock is 0.2.
(v) The stock pays quarterly dividends of 1. The next dividend will be paid 3 months from now.
(vi) The continuously compounded risk-free interest rate is 0.04.
A 1-year European call option on the stock has strike price 100. The payoff on the option will be based on
the ex-dividend price.
Simulate the stock’s price using the following standard normal random numbers in the order given to simulate the change in price for each quarter.
−1.2 1.8 0 −0.3
Calculate the simulated present value of the option in this run.
-------------------------------------------------------------------------
My problem is with "The prices of the prepaid forward on the stock follow a lognormal model".
I think since the prepaid foward price is actually ex-dividend stock price for discrete dividends paying stocks. Therefore, "prepaid forward follows lognormal model" can be translated into the change of "the ex-dividend" stock price follow the lognormal model.
Assume the ex-dividend stock price is St at time t and St+1 at time t+1. My understanding is St+1 / St ~ lognormal ( m, v^2) .
Therefore, this monte carlo problem should be solved as St+1/St = e^n
(where n = m + v * z, and z is the std. normal random number given)
The solution of this problem also set St and St+1 as ex-dividend prices. However, on the solution it shows:
St+1 + Dividend / St = e^n ( which is cum-dividend stock price / ex-dividend stock price = e^n ).
I am always confused with those discrete dividends problems. Could somebody explain why the solution is doing like that ?
and what is the right way to understand the " prepaid foward price follows lognormal ?
Thanks
Question on prepaid forward price - ASM lesson 15 Monte Carlo Exercise 15.5
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