Consider an American call option with ex-dividend dates in 3 months and 5 months, and has an expiration date of 6 months. The dividend on each ex-dividend date is expected to payout $0.70. Additional information is presented below. Find the value of the American call option. :\begin{align} S_0 &= \$40 \\ X &= \$40 \\ \sigma &= 30\% \; p.a. \\ r &= 10\% \; p.a. \\ T &= 6 \; months = .5 \; years\\ D &= \$0.70 \\ \end{align} First, we need to calculate based on the two methods provided above in the methods section. Here we will calculate both of the parts: :
(1) This is the first method calculation, which states: :A European call with the same maturity as the American call being valued, but with the stock price reduced by the present value of the dividend. : :\begin{align} PV &= D_1 e^{-(r)(\frac{\Delta t_1}{m})} + D_2 e^{-(r)(\frac{\Delta t_2}{m})} \end{align} : :where : ::PV is the net present value of the dividends at the ex-dividend dates (we use the ex-dividend dates because on this date the stock price declines by the amount of the dividend) ::D_{1,2} are the dividends on the ex-dividend dates ::r is the risk-free rate of the market, which we will assume to be constant for this example ::\Delta t_{1,2} amount of time until the ex-dividend date ::m a division factor to bring the Δt to a full year. (example \Delta t = 2 months, m = 12 months, therefore \frac{\Delta t}{m} = 2/12 = .166667) ::e is the
exponential function. :Applying this formula to the question: :\begin{align} 0.7e^{-(.1)(\frac{3}{12})} + 0.7e^{-(.1)(\frac{5}{12})} = 1.3541 \end{align} : :The option price can therefore be calculated using the
Black-Scholes-Merton model where will discount the dividends from S_0 which I will denote by S_0 ' for the new value: :S_0 ' = 40 - 1.3541 = 38.6459 :The rest of the variables remain the same. Now we need to calculate d1 and d2 using these formula's :\begin{align} C &= S_0 N(d_1) - Xe^{-r(T)} N(d_2) \\ d_1 &= \frac{\left[\ln\left(\frac{S_0}{X}\right) + \left(r + \frac{\sigma^2}{2}\right)(T)\right]}{\sigma\sqrt{T}} \\ d_2 &= d_1 - \sigma\sqrt{T} \end{align} :where, ::N(\cdot) is the
cumulative distribution function of the
standard normal distribution ::T is the time to maturity ::S_0 is the current price of the underlying asset ::X is the strike price ::r is the
risk free rate (annual rate, expressed in terms of
continuous compounding) ::\sigma is the
volatility of returns of the underlying asset :Inputting the values we get: :\begin{align} d_1 &= \frac{\left[\ln\left(\frac{38.6459}{40}\right) + \left(0.1 + \frac{0.3^2}{2}\right)(0.5)\right]}{0.3\sqrt{0.5}} = 0.1794 \\ d_2 &= 0.1794 - 0.3\sqrt{0.5} = -0.0327 \\ N(d_1) &= 0.5712 \\ N(d_2) &= 0.4870 \\ C &= 38.6459(0.5712) - 40e^{-0.1(0.5)} (0.4870) = 3.5446 \approx \$ 3.54 \end{align}
(2) This is the second method calculation, which states: :A European call that expires on the day before the dividend is to be paid. :This method begins just like the previous method except that this options maturity is set to the last maturity before the last dividend (meaning the second dividend in the fifth month): :\begin{align} PV &= D_1 e^{-(r)(\frac{\Delta t_1}{m})} \end{align} :For the most part, the variables remain same except for the time to maturity, which equals: :\begin{align} T &= 5 \; months = .4167 \; years \end{align} :\begin{align} PV &= 0.7 e^{-(0.1)(\frac{3}{12})} = 0.6827 \\ S_0 ' &= 40 - 0.6827 = 39.3173 \\ d_1 &= \frac{\left[\ln\left(\frac{39.3173}{40}\right) + \left(0.1 + \frac{0.3^2}{2}\right)(0.4167)\right]}{0.3\sqrt{0.4167}} = 0.2231 \\ d_2 &= 0.2231 - 0.3\sqrt{0.4167} = 0.0294 \\ N(d_1) &= 0.5883 \\ N(d_2) &= 0.5117 \\ C &= 39.3173(0.5883) - 40e^{-0.1(0.4167)} (0.5117) = 3.4997 \approx \$ 3.50 \end{align} Recalling method (1) price of \$ 3.54 > \$ 3.50 from method (2), we see that the price of the American call option, as per Fisher Black's approximation, is the greater of the two methods, therefore, the price of the option = \$ 3.54 . ==References==