配当を含める株式オプションの価格計算
ブラック・スコールズの計算式を使用して株式オプションの値段を算出します。でも配当落ちがオプションの契約期間内にあれば計算がちょっと変わります。
The problems I wished to address were:
1. For discrete dividends, can the option price be computed via the Black-Scholes formula (i.e. without using binomial trees or some other approximation)?
2. For continuous dividends, what is the intuition behind the formula?
For question 1, the intuition is based on the forward price of a stock with a single ex-dividend date between the settlement of the option and its expiry.
This forward price is based on the value of a portfolio containing the stock and short a risk-free bond maturing to D (dividend amount) at the ex-dividend date. The value of this portfolio at the settlement time (t) is:
S(t) - De^*1, where t1 is the ex-dividend date. After t1, this portfolio is worth S(t), where t>t1. This is because we have repaid the bond amount and are left with only the stock (whose price has been reduced by D). The value of the portfolio immediately before the dividend payment is exactly the same as the value after the dividend payment, and the value of the portfolio at T(option expiry) is the same as the value of the stock. The forward price is thus:
(S(t)-De^*2. We can generalize the above logic to many dividend payments, so that the forward price is always the current price of the stock minus the present value of all dividend payments moved forward.
Based on the above logic, we can compute the Black-Scholes price of the option, replacing the current price of the stock S(t) with the current price minus the present value of all dividends:
S* = S(t) - SUM(PV(Di))
This is because, as we can compute the forward price by moving the above amount forward at the risk free rate, we know that a dividend-paying stock priced at S behaves like a non-dividend paying stock priced at S-SUM(PV(Di))
However, we also need to adjust volatility:
SIGMA* = (S/S*)SIGMA, to account for the fact that the initial price is reduced due to the dividend.
The formula for calls is:
C = S*N(d1)-e^(-r(T-t))N(d2)K
d1 = (log(S*/K)+(r+(SIGMA*)^2/2)(T-t))/(SIGMA*(sqrt(T-t)))
d2 = d1-SIGMA*(sqrt(T-t))
For question 2, it seems that one simply replaces S(t) with S(t)*e^(-q(T-t)). Here q is the continuously compounded dividend rate. I am unsure why the volatility SIGMA is not adjusted, but I can only suspect that the SIGMA* value resulting from (S(t)/(S(t)e^(-q(T-t))) is close enough to 1 that no change is needed.
