Option price on asset with continuous dividend D

Question

I know that the European call option price fulfills the following equation: $$ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - rV = 0, $$ where $V(S_t,t) = C(S_t, t) = S_t N(d_1) -K e^{-r(T-t)} N(d_2) $ for:

• $ d_1 =\frac{\log(S/K) + (r+\sigma^2/2)T}{\sigma \sqrt{T}} $
• $d_2 = d_1 - \sigma \sqrt{T}$

Moreover I know the formula for the European call option price which pays dividend D continuously: $$ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-D) S \frac{\partial V}{\partial S} - rV = 0, $$ Now my goal is to find the formula on $C(S_t, t)$ which fulfill the above equation.

I know that the result is: $V(S_t,t) = C(S_t, t) = S_t e^{-D(T-t)} N(d_1) -K e^{-r(T-t)} N(d_2) $ for:

• $ d_1 =\frac{\log(S/K) + (r-D+\sigma^2/2)T}{\sigma \sqrt{T}} $
• $d_2 = d_1 - \sigma \sqrt{T}$

But I do not know how to derive it.

Answer 1

$$\frac{\partial V}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}+(r-D)S\frac{\partial V}{\partial S} - rV=0 \tag{1}$$

Let $$W(S_t, t) = V(S_t, t) e^{D(T-t)}, \tag{2}$$ that is

$$V=e^{-D(T-t)}W. \tag{3}$$

Compute the partial derivatives:

$$\frac{\partial V}{\partial t}=De^{-D(T-t)}W+e^{-D(T-t)}\frac{\partial W}{\partial t} \tag{4}$$

$$\frac{\partial V}{\partial S}=e^{-D(T-t)}\frac{\partial W}{\partial S} \tag{5}$$

$$\frac{\partial^2 V}{\partial S^2}=e^{-D(T-t)}\frac{\partial^2 W}{\partial S^2} \tag{6}$$

Now substitute $(3), (4), (5),$ and $ (6)$ into $(1)$:

$$De^{-D(T-t)}W+e^{-D(T-t)}\frac{\partial W}{\partial t} + \frac12 \sigma^2 S^2 e^{-D(T-t)}\frac{\partial^2 W}{\partial S^2} +(r-D)Se^{-D(T-t)}\frac{\partial W}{\partial S}- re^{-D(T-t)}W=0 \tag{7}$$

Multiply everything by $e^{D(T-t)}$:

$$DW+\frac{\partial W}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 W}{\partial S^2} +(r-D)S\frac{\partial W}{\partial S}- rW=0 \tag{8}$$

Collecting terms with $W$.

$$\frac{\partial W}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 W}{\partial S^2} +(r-D)S\frac{\partial W}{\partial S}- (r-D)W=0 \tag{9}$$

$(9)$ is just Black-Scholes equation with interest rate $r-D$, of which we know the solution.

That is $$W(S_t,t)=S_tN(d_1)-Ke^{-(r-D)(T-t)}N(d_2) \tag{10}$$

• $d_1=\frac{\log(S/K)+(r-D+\sigma^2/2)T}{\sigma \sqrt{T}}$
• $d_2 = d_1 - \sigma \sqrt{T}$

To recover $V$, multiply by $e^{-D(T-t)}$,

$$V(S_t,t)=S_te^{-D(T-t)}N(d_1)-Ke^{-r(T-t)}N(d_2)$$