# What is the analogue used by Hull to price European calls with known cash dividends?

From The Book by Hull: And Hull's comment:

1. This rule is analogous to the one developed in Section 14.12 for valuing a European option on a stock paying known cash dividends. (In that case we concluded that it is correct to reduce the stock price by the present value of the dividends; in this case we discount the stock price at the dividend yield rate.)

What 14.12 says is if there is a known future cash dividend $(D,\tau)$ then the European call price at the beginning is the value obtained by replacing $S_0$ with $S_0-D_0$ in the BS formula where $D_0=De^{-r\tau}$ represents the present value of the cash dividend at time $0$.

However, I'm quite confused how Hull has developed this analogue.

My attempt to analogise his argument for continuous dividend yield: in the case of a known cash dividend $(D,\tau)$ suppose our stock grows from $S_0$ at time $0$ to $S_T$ at time $T$; then in the absence of this dividend I think the stock would grow to $S_T + De^{r(T-\tau)}$ (a bit dubious); the ratio of growth is thus $(S_T + De^{r(T-\tau)})/S_0$. Now, without any dividend, what initial stock price would grow to $S_T$ following this ratio? It must be $S_T/((S_T + De^{r(T-\tau)})/S_0)$. So it means that replacing $S_0$ with this amount in the BS formula gives the price in case of a known cash dividend, which is completely ridiculous, as the correct formula Hull gives replaces $S_0$ with $S_0-De^{-r\tau}$ instead.

So what is wrong with my reasoning and what should be the correct way to prove this analogue?

• Is the question why the SDE corresponding to case 1 $$dS_t/S_t = (r-q)dt + \sigma dW_t,\,\,\, S(0)=S_0$$ provides the same solution for $S_T$ as the SDE for case 2 $$dS_t/S_t = r dt + \sigma dW_t,\,\,\, S(0) = S_0 e^{-qT}$$ (note that they do not agree on the full dynamics of $(S_t)_{t \geq 0}$ but this is not important for European options, since they are not path dependent)? – Quantuple Apr 28 '17 at 9:22
• And indeed the equation for case 2 can be used with $S(0) = S_0 - De^{-r\tau}$ for a single expected div $(D, \tau)$, this is known as the escrowed dividend model. – Quantuple Apr 28 '17 at 9:25
• @Quantuple no, I understand why the two SDEs are consistent. My question is about the discrete dividend, or the escrowed dividend model, how can one in this case make an analogue to the continuous dividend yield analysis? – Vim Apr 28 '17 at 12:21

Remember that Black-Scholes formula applies to lognormally distributed (under $\Bbb{Q}$) terminal asset prices $S_T$. It is convenient to write this assumption $$S_T \underset{\Bbb{Q}}{\sim} \ln \mathcal{N}\left( \ln(F(0,T))-\frac{1}{2}\sigma^2 T, \sigma^2 T \right) \tag{A}$$ since it shows that the forward price is the risk-neutral expectation of the future asset price $$\Bbb{E}_0^\Bbb{Q} [ S_T ] = F(0,T)$$

When $(A)$ holds, the price of a European call of strike $K$ and maturity $T$ reads (Black-Scholes formula) $$C(K,T) = DF(0,T) \left( F(0,T) N(d_+) - K N(d_-) \right)$$ $$d_{\pm} = \frac{\ln\left( \frac{F(0,T)}{K}\right) \pm \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}}$$

Now think of how dividends, impact the forward price in [MODEL 1] (dividend yield model) and [MODEL 2] (escrowed model).

• [MODEL 1]: Solving the corresponding SDE $$dS_t/S_t = (r-q)dt + \sigma dW_t^\Bbb{Q},\,\,\, S(0)=S_0$$ yields $S_T = S_0 e^{(r-q)T} \mathcal{E}\left[\sigma W_T^\Bbb{Q}\right]$ (lognormal), hence a forward price $$F(0,T) = S_0 e^{(r-q)T} = \underbrace{S_0 e^{-qT}}_{S_0^*} e^{rT} \tag{B}$$

• [MODEL 2]: Solving the corresponding SDE $$dS_t/S_t = r dt + \sigma dW_t^\Bbb{Q},\,\, S(0)=S_0-De^{-r\tau}$$ yields $S_T = S(0)e^{(r)T} \mathcal{E}\left[\sigma W_T^\Bbb{Q}\right]$ (lognormal), hence a forward price $$F(0,T) = \underbrace{\left( S_0 -De^{-r\tau} \right)}_{S_0^*} e^{rT} \tag{C}$$

This shows that, under both of these models, one can use BS formula $(A)$ by provided one replaces the forward price by what it is under each respective modelling assumption, which is mathematically equivalent (looking at the BS formula only) to using the spot value $S_0^*$ (see $(B)$ and $(C)$) instead of $S_0$.

• Thanks for this detailed answer. But my real question is, why in the case of escrowed model, are we entitled to replace $S_0$ by $S_0-D_0$. (In fact, this corresponds to the second case in another question of mine which you just addressed. And your last comment there actually answers my question here: this corresponds to the view that the stock price is the current capital plus pv of future dividend value.) – Vim Apr 28 '17 at 13:29
• Yes but, this is only a "model" i.e. a view. So it is really a "definition" or "convention" nothing more. Glad it helped. – Quantuple Apr 28 '17 at 13:34