Classical equity models don’t handle very well dividends, in particular constant ones. We present in this post an approach to model equities with (deterministic) dividends.
In a first time, we will show the necessary form of an equity in order there is no arbitrage with its forward. Then, we will use this form to extend the classical models to handle constant dividends. To finish, we will see how it work on an important example: the local volatility model.
The approach developed here is mainly based on: Hans Buehler, Volatility and Dividends.
Let $S$ be our asset. We assume that $S$ pay continuously the dividend yield $q$.
Let
be the dividend schedule (the ex-dividend dates) and $c_i$ (resp. $p_i$) the absolute (resp. proportional) dividend payed at $t_i$, ie the dividend payed at $t_i$ is
\[ c_i + p_iS(t_i^-) \]where $S(t_i^-)$ is the value of $S$ just before the ex-dividend date (for more convenience, we assume that the ex-dividend dates are equal to the payment dates).
We introduce the following notations:
The detention of $S$ between $t$ and $T$ is equivalent to:
The value of this contract is obviously $S(t)$. But it is also the value of the equivalent contract, because of absence of arbitrage.
Thus, $\forall 0 \leq t \leq T$:
\[ S(t) = Term_1(t, T) + Term_2(t, T) + Term_3(t, T) \]where
$$ \begin{align*} Term_1(t, T) &= P(t, T)F(t, T)\\ Term_2(t, T) &= \sum_{t_i \in (t; T]} \left[c_i + p_iF(t, t_i^-)\right]P(t, t_i)\\ Term_3(t, T) &= \int_t^T P(t, s)F(t, s)q(s)ds \end{align*} $$Let $f_t(T) = P(t, T)F(t, T)$. We have:
$$ \begin{align*} \partial_T f_t(T) + f_t(T) q(T) &= 0, \hspace{10pt} \forall i \in \mathbb N, \forall T \in [t_{i-1}; t_i)\\ f_t(t) & = S(t)\\ f_t(t_i) - f_t(t_i^-) & = - \left[c_i + p_iF(t, t_i^-)\right]P(t, t_i) \end{align*} $$This ODE can be solved piecewise on $[t_{i-1}; t_i)$, what leads to:
$$ \begin{equation} \label{eq:s} \bar P(t, T)F(t, T) = S(t) - D(t, T) \end{equation} $$where
$$ \begin{align*} \bar P(t, T) &= \frac{P(t, T)}{Q(t, T)}\\ Q(t, T) &= e^{-\int_t^Tq(s)ds}\prod_{t_i \in (t; T]} (1-p_i)\\ D(t, T) &= \sum_{t_i \in (t; T]}c_i \bar P(t, t_i) \end{align*} $$We also introduce the notation $D(t) = D(t, t_n) = D(t, +\infty)$.
Now, we will see the consequences on the asset price. For the following, we assume the positiveness of all maturing forwards.
\eqref{eq:s} at $T = t_n$ implies that:
$$ \begin{equation} \label{eq:sgreaterthan} \forall t, S(t) \geq D(t) \end{equation} $$As $P(\cdot, T)F(\cdot, T)$ is a tradable asset, $\frac{P(\cdot, T)F(\cdot, T)}{C}$ must be a $\mathbb Q$ - martingale. Then there exists a $\mathbb Q$ martingale $X_T$, $X_T(0) = 1$, such that for $0 \leq t \leq T$,
\[ P(t, T)F(t, T) = P(0, T) F(0, T) C(t) X_T(t) \]We can rewrite \eqref{eq:s}:
\[ P(t, T)F(t, T) = Q(t, T) \left[ S(t) - D(t, T) \right] \]Thus:
\[ S(t) = \frac {P(0, T)F(0, T)}{Q(t, T)}C(t)X_T(t) + D(t, T) \]Setting $t = 0$ implies:
\[ Q(0, T) = \frac{P(0, T)F(0, T)}{S(0) - D(0, T)} \] \[ \Rightarrow Q(t, T) = \frac{Q(0, T)}{Q(0, t)} = \frac{1}{Q(0, t)}\frac{P(0, T)F(0, T)}{S(0) - D(0, T)} \]And then:
\[ S(t) = Q(0, t) \left[ S(0) - D(0, T) \right] C(t)X_T(t) + D(t, T) \]Set $T = t_n$: $\forall t \leq t_n$,
$$ \begin{equation} \label{eq:s1} S(t) = Q(0, t)\left[S(0) - D(0) \right]C(t) X_{t_n}(t) + D(t) \end{equation} $$For $t \gt t_n$, there is no more constant dividends, we can use the classical approach: there exist a $\mathbb Q$-martingale $Y$, $Y(t_n) = 1$,
$$ \begin{align*} S(t) &= Q(t_n, t) \frac{C(t)}{C(t_n)} S(t_n)Y(t)\\ &=Q(0, t) \left[ S(0) - D(0, T) \right] X_{t_n}(t_n)Y(t) + D(t) \end{align*} $$The calculus has been shorted for readability. The result comes replacing $S(t_n)$ with its expression in \eqref{eq:s1} and using $t \geq t_n \Rightarrow D(t) = 0$.
We define the $\mathbb Q$-martingale $X$ with
\[ \left\{ \begin{array} {c l} \forall t \leq t_n, &X(t) = X_{t_n}(t)\\ \forall t \gt t_n, &X(t) = X_{t_n}(t_n)Y(t) \end{array} \right. \]What leads to the main result:
\[ \forall t, S(t) = Q(0, t)\left[S(0) - D(0) \right]C(t) X(t) + D(t) \]We note that on a deterministic rate model, we have:
\[ S(t) = \left[F(0, t) - D(t)\right] X(t) + D(t) \]We can show using \eqref{eq:sgreaterthan} that $X$ must be positive.
Although the preceding applies to stochastic rates, we restrict this section to deterministic rate case. Basically, instead of modeling $S$, we now apply the model to $X$. An advantage that come directly is that we don’t care of interest rate in $X$, what simplify models. For instance, in the Heston case:
$$ \begin{align*} dX &= \sqrt VXdW^X\\ dV &= \kappa(\theta - V)dt + \xi \sqrt VdW^V \end{align*} $$To calibrate the model on $X$, we need to have call prices on $X$, that can be deduced from the call prices on $S$.
Let $C^S(T, K)$ (resp. $C^X(T, K)$) be the price of the call on $S$ (resp. $X$) maturing at $T$, with strike $K$.
$$ \begin{align*} C^S(T, K) &= P(0, T)\mathbb E^{\mathbb Q} \left[ (S(T) - K)^+ \right]\\ &= P(0, T)(F(0, T) - D(T)) \mathbb E^{\mathbb Q} \left[ \left(X(T) - \frac{K - D(T)}{F(0, T) - D(T)} \right)^+ \right]\\ \end{align*} $$The equalities hold because we assumed deterministic rates.
Thus,
\[ C^S(T, K) = P(0, T)(F(0, T) - D(T)) C^X\left(T, \frac{K - D(T)}{F(0, T) - D(T)}\right) \] $$ \begin{equation} \label{eq:cx} C^X(T, K) = \frac{C^S(T, (F(0, T) - D(T))K + D(T))}{P(0, T)(F(0, T) - D(T))} \end{equation} $$The local volatility model will then be:
$$ \begin{align*} dX(t) &= \sigma(t, X(t))X(t)dW(t)\\ X(0)&=1\\ S(t) &= (F(0, t) - D(t))X(t) + D(t) \end{align*} $$$\sigma$ can be expressed using the Dupire’s formula:
\[ \sigma(t, x)^2 = \frac{2\partial_t C^X(t, x)}{x^2\partial^2_x C^X(t, x)} \]Where $C^X$ can be computed using \eqref{eq:cx}.
We shown with few assumptions and using no arbitrage theory that taking dividends into account is not that difficult, and that the value of the asset must be:
\[ S(t) = Q(0, t)\left[S(0) - D(0) \right]C(t) X(t) + D(t) \]or in the case of deterministic rates:
\[ S(t) = \left[F(0, t) - D(t)\right] X(t) + D(t) \]We also shown, in the deterministic case, that the adaptation of the modeling and calibration is easy, using the relation
\[ C^S(T, K) = P(0, T)(F(0, T) - D(T)) C^X\left(T, \frac{K - D(T)}{F(0, T) - D(T)}\right) \]This methodology has been chosen to be implemented in LexiFi Apropos.