0.1 Intuition
0.1.1 Risk-Neutral Pricing Approach
Continuous-time dividend yield is always denotes as \(\delta\), meaning that in an infinitesimal time interval \([t,t+dt]\), one gets \(\delta S_{t}dt\) dividend payment from the stock and so stock value decreases as dividends are paid. This is equivalent to stating that the number of shares investors hold increases at rate \(\delta\) continuous in time - i.e. if one is holding 1 share of stock at time 0 and the stock has dividend yield \(\delta\), then one is actually holding \(e^{\delta t}\) shares of stock at time \(t\).
Clearly, the dividend payment is discounting the price of the stock as time goes forward and so has the opposite effect compared to that of the interest rate. In the risk-neutral setting we would expect to see \(\{e^{-(r-\delta)t}S_{t}\}\) as a martingale, so the dynamics under risk-neutral measure \(\mathbb{Q}\) is \[ dS_{t}=(r-\delta)S_{t}dt+\sigma S_{t}dW_{t}^\mathbb{Q} \] just replacing \(r\) with \(r-\delta\) in the dynamics. #### PDE Approach
The Black-Scholes PDE (BSPDE) is given by \[ \partial _{t}u+rx\partial _{x}u+ \frac{1}{2}\sigma^2x^2\partial_{x x}u-ru=0. \] We do not simply replace \(r\) with \(r-\delta\) as above and we must be careful to understand the meaning of the \(r\) values.
- The term \(ru\) comes from the fact that \(\{e^{-rt}V_{t}\}\) is a martingale under \(\mathbb{Q}\) which still holds in the dividend paying situation hence this doesn’t change.
- The term \(rx\delta_{x}u\) comes from the Ito formula expansion of \(dV_{t}\) which is directly related to \(dS_{t}\) under \(\mathbb{Q}\) hence this does change.
The result is the Black-Scholes Equation with Dividends \[ \partial_{t}u+(r-\delta)x\partial_{x}u+ \frac{1}{2}\sigma^2x^2\partial_{x x}u-ru=0. \]