Risk-Neutral Drift with Discrete Dividend Payments

# Black-Scholes model with discrete dividend payments


# Black-Scholes model with discrete dividend payments












Consider the Black-Scholes model with discrete dividends in the interval $[0,T]$. This means that there's a sequence of dates such that, $$0 < t_1 < \dots < t_k < \dots < t_n < T $$ where the dividend that is paid out to the owner of the risky asset is given by, $$d_k S_{t^-_k} $$ where, $$ S_{t^-_k} = \lim_{t \to t^-_k} S_t $$ is the price of the risky asset immediately before the dividend is paid out. After the dividend payment, the risky asset price is given by, $$S_{t_k} = (1-d_k)S_{t^-_k} $$ The sequences $(t_k)_{1 \leq k \leq n}$ and $(d_k)_{1 \leq k \leq n}$ are known and $0 < d_k < 1$ , for all ${1 \leq k \leq n}$.

In this model, it is assumed that the price of the risky asset, between two dividend payment dates, is given by the usual Black-Scholes model:

$$dS_t = \mu S_t dt + \sigma S_t dW_t, \space t_k \leq t < t_{k+1} $$

for $k = 0,\dots,n$, where, by notation convenience, we introduce the values $t_0 = 0$, $t_{n+1} = T$, $d_0 = d_{n+1} = 0$, and $\mu$,$\sigma > 0$ are constant.

The price of the risk free asset, $(B_t)_{t \geq 0}$, is given by, $$dB_t = rB_tdt$$ where $r>0$ is also a constant.

The objective of this problem is to prove that there exists a Brownian motion, $(\tilde W_t)$, defined in the probability space $(\Omega,\tilde P, \mathcal{F})$, such that $$dS_t = r S_t dt + \sigma S_t d \tilde W_t, \space t_k \leq t < t_{k+1} $$

My main approach to this problem was using Girsanov's theorem.

From the geometric brownian motion, $$dS_t = \mu S_t dt + \sigma S_t dW_t, \space t_k \leq t < t_{k+1} $$ it is known that $$S_t = S_0 e^{\big(\mu-\frac{\sigma^2}{2}\big)t - \sigma W_t} $$ I also know that, $dB_t = rB_t dt$, has the following solution: $B_t = e^{rt} $. By multiplying $B_t$ on both sides of the last equation, $$B_t S_t = S_0 e^{\big(r + \mu-\frac{\sigma^2}{2}\big)t - \sigma W_t} $$ which has the following differential, $$ d(B_t S_t) = (r + \mu)B_t S_t dt + \sigma B_t S_t dW_t $$ which we can rewrite as, $$ d(B_t S_t) = \sigma B_t S_t \Bigg[dW_t + \frac{r + \mu}{\sigma}dt\Bigg] $$ Now using Girsanov's theorem, $$ dW_t = -\Bigg(\frac{r + \mu}{\sigma}\Bigg)dt + d\tilde W_t $$ Making the substitution in the starting geometric brownian motion, $$ dS_t = \mu S_t dt + \sigma S_t \Bigg[-\Bigg(\frac{r + \mu}{\sigma}\Bigg)dt + d\tilde W_t \Bigg] = \mu S_t dt - (r + \mu)S_t dt + \sigma S_t d\tilde W_t = -rS_tdt + \sigma S_t d\tilde W_t$$

As you can see, my calculations are not giving out the final solution desired.

My question is, did I miss a crucial step on my draft resolution that is stopping me from obtaining the desired equation? Am I not seeing something?

Furthermore, is there an easier way to solve this problem?

Thank you in advance.