Self-Financing Wealth Dynamics with Discrete Dividends

# Wealth process in the Black-Scholes model with discrete dividends


# Wealth process in the Black-Scholes model with discrete dividends












Good evening,

The following problem is the sequel of a previous post I made here a few days ago.

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.

I was able to derive the following formula for the price of an european call in this model with discrete payments, whose payoff is: $U_T = (S_T - K)^{+}$ $$U_t = \Bigg[ \prod_{k = 1}^{n} (1-d_k) \Bigg]S_0e^{rt}N(d_1) - Ke^{-r(T-t)}N(d_2) $$ where $$ d_1 = \frac{\ln\Big(\frac{S_0 \prod_{k = 1}^{n} (1-d_k)}{K}\Big) + \Big(r + \frac{\sigma^2}{2}\Big)T }{\sigma \sqrt{T}} $$ $$ d_2 = \frac{\ln\Big(\frac{S_0 \prod_{k = 1}^{n} (1-d_k)}{K}\Big) + \Big(r - \frac{\sigma^2}{2}\Big)T }{\sigma \sqrt{T}} $$

Having the valuation formula for this european call, I was also able to derive the wealth process between two dividend payment dates, for a short position in the replica portfolio for the previous call. Therefore, for some $t_k \leq t < t_{k+1}$, we have the following portfolio, $$\phi_t = \Bigg(\frac{X_t - \Delta_t S_t}{B_t}, \Delta_t\Bigg) $$ where $(\Delta_t)_{0 \leq t \leq T}$ is the option delta, and it's adapted to the standard brownian filtration, $(W_t)_{t \geq 0}$. Now deriving the wealth process, given that the portfolio is self-financed, $$X_t = b_tB_t + s_tS_t \implies dX_t = b_tdB_t + s_tdS_t $$ Given that $b_t = \frac{X_t - \Delta_t S_t}{B_t}$, $s_t = \Delta_t$, and that $dB_t = rB_tdt$, we have that, $$dX_t = \Bigg(\frac{X_t - \Delta_t S_t}{B_t}\Bigg)rB_tdt + \Delta_t dS_t = (X_t - \Delta_t S_t)rdt + \Delta_t dS_t$$ Having this equation, that is valid for $t_k \leq t < t_{k+1}$, is it possible to argue that the equation for this previous wealth process is in fact valid for any transaction date, $t \in [0,T]$ ?