Black-Scholes formula with deterministic discrete dividend (Musiela approach)

Question

For deterministic discrete dividend, there are two approach

• Musiela approach, works when every dividend are paid at maturity of the option.
• Hull approach, works when every dividend are paid immediately after ex-dividend date.

I spend 1 day to understand the Musiela approach, but I can not understand his formula. In his book "Martingal Method for Financial Modelling 2nd Edit" $3.2.2, his first approach firstly define quantity :

• Timeline $0 < T_1 <T_2 … < T_m <T$ and dividend cash flow $q_1, q_2, .. q_m$

• Value of all posterior-t dividend compounded to Maturity time : $$ I_t = \sum^m_{i=1} q_i e^{r(T-T_i)} \mathbf{1}_{[0,T_i]}(t) $$ Note that $I_t$ decrease in time $t$ and piecewise constant. At each time $T_i$, $I_t$ drop down $q_i$

• Value of all anterior-t dividend compound to time $t$. $$ D_t = \sum^m_{i=1} q_i e^{r(t-T_i)} \mathbf{1}_{[T_i,T]}(t) $$ Here, $D_t$ increase in time $t$. At each time $T_i$, $D_t$ jump up $q_i$
• He define the capital gain process $$ G_t = S_t + D_t $$

Note that $$ D_T=I_0 \hspace{1cm} G_0=S_0 \hspace{1cm} G_T=S_T+D_T=S_T+I_0 $$
And all jump in price process $S_t$ are separated to $D_t$, he can model $G_t$ by the geometric brownian as usual, i.e under risk-neutral measure $$ \frac{dG_t}{G_t} = rdt + \sigma dW_t $$ Now, he can give the B&S formula for European Call option at time zero $$ C_0 = e^{-rT}\mathbb{E}[(S_T-K)^+] = e^{-rT}\mathbb{E}[(G_T-(K+I_0))^+] $$ Since the modelled process is $G_t$, this price at time $0$ is easily found by Black-Scholes calculation routine. $$ C_0 = S_0 \mathcal{N}(d_+) - e^{-rT}K \mathcal{N}(d_{-}) $$ with $$ d\pm = \frac{ \text{ln}\frac{S_0}{K+I_0} + (r\pm\frac{\sigma^2}{2})T } {\sigma \sqrt{T}} $$ For this price formula at time $0$, I can understand it. An then I tried to compute for an arbitrary time $t$ $$ C_t = e^{-r(T-t)}\mathbb{E}[(S_T-K)^+|\mathcal{F}_t] = e^{-r(T-t)}\mathbb{E}[(G_T-(K+I_0))^+|\mathcal{F}_t] $$ Again, the calculation routine of Black-Scholes should give $$ C_t = G_t \mathcal{N}(d_+) - e^{-r(T-t)} (K+I_0) \mathcal{N}(d_{-}) $$ with $d\pm$ should be $$ d\pm = \frac{ \text{ln}\frac{G_t}{K+I_0} + (r\pm\frac{\sigma^2}{2})(T-t) } {\sigma \sqrt{T-t}} $$ But in the Musiela's book, he give the different result without detail proof. His result is $$ C_t = S_t \mathcal{N}(\hat{d}_+) - e^{-r(T-t)} (K+I_t) \mathcal{N}(\hat{d}_{-}) $$ with $$ \hat{d}\pm = \frac{ \text{ln}\frac{S_t}{K+I_t} + (r\pm\frac{\sigma^2}{2})(T-t) } {\sigma \sqrt{T-t}} $$ So the annoying differences are

• He have strike term as $K+I_t$, I have $K+I_0$
• He have random process as $S_t$, I have $G_t$

Can anyone help please. I've spent to much time without success.

PS : one more question. There are maybe something that I am missing. The fact that he use $D_t$ to model the dividend, but in the result, he use $I_t$, that seems strange.

Answer 1

The derivation in the book appears wrong. However, the results make sense as the option price at time $t$ should not be impacted by prior dividend payments. It may be out-of topic, I would like to provide some justification of the Musiela-Rutkowski formula.

Let $\{H_t \mid t >0\}$, where \begin{align*} H_t = \sum_{0 < T_i \leq t} q_i, \end{align*} be a step process. Moreover, we assume that, under the risk-neutral probability measure, the stock price $S_t$ satisfies an SDE of the form \begin{align*} dS_t = S_{t-}(r dt + \sigma dW_t) - dH_t, \end{align*} where $\{W_t\mid t>0\}$ is a standard Brownian motion. For $0<t \leq T$, assuming that \begin{align*} t < T_{i_0} < \cdots < T_m \leq T. \end{align*} Then, \begin{align*} S_{T_{i_0}} &= S_te^{\int_t^{T_{i_0}}(r-1/2\sigma^2)ds +\int_t^{T_{i_0}}\sigma dW_s}-q_{i_0}\\ S_{T_{i_0+1}} &= \bigg(S_te^{\int_t^{T_{i_0}}(r-1/2\sigma^2)ds +\int_t^{T_{i_0}}\sigma dW_s}-q_{i_0}\bigg)e^{\int_{T_{i_0}}^{T_{i_0+1}}(r-1/2\sigma^2)ds +\int_{T_{i_0}}^{T_{i_0+1}}\sigma dW_s} -q_{i_0+1}\\ &\approx S_te^{\int_t^{T_{i_0+1}}(r-1/2\sigma^2)ds +\int_t^{T_{i_0+1}}\sigma dW_s}-q_{i_0} e^{\int_{T_{i_0}}^{T_{i_0+1}}rds} - q_{i_0+1}\\ & \ldots\ldots\\ S_T &\approx S_te^{\int_t^{T}(r-1/2\sigma^2)ds +\int_t^{T}\sigma dW_s}-\sum_{i=i_0}^m q_{i} e^{\int_{T_{i}}^{T}rds}\\ &= S_te^{\int_t^{T}(r-1/2\sigma^2)ds +\int_t^{T}\sigma dW_s}-I_t. \end{align*} Now, the Musiela-Rutkowski formula follows immediately.

Answer 2

I agree that your derivation makes sense.

To me, the only way to explain the book's price is if, as time goes by, the model is constantly modified, so that at time $\tau$, $\hat{G_t} = S_t + D_t - D_\tau$ is assumed to be a geometric brownian process with volatility $\sigma$.

The book's price wouldn't be equal to the price of the option if the world really behaved under the diffusion that was assumed at time $0$, but I can see how it could still be somewhat justified: in practice, such a recalibration would be what one would do when risk managing this option (it doesn't really make sense to consider at time $\tau$ that the volatility of the stock's price depends on past dividends, which is what this model implies).

Of course, for risk management purposes, the volatility would also be updated over time, so the true price as observed at time $\tau$ would use some $\sigma_\tau$ as its volatility, not necessarily equal to the original $\sigma$.

I haven't read the book, so I can't comment on why the formula was written this way, but in the end I think both formulas could be argued for and against depending on the point of view (i.e. are you talking about the price at time $\tau$ as a pure random variable seen from time $0$, or as a known value observed on time $\tau$).