Question: Consider the one-period binomial model with a stock that pays continuous dividend $\delta$. I want to show that the risk-neutral probability is given by $$p=\frac{\exp((r-\delta)\Delta t)-d}{u-d}$$. Hint: The value of stock at time $\Delta t$ is: $S_0uexp(-\delta\Delta t)$ for stock moving up and $S_0dexp(\delta\Delta t)$ for stock moving down.
Approach: Here is how I started the proof. First I form portfolio at t=0. $\Delta S-O_{t=0}$ to find $\Delta$ which is; $$\Delta S_0uexp(\delta\Delta t)-O_{up}=\Delta S_0dexp(\delta\Delta t)-O_{down}$$ where $$\Delta=\frac{O_{up}-O_{down}}{S_0exp(\delta\Delta t)(u-d)}$$ after finding this $\Delta$ I'm sure we are supposed to substitute it into the initial portfolio but I'm confused as to how am I supposed to get rid of the O's and how to derive p from this. The professor gave me this much hint.
Any help will be greatly appreciated! Thank you!!
Sometimes, its more beneficial to work via first principles, than plugging into an equation.
Let $X$ denote a claim -- in this case $X$ denotes the payoff on the stock held at time $t$. Lets simplify the notation, so that without loss of generality let $t=0$ and $t=1$, and suppose the stock has a value $u$ or $d$ at $t=1$.
What we want is the value of the claim $X$ at time $t=0$, under the risk neutral measure. Denote this measure by $p$. Thus we want to find $p$, given that we know the possible prices of the stock. That is if $X_{1}(u),X_{1}(d)$ represent the claim at time t=1, we have that.
\begin{equation} V(X) = pX_{1}(u) + (1-p)X_{1}(d) \end{equation}
Now $V(X) = S_{0}exp(r\Delta t)$, since the stock price at $t=0$ is known. Thus we have, \begin{equation} S_{0}exp(r\Delta t) = pX_{1}(u) + (1-p)X_{1}(d) \end{equation} $X_{1}(u)$ is the claim at t=1, when the market is in state $u$. That is use the hint given to calculate $X_{1}(u)$ and $X_{1}(d)$, plug it into the equation above and solve for $p$. Hope this helps, let me know if you need more details.