I am in the following continuous time market:
- $S_t^0 = rS_t^0dt$
- $S_t^1 = (\mu - \delta) S_t^1dt + \sigma S_t^1 dB_t$
where $r, \mu, \delta$ and $\sigma$ are constant values in $\mathbb{R}$. $\delta$ is the dividend yield. I want to recover the risk neutral price of a future written on $S^1$ with delivery price $K$ and maturity $T$. This basically means that the payoff at time $T$ will be: $H_T = S_T^1 - K$.
In my mind, what I have to do is using universal pricing theorem: $$\pi_t(H) = S_t^0 \ \mathbb{E}_{\mathbb{Q}}\left[ \frac{H_T}{S_T^0} \ | \ \mathcal{F}_t\right]$$ Setting $\lambda = \frac{(\mu - \delta) - r}{\sigma}$ and defining $\frac{d\mathbb{Q}}{d\mathbb{P}} = \epsilon_T(-\lambda \cdot B_t)$ I obtain that the process $\tilde{B_t} = B_t + \lambda t$ is a $\mathbb{Q}$ standard brownian motion. This follows by the fact that my radon-nykodym density is well defined since my $\lambda$ fulfills Novikov condition.
Using this approach I can recover that the new assets dynamics (under the new measure $\mathbb{Q}$) are given by:
- $S_t^0 = rS_t^0dt$
- $S_t^1 = r S_t^1dt + \sigma S_t^1 d\tilde{B_t}$
- $d\frac{S_t^1}{S_t^0} = \sigma \frac{S_t^1}{S_t^0}d\tilde{B_t}$
So that discounted process is a martingale under the new measure $\mathbb{Q}$. I have that the price of the forward is given by: $$\pi_t(H) = S_t^0 \ \mathbb{E}_{\mathbb{Q}}\left[ \frac{H_T}{S_T^0} \ | \ \mathcal{F}_t\right] = S_t^0 \ \mathbb{E}_{\mathbb{Q}}\left[ \frac{S_T^1}{S_T^0} \ | \ \mathcal{F}_t\right] - \frac{S_t^0}{S_T^0}K $$ now, using the fact that discounted process is a martingale, $$\pi_t(H) = S_0^1 - \frac{S_t^0}{S_T^0}K$$ It seems to be mathematically correct. However, I have the sensation that I am not properly considering the effect of dividends. Any idea where I am wrong and how to proceed?
