# Future price in continous time

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?

• The second term is more commonly written $Ke^{-r(T-t)}$
– dm63
Mar 22, 2021 at 12:36
• Yes, but my problem is that this price is not so consistent with arbitrage theory. It seems that the true arbitrage price consider the discounted value of the dividends, and I cannot fully understand why is that! Mar 23, 2021 at 7:30