# Payoff from an option contract

In period 1 the consumer of type $$\theta$$ selects an option contract consisting of an up-front fee, $$B>0$$, and exercise price, $$\bar{R}$$. The consumer pays $$B$$ at the end of the first period. In period 2, he realises his valuation, $$\theta$$, distributed on $$[0,1]$$ by CDF $$G(\theta)$$ with density $$g>0$$. His expected payoff, from choosing contract $$(B, \bar{R})$$, should thus be $$-B+\int_{\bar{R}}^{1}(\theta-\bar{R})g(\theta)d(\theta).$$ However, in the paper I am told that it is instead: $$-B+\int_{\bar{R}}^{1}(1-G(\theta))d\theta.$$ Are these two expressions equivalent? What does it mean to integrate a CDF in this fashion? Thank you.

The CDF satisfies $$G(0) = 0$$, $$G(1) = 1$$ and $$G'(\theta) = g(\theta)$$.
Integrating by parts, with $$u = 1- G(\theta)$$ and $$dv = d\theta$$, we have $$du = -G'(\theta)\,d\theta = - g(\theta)\,d\theta$$, $$v = \theta$$, and
$$\int_{\bar{R}}^1(1- G(\theta)) \, d\theta = \left.(1- G(\theta))\theta\right|_{\bar{R}}^1 + \int_{\bar{R}}^1 \theta\,g(\theta) \, d\theta = -(1- G(\bar{R}))\bar{R} + \int_{\bar{R}}^1 \theta\, g(\theta) \, d\theta\\=-\bar{R} \int_{\bar{R}}^1 g(\theta)\, d\theta+ \int_{\bar{R}}^1 \theta\, g(\theta) \, d\theta = \int_{\bar{R}}^1 (\theta-\bar{R}) \, g(\theta) \, d\theta$$