# Stochastic Interest Rates in Option pricing

My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to t. Surely, it's a negative integral from t to T? Her notes are full of mistakes so I cannot figure out if I'm not getting something or if she's made a mistake.

$$Value(t)$$ is the value at time $$t$$ of receiving \$1 at time $$T$$. Thus, indeed $$Value(t)=\exp\left(-\int_t^T r_u\mathrm{d}u\right).$$ This expression is also known as (stochastic) discount factor. Let $$Wealth(t)=\exp\left(\int_0^t r_u\mathrm{d}u\right)$$ be the value at time $$t$$ of investing \$1 at time $$0$$. Your \1 grows at the stochastic rate $$r_t$$. Then, \begin{align*} Value(t)&=\frac{Wealth(t)}{Wealth(T)} \\ &= \frac{\exp\left(\int_0^t r_u\mathrm{d}u\right)}{\exp\left(\int_0^T r_u\mathrm{d}u\right)} \\ &= \exp\left(\int_0^t r_u\mathrm{d}u-\int_0^T r_u\mathrm{d}u\right) \\ &= \exp\left(-\int_t^T r_u\mathrm{d}u\right) \end{align*} It's nothing else than discounting (computing present values). • So how is the value ofB^T(t)$equal to$exp(-\int_0^t r(t) dt)$. Shouldn't it be$exp(-\int_t^T r(t) dt)$– s5s May 7 '20 at 15:12 • @s5s Do you know the fundamental theorem of asset pricing? If there’s no arbitrage, aaset prices are conditional expectations of discounted payoffs. The bond’s payoff is \$1 and the discount factor is $Value(t)$. May 7 '20 at 15:14
• yes, and Value(t) is $exp(-\int^T_t r(t) dt)$ not $exp(-\int_0^t r(t) dt)$. So why are her notes saying that $B^T(t)$ is the conditional expectation from 0 to t? This is not even a random variable, it's realised.
• @s5s you’re absolutely right. Her formula is wrong. As you said, it doesn’t make sense and there is indeed no random variable at time $t$. I’m sorry, I was looking at the first two equations where you put “not correct”. Her bond pricing formula is clearly wrong. It should be, as you said the conditional expectation of the discount factor ($Value$). May 7 '20 at 15:27