# 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 thereâ€™s no arbitrage, aaset prices are conditional expectations of discounted payoffs. The bondâ€™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 youâ€™re absolutely right. Her formula is wrong. As you said, it doesnâ€™t make sense and there is indeed no random variable at time $t$. Iâ€™m sorry, I was looking at the first two equations where you put â€śnot correctâ€ť. 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