Why No-Arbitrage Does Not Determine Yield Curve Shape

# Shape of Yield curve of ZCB under no-arbitrage


# Shape of Yield curve of ZCB under no-arbitrage












Sorry if the question is somewhat elementary, but I have thought about it for a while and I cannot figure out where my mistake is.

Suppose we are in are in an arbitrage-free market in which risk-free zero coupon bonds (ZCB) are exchanged continuously. Assume also that the prices of such ZCBs are always strictly positive. Denote by $v(t,s)$ the spot price agreed in $t$ of a ZCB purchased in $t$ paying 1 unit of cash at its maturity $s$. If I am not wrong, when we assume that there are no arbitrages, at each fixed time $t$ in which we are "observing" the market, the term structure $s \mapsto v(t,s)$ is monotone increasing with respect to the maturity $s$. Consider the term structure of spot rates (in compound interest) given by $$i(t,s) = v(t,s)^{-\frac{1}{s-t}}-1.$$ Now, given the expression of $i(t,s)$, it follows from the monotonicity property of $v(t,s)$ that $i(t,s)$ must be monotone decreasing with respect to $s$. However, I cannot find any evidence of this fact in the literature I am currently consulting. Moreover, I have read from more than one source that under no-arbitrage many shapes of the yield curve are allowed (normal, flat, inverted, etc.). Can someone please point out to me what am I missing?

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/78866

There are some not-quite-correct statements in your question: "the term structure s↦v(t,s) is monotone increasing with respect to the maturity s" is generally incorrect. Given an observation time t, the prices of zero coupon bonds are usually decreasing as a function of the maturity date s. This observation relies on interest rates always being positive, which is not always true but usually is true. You then go on to say that the montonicity of v(t,s) implies the monotonicity of i(t,s) as you defined it, which is definitely not true. Using your notation I was able to derive $$ s^2 di/ds = (1+i) (ln v - s/v (dv/ds))$$ which indicates that the sign of $di/ds$ may be different to the sign of $dv/ds$ which is intuitive ly clear because we know yiled curves can be upward or downward sloping.

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/78865

The yield curve shape has annualized interest rates. You need to annualize each rate and then, any shape is allowed.

$discount factor = 1/(1+r*(delta_t))$

where $r$ is the annualized rate.