Monte Carlo, convexity and Risk-Neutral ZCB Pricing

Question

I've built a simplistic Excel monte carlo model to price a zero-coupon bond, but it came up with a slightly unepxected result so I wanted to confirm whether my maths is just a little rusty or my model is wrong.

Suppose we have a 40 year ZCB with a price $P_t$ that equates to a flat discount rate of $z_t$ at time $t$; i.e. the price $P_t = \exp(-(40 - t) \cdot z_t)$, and let's say $z_0 = 2\%$ in this case . (This is obviously different from the usual zero coupon bond vs rate pricing formula, $P_t = \mathbb{E}[\exp(-\int_t r_t)]$, as it refers to a single "average" rate over the remaining life of the bond)

The monte carlo starts with this spread $z_0$, which evolves randomly after each period with no drift, according to $z_{t+\Delta t} = z_t + (\sigma \cdot \sqrt{\Delta t} \cdot \omega)$ where $\sigma$ is a vol parameter and $\omega \sim \text{N}(0,1)$.

At year 30, I look at the simulated value of $z_{30}$, and then infer the price of the bond as $P_{30} = \exp (-(40 - 30) \cdot z_{30})$.

Having run this simulation for a number of Monte Carlo runs, I took the average spread $z_t$ at $t = 30$, which was unsurprisingly equal to $2\% = z_0$, and the average price $P_{30}$ across all of these runs.

I was expecting that this average price would be approximately $\exp (-(40 - 30) \cdot z_{0}) = \exp(-10 \cdot 2\%)) = \exp (-(40 - 30) \cdot \mathbb{E}(z_{30}))$ but in fact the value was consistently higher than .

To some extent I can justify this to myself; the bond price is convex, so in absolute terms you'll see bigger absolute positive gains than negative losses when the spread moves tighter or wider by the same $\%$ amount (assuming the avg spread is still centered around $z_0 = 2\%$).

But on the other hand, this means your expected bond price in the monte carlo is a function of volatility: $\sigma = 0$ gives you $z_{30} = z_0$ in all cases and therefore a lower expected price than for some $\sigma > 0 $. I wouldn't intuitively expect that the expected bond price over time is dependent on vol even with zero drift; it's not an option-type payoff after all, and I've never previously seen bond prices contemplated as a function of vol. Discussions such as this one make me think that the $+\sigma^2/2$ lognormal expectation effect, skewing gains over losses, should be offset by $-\sigma^2/2$ term in the expected Brownian motion path (though admittedly this is a slightly different security from the one in that link), but my model appears to suggest otherwise.

What am I missing here? Is my model wrong, am I trying to reconcile 2 fundamentally different quantities or should the expected bond price with nonzero vol genuinely be higher than the bond price discounted at the expected spread?

Answer 1

There is no mystery here.

If $X$ is Gaussian, $E[e^X] = e^{E[X]+\frac{1}{2}V[X]}$

In your case, $$P_{30} = E[e^{-(40-30)z_{30}}] = e^{-E[(40-30)z_{30}] +\frac{1}{2}V[(40-30)z_{30}]]} > e^{-(40-30)E[z_{30}]} = e^{-(40-30)z_0}$$

The fact that you are simulating a whole path is irrelevant since you could just as well simulate $z_{30}$ exactly with one step. I don't understand your point about the expected Brownian path.

Answer 2

There's nothing wrong with your formulation, in my opinion. If you model the rate z_30 with a fixed mean, then indeed the forward ZCB price is long vega. This means that the forward interest rate is short vega (i.e. the 30yr into 10yr forward rate goes down when vol goes up). This is self-consistent.

In most textbooks, however, the forward interest rate and forward bond prices are taken as given. Therefore, when you measure the vega of some instrument, you leave forward rates unchanged. Hence bonds, by definition, have no vega. To produce a model like this, the mean of your forward rate distribution would have to be set so as to reprice the forward rates correctly. How this is done exactly, depends on the probability measure you are in.

As an example, a CMS contract, which has a payoff linear in the swap rate, has a non zero vega in the textbook model, but would have a zero vega in the model that you built.