Random Coupon Bonds, Replication, and Risk-Neutral Pricing

# Why are random coupons not priced using risk-neutral evaluation?


# Why are random coupons not priced using risk-neutral evaluation?












Assume a fixed coupon bond has a coupon which, randomly, is 5 % or 4 %, each occuring with a 50 % probability. The issuer flips a coin on payment date to decide which it should be.

I would value this as a weighted average of the 5% and 4% fixed coupon bonds.

But here I am using the "P" measure. Why does this not need to be priced using the "Q" measure?

## Answer by siou0107 (score 5)

https://quant.stackexchange.com/a/78757

Bouchard and Chassagneux give the following definition of a viable price. A price $p$ for a derivative with payoff $G$ is viable if buying or selling the derivative at this price does not create an arbitrage. $$ \not\exists \epsilon \in \left\{-1, 1 \right\}, V_T^{\epsilon p, \phi} - \varepsilon G \geq 0 \quad a.s. $$ where $V_T^{x, \phi}$ is the terminal value of a portfolio with initial capital $x$ and investment strategy $\phi$.

Later, they connect the interval of viable prices with risk-neutral measures.

$p \left(G\right)$ is the super-hedging price of an option with payoff $G$ ; conversely, -$p \left(-G\right)$ is the sub-hedging price of payoff $G$ (cheapest price at which you can buy the option without creating an arbitrage). Any risk-neutral measure will give you a price in this interval. If the derivative is replicable, there is a single viable price (the cost of the replication portfolio) and the risk-neutral measure is unique. In an incomplete market, such as in your case, there is a range of viable prices, corresponding to an infinity of risk-neutral measures, and the sub/superhedging correspond to the infimum/supremum of discounted expected payoffs under this risk-neutral measures set.

In your case, the sub-hedging price is the price of a 4% (deterministic) coupon bond ; the super-hedging price is the price of a 5% coupon bond. Any price within this interval can be obtained as the expected discounted value of all future coupons (and principal payment) under the risk-neutral measure. To quote the excellent Björk, "Who chooses the martingale measure? The market!"; and Joshi added: "and the market is fickle".

Hope this elaborate answer will raise your interest for these excellent books! :)

## Answer by KaiSqDist (score 3)

https://quant.stackexchange.com/a/78749

When one prices under the $\mathbb{Q}$ measure, or performs risk-neutral valuation, one is assuming that investors have risk-neutral preferences and utilizes risk-neutral probabilities along with the riskless discount rate. The concept of risk-neutral valuation is usually adopted to price complex securities by providing a manageable framework.

However, risk-neutral valuation might not reflect the actual market price of the asset (which is usually lower). This is because in the real-world, investors are risk-adverse and demand a risk-premium to be compensated for taking on a certain level of risk. By pricing under the $\mathbb{P}$ measure, using real-world probabilities and the stochastic discount factor $M$, is what is gives us the actual market price of the asset.

To answer your question: You could perform risk-neutral valuation, but then you would have to derive risk-neutral probabilities from the real-world ones by first determining the appropriate risk premium, which is a complex process. Additionally, you might not get the actual market price of the bond i.e. what is observed on the market (unless of course you back-out the risk-neutral probabilities from the market price of the bond in the first place).

## Answer by Arshdeep (score 3)

https://quant.stackexchange.com/a/78761

Think of the coupon as "float" but corresponds to a new asset whose payoff depends on a coin toss and is uncorrelated with rates.

This asset is not tradeable in the economy and therefore you cannot replicate the payoff. There is no risk neutral probability in this case. You cannot replicate it with other floating rates because the coin toss is not dependent on rates.

If the forward of this coin toss was tradeable, this would be priced using risk neutral probability.

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/78765

The above answers have pointed out (1) that the bond is not replicable using some other tradeable instrument (2) therefore any price between 4% and 5% could be consistent with a risk neutral pricing methodology. (when I say "price" at x% I mean, treat it like a x% bond.).

But isn't there a separate point to be made, that the coin flip clearly has 0% beta with the market, hence using a model such as CAPM, investors will not attribute any risk premium to it, hence will price it at 4.5% or thereabouts. This is not a derivatives pricing comment, it's an asset pricing comment.