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Do price approximations lead to arbitrage opportunities against a price computed using the exact formula?

For instance, dirty bond price uses a linear approximation to compute the accrual interest: $$P_{d}=P_{c}+\alpha t$$ Whereas the exact formula gives a slightly higher value for $0<t<1$ (with time unit the time between payments), for both continuous or discrete compounding (assuming par): $$P(t)=ce^{i(t-1)}+e^{i(t-1)}P(0)$$ $$P(t)=\frac{c}{(1+i)^{t-1}}+\frac{P(0)}{(1+i)^{t-1}}$$ Giving the high nominal of some bonds, does the difference is ever high enough to produce arbitrage opportunities?

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No. The dirty price is the market's estimate of fair value for the bond. The clean price is just a quoting convention (so that the price doesn't jump when you pass over a coupon date).

The market doesn't try to estimate the clean price and then get the all-in (dirty) price wrong. The market estimates the all-in price, and then applies the accrued interest adjustment when it comes to submitting a quote.

The adjustment is just a convention that everyone agrees on to make the price series nicer. Since the adjustment has no economic impact whatsoever, it doesn't matter what adjustment is used, and you might as well use the simpler, linear adjustment for days accrued, rather than a more complex adjustment that takes compounding into account.

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