Covered Interest Rate Parity with FX Spot-Adjustment

Question

The Covered Interest Rate Parity for FX is often quoted simplistically as $$ X_T \quad=\quad X_S \cdot \frac{D^{base}_T}{D^{quote}_T} $$ where $X_t$ is the (projected) FX rate at time $t$ (denoted as $1$ base = $X$ quote), $D_t^{ccy}$ is the discount factor for time $t$ in currency $ccy$, and $X_S$ denotes the current FX Spot Rate.

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To derive this parity one can note a cashflow of 1 USD at $T$ can be discounted in USD and then converted at spot $X_S$ to JPY. Or equivalently, convert at $X_T$ and discount in JPY. So $$ \text{NPV in JPY} \quad=\quad (1\text{ USD} \cdot D_T^{USD}) \cdot X_S \quad=\quad (1\text{ USD} \cdot X_T) \cdot D_T^{JPY} $$ which implies the parity.

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In practice the above parity does not hold since spot $\ne$ today (e.g. when spot = today + 2 days). So we get a more comprehensive parity: $$ X_{T} \quad=\quad X_{S} \cdot \frac{D^{base}_{T}}{D^{quote}_{T}} \cdot \color{blue}{\frac{D^{quote}_{S}}{D^{base}_{S}}} $$ E.g. see QuantLib's implementation: github.com/.../QuantLib/.../ratehelpers.cpp#L995-L1012

I am struggling to properly incorporate this $\color{blue}{\text{spot adjustment}}$ in the non-arbitrage/replication argument, which I outlined above. Can anybody help?

Answer 1

A simple trick is being used to come up with the right discount factors.

Since $D_T=\frac{1}{1+r_1}\frac{1}{1+r_2}\frac{1}{1+r_3}\cdots\frac{1}{1+r_T}$

and $D_S=\frac{1}{1+r_1}\frac{1}{1+r_2}$

we can form the desired discount factor from two days hence to date $T$ as follows

$\frac{D_T}{D_S}=\frac{1}{1+r_3}\cdots\frac{1}{1+r_T}$ (which we might call $D_{2,T}$)

The same trick is being used for both the base and quote currencies to eliminate the unwanted discounting for the first two days which would occur in a naive day 0 to day $T$ discounting. It is not so much a different arbitrage argument as a refinement of the calculation of $D^{base}$ and $D^{quote}$ in your first equation, reflecting that $D_{2,T}$ and not $D_{0,T}$ type of calculation is needed.