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Correlation

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Last updated 1 year ago

Lending and borrowing positions in Infinity are not exposed to (FX,FX) or (FX,Rates) correlation risks. For example, with an ETH-1D fixed rate lending position, the present value of your 1-day ETH lending position could be expressed in either USD, PV(USD), or ETH, PV(ETH), terms:

PVUSD=PVETH×FX(0)=1+X⋅T(1+r(0,T))T⋅FX(0)PV_{USD}=PV_{ETH}\times FX(0) = \frac{{1 + X \cdot T}}{{(1 + r(0,T))^T}} \cdot FX(0)PVUSD​=PVETH​×FX(0)=(1+r(0,T))T1+X⋅T​⋅FX(0)

"X" is the ETH fixed rate and T= 1 / Actual Days (365, or 366), where:

  • PV(USD) is a function of PV(ETH) and FX(0), where PV(ETH) is the zero coupon rate r(0,T), both observed at T=0

    • => PV(USD) is not sensitive to the (Rates, FX) correlation

  • PV(ETH) is a linear combination of PV in a similar format to the PV(USD) above

    • => PV(ETH) is not sensitive to the (FX, FX) correlation

Correlation between assets (FX,FX) or (FX,Rates) will be applied during the liquidation order. Similarly, for margining, in particular your , which depends on the stresses applied and (across the yield curve).

Note: stressing FX and Rates separately and then aggregating their impact is different from stressing Rates and FX simultaneously. To illustrate this point, let’s denote PV=f(r,FX):

  • If we bump Rates and FX separately, the impact of dPV:

dPV=df=∂f∂x⋅dx+∂f∂y⋅dydPV = df = \frac{\partial f}{\partial x} \cdot dx + \frac{\partial f}{\partial y} \cdot dy dPV=df=∂x∂f​⋅dx+∂y∂f​⋅dy

  • If we bump both Rates and FX simultaneously, the impact of dPV is:

dPV=df=∂f∂x⋅dx+∂f∂y⋅dy+∂2f∂x∂y⋅dxdydPV =df= \frac{\partial f}{\partial x} \cdot dx + \frac{\partial f}{\partial y} \cdot dy + \frac{\partial^2f}{\partial x \partial y} \cdot dxdydPV=df=∂x∂f​⋅dx+∂y∂f​⋅dy+∂x∂y∂2f​⋅dxdy

In our case:

∂2f∂x∂y ∼−T∗(1+XT)<0\frac{\partial^2f}{\partial x \partial y} \ \sim -T*(1+XT)<0∂x∂y∂2f​ ∼−T∗(1+XT)<0

To maximize the impact (and apply a larger stress for StrNAV), we want dxdy>0, where Rates (r) and FX move in the same direction. In this case,

  • The impact of bumping both FX and Rates simultaneously will be greater than adding the individual impacts of bumping FX and Rates in isolation, and vice versa where FX and Rates move in the opposite direction. Though, in practice, one would assume that a rise in the ETH/USD FX would be positively correlated with ETH Rates.

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