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How to Calculate Cross Currency Pair Exchange Rates?

上海子午线About 4 min

How to Calculate Cross Currency Pair Exchange Rates?

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Cross Currency (Spot) Price Calculation

The formulas for calculating the bid (BID) and ask (ASK) prices for Currency1/Currency2 depend on how Currency1 and Currency2 are quoted.

The table below summarizes the spot price calculation rules:

Currency1 QuotationCurrency2 QuotationBID12ASK12
DIRECT*DIRECTqu12×(Bid2Ask1)×(qu1qu2)qu_{12}\times\left(\frac{Bid_{2}}{Ask_{1}}\right)\times\left(\frac{qu_{1}}{qu_{2}}\right)qu12×(Ask2Bid1)×(qu1qu2)qu_{12}\times\left(\frac{Ask_{2}}{Bid_{1}}\right)\times\left(\frac{qu_{1}}{qu_{2}}\right)
DIRECTINDIRECTqu12(Ask1×Ask2)×(qu1×qu2)\frac{qu_{12}}{(Ask_1\times Ask_2)\times(qu_1\times qu_2)}qu12(Bid1×Bid2)×(qu1×qu2)\frac{qu_{12}}{\left(Bid_1\times Bid_2\right)\times\left(qu_1\times qu_2\right)}
INDIRECT**DIRECTqu12×(Bid1×Bid2)qu1×qu2\frac{qu_{12}\times\left(Bid_{1}\times Bid_{2}\right)}{qu_{1}\times qu_{2}}qu12×(Ask1×Ask2)qu1×qu2\frac{qu_{12}\times\left(Ask_1\times Ask_2\right)}{qu_1\times qu_2}
INDIRECTINDIRECTqu12×(Bid1Ask2)×(qu2qu1)qu_{12}\times\left(\frac{Bid_{1}}{Ask_{2}}\right)\times\left(\frac{qu_{2}}{qu_{1}}\right)qu12×(Ask1Bid2)×(qu2qu1)qu_{12}\times\left(\frac{Ask_{1}}{Bid_{2}}\right)\times\left(\frac{qu_{2}}{qu_{1}}\right)

Here:

SymbolMeaning
Bid1{Bid_{1}}Bid spot price for Currency1
Ask1{Ask_{1}}Ask spot price for Currency1
Bid2{Bid_{2}}Bid spot price for Currency2
Ask2{Ask_{2}}Ask spot price for Currency2
qu12{qu_{12}}Quotation unit of Currency1/Currency2 if present in the style management*; otherwise, 1 is used
qu1{qu_{1}}Quotation unit of USD/Currency1 if present; otherwise, 1
qu2{qu_{2}}Quotation unit of USD/Currency2 if present; otherwise, 1

qu1{qu_{1}} and qu2{qu_{2}} represent the default base currency (Base Currency) as USD.

Direct Quotation (DIRECT)

When a fixed amount of foreign currency (USD) is quoted against a variable amount of domestic currency (JPY), it is called direct quotation.

For example, the USD/JPY exchange rate of 121.91 means:

  • 1 USD can be exchanged for 121.91 JPY.

USD is the base currency, and JPY is the quote currency.

Indirect Quotation (INDIRECT)

When a fixed amount of domestic currency (GBP) is quoted against a variable amount of foreign currency (USD), it is called indirect quotation.

For example, the GBP/USD exchange rate of 1.6870 means:

  • 1 GBP can be exchanged for 1.6870 USD.

GBP is the base currency, and USD is the quote currency.

Example

Assume the following quotes to calculate the EUR/GBP price:

Currency PairBIDASKQuotation Mode
EUR/USD0.99070.9910INDIRECT
GBP/USD1.55921.5598INDIRECT

Here:
qu12{qu_{12}} = 1
qu1{qu_{1}} = 1
qu2{qu_{2}} = 1

Since both direct currency pairs are INDIRECT:

DirectionFormulaCalculation Process
Bidqu12×(Bid1Ask2)×(qu2qu1)qu_{12}\times\left(\frac{Bid_{1}}{Ask_{2}}\right)\times\left(\frac{qu_{2}}{qu_{1}}\right)1×(0.99071.5598)×(11)=0.63511\times\left(\frac{0.9907}{1.5598}\right)\times\left(\frac{1}{1}\right)=0.6351
Askqu12×(Ask1Bid2)×(qu2qu1)qu_{12}\times\left(\frac{Ask_{1}}{Bid_{2}}\right)\times\left(\frac{qu_{2}}{qu_{1}}\right)1×(0.99101.5592)×(11)=0.63561\times\left(\frac{0.9910}{1.5592}\right)\times\left(\frac{1}{1}\right)=0.6356

Cross Currency (Forward Price) Calculation

The rules for calculating forward prices are essentially the same as for spot prices:

Currency1 QuotationCurrency2 QuotationBidOut12AskOut12
DIRECT*DIRECTqu12×(BidOut2AskOut1)×(qu1qu2)qu_{12}\times\left(\frac{BidOut_{2}}{AskOut_{1}}\right)\times\left(\frac{qu_{1}}{qu_{2}}\right)qu12×(AskOut2BidOut1)×(qu1qu2)qu_{12}\times\left(\frac{AskOut_{2}}{BidOut_{1}}\right)\times\left(\frac{qu_{1}}{qu_{2}}\right)
DIRECTINDIRECTqu12(AskOut1×AskOut2)×(qu1×qu2)\frac{qu_{12}}{(AskOut_1\times AskOut_2)\times(qu_1\times qu_2)}qu12(BidOut1×BidOut2)×(qu1×qu2)\frac{qu_{12}}{\left(BidOut_1\times BidOut_2\right)\times\left(qu_1\times qu_2\right)}
INDIRECT**DIRECTqu12×(BidOut1×BidOut2)qu1×qu2\frac{qu_{12}\times\left(BidOut_{1}\times BidOut_{2}\right)}{qu_{1}\times qu_{2}}qu12×(AskOut1×AskOut2)qu1×qu2\frac{qu_{12}\times\left(AskOut_1\times AskOut_2\right)}{qu_1\times qu_2}
INDIRECTINDIRECTqu12×(BidOut1AskOut2)×(qu2qu1)qu_{12}\times\left(\frac{BidOut_{1}}{AskOut_{2}}\right)\times\left(\frac{qu_{2}}{qu_{1}}\right)qu12×(AskOut1BidOut2)×(qu2qu1)qu_{12}\times\left(\frac{AskOut_{1}}{BidOut_{2}}\right)\times\left(\frac{qu_{2}}{qu_{1}}\right)

Here:

SymbolMeaning
BidOut1{BidOut_{1}}Bid1+BidSwapPoint1SwapRatio1{Bid_{1} + \frac {BidSwapPoint_{1}} {SwapRatio_{1}}}
AskOut1{AskOut_{1}}Ask1+AskSwapPoint1SwapRatio1{Ask_{1} + \frac {AskSwapPoint_{1}} {SwapRatio_{1}}}
BidOut2{BidOut_{2}}Bid2+BidSwapPoint2SwapRatio2{Bid_{2} + \frac {BidSwapPoint_{2}} {SwapRatio_{2}}}
AskOut2{AskOut_{2}}Ask2+AskSwapPoint2SwapRatio2{Ask_{2} + \frac {AskSwapPoint_{2}} {SwapRatio_{2}}}
BidSwapPoint1{BidSwapPoint_{1}}Bid Swap Point 1
AskSwapPoint1{AskSwapPoint_{1}}Ask Swap Point 1
BidSwapPoint2{BidSwapPoint_{2}}Bid Swap Point 2
AskSwapPoint2{AskSwapPoint_{2}}Ask Swap Point 2
SwapRatio1{SwapRatio_{1}}Swap point ratio of USD/Currency1; otherwise, 1
SwapRatio2{SwapRatio_{2}}Swap point ratio of USD/Currency2; otherwise, 1
qu12{qu_{12}}Quotation unit of Currency1/Currency2
qu1{qu_{1}}Quotation unit of USD/Currency1 if present; otherwise, 1
qu2{qu_{2}}Quotation unit of USD/Currency2 if present; otherwise, 1

Example

Step 1: Calculate Spot Price

Currency PairBIDASKSwapRatioQuotation Mode
GBP/USD1.60121.601810000INDIRECT
USD/JPY116.91116.97100DIRECT
GBP/USD SwapPoints-1.55-1.3
USD/JPY SwapPoints-0.72-0.12

Calculated spot price:

Calculation ProcessGBP/JPY
Bidqu12×(Bid1×Bid2)qu1×qu2=1×(1.6012×116.91)1×1=187.196292\frac{qu_{12}\times\left(Bid_{1}\times Bid_{2}\right)}{qu_{1}\times qu_{2}}=\frac{1\times\left(1.6012\times116.91\right)}{1\times1}=187.196292187.1962
Askqu12×(Ask1×Ask2)qu1×qu2=1×(1.6018×116.97)1×1=187.362546\frac{qu_{12}\times\left(Ask_{1}\times Ask_{2}\right)}{qu_{1}\times qu_{2}}=\frac{1\times\left(1.6018\times116.97\right)}{1\times1}=187.362546187.3625

Step 2: Calculate Outright Forward Price

FormulaCalculation ProcessGBP/JPY Outright
Bidqu12×(BidOut1×BidOut2)qu1×qu2\frac{qu_{12}\times\left(BidOut_{1}\times BidOut_{2}\right)}{qu_{1}\times qu_{2}}1×(1.601045×116.9028)1×1=187.16664\frac{1\times\left(1.601045\times116.9028\right)}{1\times1}=187.16664187.16664
Askqu12×(AskOut1×AskOut2)qu1×qu2\frac{qu_{12}\times\left(AskOut_1\times AskOut_2\right)}{qu_1\times qu_2}1×(1.60167×116.9688)1×1=187.34541\frac{1\times\left(1.60167\times116.9688\right)}{1\times1}=187.34541187.34541

Step 3: Calculate Swap Points

FormulaCalculation ProcessGBP/JPY Swap Points
Bid(BidOut12−Bid12)×SwapRatio12({BidOut_{12}}-{Bid_{12}}) \times {SwapRatio_{12}}=(187.1666434-187.196292)*100=-2.9560-2.96485
Ask(AskOut12−Ask12)×AskRatio12({AskOut_{12}}-{Ask_{12}}) \times {AskRatio_{12}}=(187.3454179-187.362546)*100=-1.7090-1.71281

Cross Currency (Swap Points) Calculation

For currencies converted to USD (the base currency), forward exchange rates are typically quoted not as absolute rates but as points, which represent deviations from the corresponding spot rate.

These points are called forward points or swap points. The forward cross exchange rate between two non-USD currencies can be quoted as forward points or absolute rates. In the former case, the forward points are called cross swap points. The forward rate corresponding to the exchange rate between two forward dates is called the forward/forward rate. Such rates are usually also quoted as forward points.

Calculation Process

Step 1: Calculate Spot Price

Refer to the above section.

Step 2: Calculate Outright Forward Price

Refer to the above section.

Step 3: Calculate Swap Points Based on Forward and Spot Prices

Formula
Bid(BidOut12−Bid12)×SwapRatio12({BidOut_{12}}-{Bid_{12}}) \times {SwapRatio_{12}}
Ask(AskOut12−Ask12)×AskRatio12({AskOut_{12}}-{Ask_{12}}) \times {AskRatio_{12}}
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