Choosing the Right Spread
Consistent Modelling of Funding and Tenor Basis
Sebastian Schlenkrich
QuantLib Workshop
Düsseldorf, December 4, 2014
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Agenda
1. Refresher Multi-Curve Pricing
›
Tenor- and Funding-Specific Yield Curves
›
Why Do We Need to Model the Basis?
2. Modelling Deterministic Tenor and Funding Basis
›
Continuous Compounded Funding Spreads
›
Simple and Continuous Compounded Tenor Spreads
3. Consistent Payoff-Adjustments for Multiple Funding Curves
›
Why Not Just Substitute Discount Curves?
›
What Can Go Wrong with Simple Compounded Spreads?
4. Deterministic Tenor and Funding Basis in QuantLib
› Where Is the “Best” Place to Model the Basis?
› Instruments, Models or Pricing Engines
5. Summary and References
2014-12-04 | Choosing the Right Spread
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Refresher Multi-Curve Pricing
›
Tenor- and Funding-Specific Yield Curves
›
Why Do We Need to Model the Basis?
2014-12-04 | Choosing the Right Spread | Refresher Multi-Curve Pricing
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Pre-Crisis Yield Curve Modelling
Single-Curve Setting
» Discount Factors
Interest Rate
𝑃 𝑡, 𝑇 =
𝑇
𝑒 − 𝑡 𝑓 𝑡,𝑠 𝑑𝑠
= 𝑒 −𝑧
𝑇 𝑇
» Forward Libor Rates
Yield Curve with 𝑃(𝑡, 𝑇)
𝑃(𝑡, 𝑇 ′ )
1
𝐿 𝑡, 𝑇 , 𝑇 =
−1
𝑃(𝑡, 𝑇)
Δ
′
» (Discounted) Libor Coupons
Maturity
𝐿 𝑡, 𝑇 ′ , 𝑇 ⋅ Δ ⋅ 𝑃(𝑡, 𝑇) = 𝑃(𝑡, 𝑇′) - 𝑃(𝑡, 𝑇)
» Derivative payoffs are expressed in terms of single interest rate curve
» Term structure models describe single interest rate curve dynamics, e.g. in terms of
› Continuous compounded forward rate
𝑓 𝑡, 𝑇 ,
› Short rate
𝑟 𝑡 = 𝑓 𝑡, 𝑡 , or
› Simple compounded (Libor) forward rate
𝐿(𝑡, 𝑇 ′ , 𝑇)
2014-12-04 | Choosing the Right Spread | Refresher Multi-Curve Pricing
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Differentiating Forwarding and Discounting Curves
Incorporate Tenor Forwarding Curve (e.g. 3M/6M Libor Curves)
Interest Rate
Forwarding Curve with 𝑃Δ (𝑡, 𝑇)
Discount Curve with 𝑃(𝑡, 𝑇)
» (Pseudo) Discount Factors 𝑃Δ 𝑡, 𝑇
» Forward Libor Rates
𝑃Δ (𝑡, 𝑇 ′ )
1
Δ
′
𝐿 𝑡, 𝑇 , 𝑇 = Δ
−1
Δ
𝑃 (𝑡, 𝑇)
» (Discounted) Libor Coupons
𝐿Δ 𝑡, 𝑇 ′ , 𝑇 ⋅ Δ ⋅ 𝑃(𝑡, 𝑇)
Maturity
» Derivative payoffs are expressed in terms of two interest rate curves
› Discount Curve
› Tenor-specific Forwarding Curve
» Often some deterministic spread assumption is applied to allow using available models
2014-12-04 | Choosing the Right Spread | Refresher Multi-Curve Pricing
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Differentiating Funding Curves
Incorporate Funding-specific Discounting Curve (e.g. Cross-Currency Funding)
Interest Rate
Forwarding Curve with 𝑃Δ (𝑡, 𝑇)
OIS Discounting Curve with 𝑃𝑂𝐼𝑆 (𝑡, 𝑇)
XCY Discounting Curve with 𝑃 𝑋𝐶𝑌 (𝑡, 𝑇)
» Forward Libor Rates 𝐿Δ 𝑡, 𝑇 ′ , 𝑇
» Cash Collateralizes Discounting with
𝑃𝑂𝐼𝑆 𝑡, 𝑇
» Cross-Currency Collateralizes
Discounting with 𝑃 𝑋𝐶𝑌 (𝑡, 𝑇)
Maturity
» Use Case:
› Calibrate model to cash collateralized (Eonia/OIS discounting) swaptions based on 6M
Euribor Forwards
› Use model to price a USD cash collateralized (XCY discounting) derivative
» Discounting spread could also originate from uncollateralised discounting or credit spread
2014-12-04 | Choosing the Right Spread | Refresher Multi-Curve Pricing
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Why Do We Need to Model the Basis?
Linear Products (e.g. Swaps)
Interest Rate
Forwarding Curve with 𝑃Δ (𝑡, 𝑇)
OIS Discounting Curve with 𝑃𝑂𝐼𝑆 (𝑡, 𝑇)
XCY Discounting Curve with 𝑃 𝑋𝐶𝑌 (𝑡, 𝑇)
» Only need 𝐿Δ 𝑡, 𝑇 ′ , 𝑇 , 𝑃𝑂𝐼𝑆 𝑡, 𝑇 ,
𝑃 𝑋𝐶𝑌 (𝑡, 𝑇)
» Require multi-curve bootstrapping
» Relation between curves irrelevant
Maturity
Classical Term Structure Models
» Describe dynamics of only one curve
» Payoffs of Exotics may depend on various curves
Relation between curves required to evaluate exotic rate option payoffs
2014-12-04 | Choosing the Right Spread | Refresher Multi-Curve Pricing
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Tenor and Funding Basis Spreads
Forwarding Curve with 𝑃Δ (𝑡, 𝑇)
Interest Rate
Tenor basis spread
OIS Discounting Curve with 𝑃𝑂𝐼𝑆 (𝑡, 𝑇)
Backbone curve
Funding basis spread
XCY Discounting Curve with 𝑃 𝑋𝐶𝑌 (𝑡, 𝑇)
Maturity
Multi-curve modelling via backbone curve plus basis spreads
2014-12-04 | Choosing the Right Spread | Refresher Multi-Curve Pricing
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Our Focus: Relation Between Tenor and Funding Basis Spreads
Tenor “plus” funding
spread
Interest Rate
Tenor basis spread
Backbone curve
Funding basis spread
Maturity
Multi-curve modelling requires consistent treatment of basis spreads
2014-12-04 | Choosing the Right Spread | Refresher Multi-Curve Pricing
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Modelling Deterministic Tenor and Funding Basis
›
Continuous Compounded Funding Spreads
›
Simple and Continuous Compounded Tenor Spreads
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Continuous Compounded Funding Basis
Interest Rate
Discount Factors
OIS Discounting Curve with 𝑃𝑂𝐼𝑆 (𝑡, 𝑇)
𝑃
Funding basis spread
𝑂𝐼𝑆
𝑡, 𝑇 =
𝑡, 𝑇 =
𝑋𝐶𝑌 𝑡,𝑠 𝑑𝑠
𝑒− 𝑡 𝑓
XCY Discounting Curve with 𝑃 𝑋𝐶𝑌 (𝑡, 𝑇)
𝑃
𝑋𝐶𝑌
𝑇
𝑂𝐼𝑆
𝑒 − 𝑡 𝑓 𝑡,𝑠 𝑑𝑠
𝑇
Maturity
Continuous Compounded Funding Spread
𝒔 𝒕, 𝑻 = 𝒇𝑿𝑪𝒀 𝒕, 𝑻 − 𝒇𝑶𝑰𝑺 𝒕, 𝑻
Multiplicative Discount Factor Relation
𝑃 𝑋𝐶𝑌 𝑡, 𝑇 =𝑃𝑂𝐼𝑆 𝑡, 𝑇 ⋅ 𝐷(𝑡; 𝑡, 𝑇) with
𝐷 𝑡; 𝑇′, 𝑇 =
𝑇
𝑒 − 𝑇′ 𝑠 𝑡,𝑠 𝑑𝑠
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Deterministic Funding Basis
Continuous Compounded Funding Spread
𝑠 𝑡, 𝑇 = 𝑓 𝑋𝐶𝑌 𝑡, 𝑇 − 𝑓 𝑂𝐼𝑆 𝑡, 𝑇
Assumption: 𝒔(𝒕, 𝑻) is a deterministic function of t for all T
Forward Rate Modelling Relation
𝑑𝑓 𝑋𝐶𝑌 𝑡, 𝑇 = 𝑑𝑓 𝑂𝐼𝑆 𝑡, 𝑇 +
𝜕𝑠 𝑡, 𝑇
𝑑𝑡
𝜕𝑡
» Equivalent forward rate volatility for OIS and XCY curve
» Equivalent volatilities of OIS and XCY zero coupon bonds 𝑃𝑂𝐼𝑆 𝑡, 𝑇 and 𝑃 𝑋𝐶𝑌 𝑡, 𝑇
» 𝑇-forward meassure associated to numerairs 𝑃𝑂𝐼𝑆 𝑡, 𝑇 and 𝑃 𝑋𝐶𝑌 𝑡, 𝑇 coincide, i.e.,
𝑬𝑻,𝑿𝑪𝒀 ⋅ = 𝑬𝑻,𝑶𝑰𝑺 [⋅]
No convexity adjustment for switching between OIS and XCY discounting
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Term Structure Models Using Deterministic Funding Basis
Forward Rate Modelling Relation
𝑑𝑓 𝑂𝐼𝑆 𝑡, 𝑇 = ⋅ 𝑑𝑡 + 𝜎 𝑡, 𝑇 𝑑𝑊(𝑡)
𝑑𝑓 𝑋𝐶𝑌 𝑡, 𝑇 = 𝑑𝑓 𝑂𝐼𝑆 𝑡, 𝑇 +
𝜕𝑠 𝑡, 𝑇
𝑑𝑡
𝜕𝑡
Derivative Pricing
Model Calibration
» Model OIS curve 𝑓 𝑂𝐼𝑆 𝑡, 𝑇
» Model XCY curve 𝑓 𝑋𝐶𝑌 𝑡, 𝑇
» Calibrate OIS curve based model
» Use OIS curve based vol structure
𝜎 𝑡, 𝑇
parameters
» In particular vol structure 𝜎 𝑡, 𝑇
» Substitute 𝑓 𝑂𝐼𝑆 = 𝑓 𝑋𝑌𝐶 + 𝑠
Model parameters can be reused under deterministic funding basis assumption
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Simple Compounded Tenor Basis
Interest Rate
Forwarding Curve with 𝑃Δ (𝑡, 𝑇)
Tenor basis spread
Forward Libor Rates
𝑃Δ (𝑡, 𝑇 ′ )
1
𝐿 𝑡, 𝑇 , 𝑇 = Δ
−1
Δ
𝑃 (𝑡, 𝑇)
Δ
′
OIS Discounting Curve with 𝑃𝑂𝐼𝑆 (𝑡, 𝑇)
OIS Forwards
𝐿OIS
OIS
′
𝑃
(𝑡,
𝑇
)
1
𝑡, 𝑇 ′ , 𝑇 = OIS
−1
Δ
𝑃 (𝑡, 𝑇)
Maturity
Simple Compounded Tenor Spread
𝑩 𝒕, 𝑻 = 𝑳𝚫 𝒕, 𝑻′ , 𝑻 − 𝑳𝑶𝑰𝑺 𝒕, 𝑻′ , 𝑻
Assumption: 𝑩(𝒕, 𝑻) is a deterministic function of t for all T
Payoff adjustment at event date 𝑡𝑒
𝐿Δ 𝑡𝑒 , 𝑇 ′ , 𝑇 = 𝐿OIS 𝑡𝑒 , 𝑇 ′ , 𝑇 + 𝐵 𝑡𝑒 , 𝑇
Tenor basis results in static payoff adjustment, e.g. shift in strike for caplets
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Simple Compounded Tenor Basis with XCY Discounting
Tenor “plus” funding spread
Tenor Basis
Interest Rate
𝐵 𝑡, 𝑇 = 𝐿Δ 𝑡, 𝑇 ′ , 𝑇 − 𝐿𝑂𝐼𝑆 𝑡, 𝑇 ′ , 𝑇
Funding Basis
𝑃 𝑋𝐶𝑌 𝑡, 𝑇 =𝑃𝑂𝐼𝑆 𝑡, 𝑇 ⋅ 𝐷(𝑡; 𝑡, 𝑇)
Maturity
Payoff adjustment at event date 𝑡𝑒
𝐿Δ 𝑡𝑒 , 𝑇 ′ , 𝑇 = 𝐿OIS 𝑡𝑒 , 𝑇 ′ , 𝑇 + 𝐵 𝑡𝑒 , 𝑇
′
XCY
= 𝐷 𝑡𝑒 , 𝑇 , 𝑇 ⋅ 𝐿
𝐷 𝑡𝑒 , 𝑇 ′ , 𝑇 − 1
𝑡𝑒 , 𝑇 , 𝑇 + 𝐵 𝑡𝑒 , 𝑇 +
Δ
′
Both tenor and funding basis required for static payoff adjustment
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Continuous Compounded Tenor Basis
Continuous Compounded Tenor Forward Rates
𝑓 Δ 𝑡, 𝑇 = −
𝜕
ln 𝑃Δ (𝑡, 𝑇)
𝜕𝑇
Continuous Compounded Tenor Spread
𝒃 𝒕, 𝑻 = 𝒇𝚫 𝒕, 𝑻 − 𝒇𝑶𝑰𝑺 𝒕, 𝑻
Assumption: 𝒃(𝒕, 𝑻) is a deterministic function of t for all T
Multiplicative Discount Factor Relation
Δ
𝑃 𝑡, 𝑇 = 𝑃
OIS
𝑡, 𝑇 ⋅ 𝐷𝑏 𝑡, 𝑡, 𝑇
−1
with 𝐷𝑏 𝑡, 𝑡, 𝑇 = 𝑒
𝑇
𝑇′ 𝑏
𝑡,𝑢 𝑑𝑢
Payoff adjustment at event date 𝑡𝑒
Δ
′
′
OIS
𝐿 𝑡𝑒 , 𝑇 , 𝑇 = 𝐷𝑏 𝑡𝑒 , 𝑇 , 𝑇 ⋅ 𝐿
𝐷𝑏 𝑡𝑒 , 𝑇 ′ , 𝑇 − 1
𝑡𝑒 , 𝑇 , 𝑇 +
Δ
′
Payoff adjustment results in affine transformation of OIS forwards
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Continuous Compounded Tenor Basis with XCY Discounting
Tenor “plus” funding spread
Tenor Basis
Interest Rate
𝑃Δ 𝑡, 𝑇 = 𝑃OIS 𝑡, 𝑇 ⋅ 𝐷𝑏 𝑡, 𝑡, 𝑇
−1
Funding Basis
𝑃 𝑋𝐶𝑌 𝑡, 𝑇 =𝑃𝑂𝐼𝑆 𝑡, 𝑇 ⋅ 𝐷(𝑡; 𝑡, 𝑇)
Maturity
Payoff adjustment at event date 𝑡𝑒
Δ
′
′
OIS
𝐿 𝑡𝑒 , 𝑇 , 𝑇 = 𝐷𝑏 𝑡𝑒 , 𝑇 , 𝑇 ⋅ 𝐿
′
𝐷𝑏 𝑡𝑒 , 𝑇 ′ , 𝑇 − 1
𝑡𝑒 , 𝑇 , 𝑇 +
Δ
′
′
XCY
= 𝐷𝑏 𝑡𝑒 , 𝑇 , 𝑇 ⋅ 𝐷 𝑡𝑒 , 𝑇 , 𝑇 ⋅ 𝐿
𝐷𝑏 𝑡𝑒 , 𝑇 ′ , 𝑇 ⋅ 𝐷 𝑡𝑒 , 𝑇 ′ , 𝑇 − 1
𝑡𝑒 , 𝑇 , 𝑇 +
Δ
′
Affine transformation payoff adjustment structure is preserved
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Continuous Compounded Tenor Basis with XCY Discounting (2)
Payoff adjustment at event date 𝑡𝑒
Δ
′
′
XCY
𝐿 𝑡𝑒 , 𝑇 , 𝑇 = 𝐷𝑏,𝑋𝐶𝑌 𝑡𝑒 , 𝑇 , 𝑇 ⋅ 𝐿
𝐷𝑏,𝑋𝐶𝑌 𝑡𝑒 , 𝑇 ′ , 𝑇 − 1
𝑡𝑒 , 𝑇 , 𝑇 +
Δ
′
with
𝐷𝑏,𝑋𝐶𝑌 𝑡𝑒 , 𝑇 ′ , 𝑇 = 𝐷𝑏 𝑡𝑒 , 𝑇 ′ , 𝑇 ⋅ 𝐷 𝑡𝑒 , 𝑇 ′ , 𝑇
=𝑒
𝑇
𝑇′
𝑏 𝑡,𝑢 𝑑𝑢
=𝑒
𝑇
𝑇′
𝑓Δ 𝑡,𝑢 −𝑓𝑂𝐼𝑆 𝑡,𝑢 𝑑𝑢
=𝑒
𝑇
𝑇′
𝑓Δ 𝑡,𝑢 −𝑓𝑋𝐶𝑌 𝑡,𝑢 𝑑𝑢
⋅ 𝑒−
𝑇
𝑇′
𝑠 𝑡,𝑢 𝑑𝑢
⋅
𝑇
𝑋𝐶𝑌 𝑡,𝑢 −𝑓 𝑂𝐼𝑆 𝑡,𝑢 𝑑𝑢
𝑒 − 𝑇′ 𝑓
Cont. Compounded Spread payoff adjustment is independent of OIS curve
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
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Summary Funding and Tenor Basis Payoff Adjustments
𝑃 𝑋𝐶𝑌 = 𝑃𝑂𝐼𝑆 ⋅ 𝐷
Funding Basis
Spread Convention
Simple Compounded
𝐿Δ = 𝐿𝑂𝐼𝑆 + 𝐵
Continuous Compounded
𝐿Δ
= 𝐷𝑏 ⋅
𝐿OIS
𝐷𝑏 − 1
+
Δ
Tenor Basis
𝐿Δ = 𝐷 ⋅ 𝐿XCY +𝐵 +
𝐷−1
Δ
2014-12-04 | Choosing the Right Spread | Modelling Deterministic Tenor and Funding Basis
𝐿Δ = 𝐷𝑏,𝑋𝐶𝑌 ⋅ 𝐿XCY +
𝐷𝑏,𝑋𝐶𝑌 − 1
Δ
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Consistent Payoff-Adjustments for Multiple Funding Curves
›
Why Not Just Substitute Discount Curves?
›
What Can Go Wrong with Simple Compounded Spreads?
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
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Modelling Funding Basis vs. Modelling Individual Discount Curves
Interest Rate
Tenor and
Funding Basis
Δ
𝑂𝐼𝑆
𝑋𝐶𝑌
Maturity
XCY discounting (e.g. pricing)
Δ
𝑂𝐼𝑆
Interest Rate
Δ
Interest Rate
Discounting
and Tenor Basis
OIS discounting (e.g. calibration)
Maturity
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
𝑋𝐶𝑌
Maturity
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Consistently Modelling Individual Discount Curves
Consistent Model Dynamics
𝑑𝑓 𝑂𝐼𝑆 𝑡, 𝑇 = ⋅ ⋅ 𝑑𝑡 + 𝜎 𝑂𝐼𝑆 ⋅ 𝑑𝑊 𝑡
𝜕𝑠 𝑡, 𝑇
𝑑𝑓 𝑋𝐶𝑌 𝑡, 𝑇 = 𝑑𝑓 𝑂𝐼𝑆 𝑡, 𝑇 +
𝑑𝑡
𝜕𝑡
= ⋅ ⋅ 𝑑𝑡 + 𝜎 𝑂𝐼𝑆 ⋅ 𝑑𝑊 𝑡
𝜎𝑋𝐶𝑌
» Invariant Volatility structure (with deterministic shift)
𝜎 𝑋𝐶𝑌 𝑓 𝑋𝐶𝑌 𝑡, 𝑇 ; 𝑡, 𝑇 = 𝜎 𝑂𝐼𝑆 𝑓 𝑂𝐼𝑆 𝑡, 𝑇 − 𝑠(𝑡, 𝑇); 𝑡, 𝑇
» In particular unchanged short rate volatility and mean reversion for Hull White model
Uniform Payoff Adjustment Function
OIS discounting
𝐿Δ = 𝐺 𝐿𝑂𝐼𝑆 , 𝑓 𝑂𝐼𝑆 , 𝑓 Δ
XCY discounting
𝐿Δ = 𝐺 𝐿𝑋𝐶𝑌 , 𝑓 𝑋𝐶𝑌 , 𝑓 Δ
» Depends on spread compounding convention
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
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Recall Funding and Tenor Basis Payoff Adjustments
Funding Basis
Spread Convention
𝑃 𝑋𝐶𝑌 = 𝑃𝑂𝐼𝑆 ⋅ 𝐷
Simple Compounded
𝑃𝑂𝐼𝑆
Continuous Compounded
𝐿Δ = 𝐿𝑂𝐼𝑆 + 𝐵
𝐿Δ = 𝐷𝑏 ⋅ 𝐿OIS +
𝐷𝑏 − 1
Δ
Tenor
Basis
𝑃 𝑋𝐶𝑌
𝐿Δ
=𝐷⋅
𝐿XCY
𝐷−1
+𝐵 +
Δ
𝐿Δ
= 𝐷𝑏,𝑋𝐶𝑌 ⋅
𝐿XCY
𝐷𝑏,𝑋𝐶𝑌 − 1
+
Δ
Simple compounded tenor basis spreads do not yield uniform payoff adjustment function
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
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Enforcing Uniform Tenor Basis Payoff Adjustments
Funding Basis
Spread Convention
𝑃𝑂𝐼𝑆
𝑃 𝑋𝐶𝑌 = 𝑃𝑂𝐼𝑆 ⋅ 𝐷
Simple Compounded
Continuous Compounded
𝐿Δ = 𝐿𝑂𝐼𝑆 + 𝐵
𝐿Δ = 𝐷𝑏 ⋅ 𝐿OIS +
𝐷𝑏 − 1
Δ
Tenor
Basis
𝑃 𝑋𝐶𝑌
𝑳𝜟
=
𝑳𝑿𝑪𝒀
+
𝑩𝑿𝑪𝒀
𝐿Δ
= 𝐷𝑏,𝑋𝐶𝑌 ⋅
𝐿XCY
𝐷𝑏,𝑋𝐶𝑌 − 1
+
Δ
Assume a deterministic basis 𝐵 𝑋𝐶𝑌 in addition to deterministic terms 𝐵 and 𝐷
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
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Contradiction of Simultanously Deterministic Terms 𝐷, 𝐵 and 𝐵 𝑋𝐶𝑌
We have
𝑋𝐶𝑌
𝐿
′
𝑂𝐼𝑆
𝑡, 𝑇 , 𝑇 − 𝐿
𝑃 𝑋𝐶𝑌 𝑡, 𝑇 ′
𝑃𝑂𝐼𝑆 𝑡, 𝑇 ′
𝑡, 𝑇 , 𝑇 =
− 𝑂𝐼𝑆
𝑃 𝑋𝐶𝑌 𝑡, 𝑇
𝑃
𝑡, 𝑇
′
1
= 𝐵 𝑡, 𝑇 − 𝐵 𝑋𝐶𝑌 (𝑡, 𝑇)
Δ
It follows
𝑒
𝑇 𝑋𝐶𝑌
𝑇′ 𝑓
𝑡,𝑢 𝑑𝑢
−𝑒
𝑇
𝑇′
𝑓𝑂𝐼𝑆 𝑡,𝑢 𝑑𝑢
= 𝐵 𝑡, 𝑇 − 𝐵 𝑋𝐶𝑌 𝑡, 𝑇
⋅Δ
Solving for the funding spread yields
𝑠 𝑡, 𝑇 = 𝑓
𝑋𝐶𝑌
Though 𝐵 𝑡, 𝑇 − 𝐵 𝑋𝐶𝑌 𝑡, 𝑇
𝑡, 𝑇 − 𝑓
𝑂𝐼𝑆
𝜕
𝐵 𝑡, 𝑇 − 𝐵 𝑋𝐶𝑌 𝑡, 𝑇
𝑡, 𝑇 =
ln 1 +
𝑇 𝑂𝐼𝑆
𝜕𝑇
𝑒 𝑇′ 𝑓 𝑡,𝑢 𝑑𝑢
⋅Δ
⋅ Δ deterministic, 𝑠 𝑡, 𝑇 depends on future forward rates 𝑓 𝑂𝐼𝑆 𝑡,⋅
Simple compounded tenor basis vs. XCY may only yield approximate payoff adjustment
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
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Example: XCY Cash Collateralized Caplet with Simple Comp. Tenor Basis
We have
𝐶𝑝𝑙𝑂𝐼𝑆 𝑡 = 𝑃𝑂𝐼𝑆 𝑡, 𝑇 𝐸 𝑂𝐼𝑆 𝐿 𝑇 ′ , 𝑇 ′ , 𝑇 − 𝑘
𝐶𝑝𝑙 𝑋𝐶𝑌 𝑡 = 𝑃 𝑋𝐶𝑌 𝑡, 𝑇 𝐸 𝑋𝐶𝑌 𝐿 𝑇 ′ , 𝑇 ′ , 𝑇 − 𝑘
+
⋅Δ
+
⋅Δ
From 𝐸 𝑋𝐶𝑌 ⋅ = 𝐸 𝑂𝐼𝑆 ⋅ follows model-independent that
𝐶𝑝𝑙 𝑋𝐶𝑌 𝑡 = 𝐷 𝑡; 𝑡, 𝑇 ⋅ 𝐶𝑝𝑡 𝑂𝐼𝑆 (𝑡)
Rewriting OIS caplet payoff as zero coupon bond put option and Hull White model
𝐶𝑝𝑙𝑂𝐼𝑆 𝑇 ′
𝐶𝑝𝑙
𝑂𝐼𝑆
1
= 1+ 𝑘−𝐵 𝑇 Δ ⋅
− 𝑃𝑂𝐼𝑆 𝑇 ′ , 𝑇
1+ 𝑘−𝐵 𝑇 Δ
𝑡 =𝑃
𝑂𝐼𝑆
+
𝑃𝑂𝐼𝑆 𝑡, 𝑇 ′
1
𝑡, 𝑇 ⋅ 1 + 𝑘 − 𝐵 𝑇 Δ ⋅ 𝐵76
,
, 𝜎 , −1
𝑃𝑂𝐼𝑆 𝑡, 𝑇 1 + 𝑘 − 𝐵 𝑇 Δ 𝑃
′
Discount
Factor
Notional
Black
Formula
Forward
ZCB
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
Strike
Bond Put
Vol Flag
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Correct vs. Simplified Payoff Adjustment XCY Cash Collateralized Caplet
We have
Correct
𝐶𝑝𝑙
𝑋𝐶𝑌
𝑇
′
= 1+ 𝑘−𝐵 𝑇 Δ ⋅
Variance Notional
Simplified
𝐶𝑝𝑙
𝑋𝐶𝑌
𝑇
′
= 1+ 𝑘−𝐵
𝑋𝐶𝑌
𝑇 Δ ⋅
𝐷 𝑇 ′ ,𝑇
1+ 𝑘−𝐵 𝑇 Δ
−𝑃
𝑋𝐶𝑌
+
′
𝑇 ,𝑇
Variance Strike
1
1+ 𝑘−𝐵 𝑋𝐶𝑌 𝑇 Δ
−𝑃
𝑋𝐶𝑌
′
𝑇 ,𝑇
+
Relative Valuation Error
𝑒𝑟𝑟𝑁𝑡𝑙
1 + 𝑘 − 𝐵 𝑋𝐶𝑌 𝑇 Δ
=
− 1 ≈ 𝐿𝑋𝐶𝑌 − 𝐿𝑂𝐼𝑆 Δ
1+ 𝑘−𝐵 𝑇 Δ
𝑒𝑟𝑟𝐵76
𝑃 𝑋𝐶𝑌
𝐵76 𝑋𝐶𝑌
𝑃
=
𝑃 𝑋𝐶𝑌
𝐵76 𝑋𝐶𝑌
𝑃
≈
Φ
2Φ
𝜎𝑃
2
𝜎𝑃
2
𝑡, 𝑇 ′
𝑡, 𝑇
𝑡, 𝑇 ′
𝑡, 𝑇
𝐷 𝑇′, 𝑇
,
, 𝜎 , −1
1+ 𝑘−𝐵 𝑇 Δ 𝑃
−1
𝐷 𝑇′, 𝑇
,
, 𝜎 , −1
1+ 𝑘−𝐵 𝑇 Δ 𝑃
𝐿𝑋𝐶𝑌 − 𝐿𝑂𝐼𝑆 Δ
⋅
⋅ 𝑘 − 𝐿Δ
𝑋𝐶𝑌
𝑂𝐼𝑆
1 + Δ𝐿
− 1 1 + Δ𝐿
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
© d-fine — All rights reserved | 26
Numerical Example XCY Cash Collateralized Caplet
» Yield curves for OIS/XCY/6M Forward flat at 100BP/50BP/200BP respectively
» Black caplet volatility 65%
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
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Summary Consistent Payoff Adjustment
Continuous Compounded Tenor Basis
» Uniform payoff adjustment formula for OIS and XCY discounting
» Consistent multi-curve pricing with Hull White model by substituting discount curve
Simple Compounded Tenor Basis
» Different payoff adjustment formula for OIS and XCY discounting
» Approximations in multi-curve pricing with Hull White model by substituting discount curve
» Valuation error depends on cross currency basis and strike
2014-12-04 | Choosing the Right Spread | Consistent Payoff-Adjustments for Multiple Funding Curves
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Deterministic Tenor and Funding Basis in QuantLib
›
Where Is the “Best” Place to Model the Basis?
›
Transforming Instruments
›
Generalising Models vs. Pricing Engines
2014-12-04 | Choosing the Right Spread | Deterministic Tenor and Funding Basis in QuantLib
© d-fine — All rights reserved | 29
QuantLib Object Model (Simplified)
Instrument
EuropeanSwaption
1
Model
PricingEngine
HullWhiteEngine
1
HullWhiteModel
Where Is the “Best” Place to Model the Basis?
2014-12-04 | Choosing the Right Spread | Deterministic Tenor and Funding Basis in QuantLib
© d-fine — All rights reserved | 30
Generalising Models
EuropeanSwaption
HullWhiteEngine
HullWhiteModel
discTermStructure_
forwTermStructure_
europSwaption()
HullWhiteModelS
HullWhiteModelC
europSwaption()
europSwaption()
Model simple
compounded Basis
Model continuous
compounded Basis
» Keep modelling assumptions and details in one place
» Manage forwTermStructure_ consistent to EuropeanSwaption→Index→TermStructure
2014-12-04 | Choosing the Right Spread | Deterministic Tenor and Funding Basis in QuantLib
© d-fine — All rights reserved | 31
Generalising Pricing Engines
EuropeanSwaption
HullWhiteEngine
swaption2BondOption()
HullWhiteModel
termStructure_
couponBondOption()
HullWhiteEngineS
HullWhiteEngineC
swaption2BondOption()
swaption2BondOption()
Model simple
compounded Basis
Model continuous
compounded Basis
» By design knows forwarding and discounting term structure (no redundant information)
» Holding modelling assumptions out of the pricing model mixed up with instrument data
2014-12-04 | Choosing the Right Spread | Deterministic Tenor and Funding Basis in QuantLib
© d-fine — All rights reserved | 32
Transforming Instruments
EuropeanSwaption
BondOption
HullWhiteEngine
BondOption(
EuropSwaption s,
TermStructure discTS,
SpreadMethod m )
HullWhiteModel
termStructure_
couponBondOption()
Model simple and
continuous
compounded Basis
» Disentangle spread modelling from existing yield curve modelling and pricing
» Requires consistency of discounting curves between BondOption construction and
HullWhiteModel
2014-12-04 | Choosing the Right Spread | Deterministic Tenor and Funding Basis in QuantLib
© d-fine — All rights reserved | 33
Summary and Reference
2014-12-04 | Choosing the Right Spread | Summary and Reference
© d-fine — All rights reserved | 34
Summary and Reference
Modelling Deterministic Tenor and Funding Basis
» Interdependencies of tenor and funding spreads
» Payoff adjustments for simple and continuous compounded tenor spreads
» Consistency for payoff adjustments with multiple funding curves
Continuous compounded spreads appear more favourable
Integrating Tenor and Funding Basis into QuantLib
» Modifying Instrument, PricingEngine or Model classes
Best practice has yet to emerge
Further Reading
S. Schlenkrich, A. Miemiec. Choosing the Right Spread. SSRN Preprint
http://ssrn.com/abstract=2400911. 2014.
2014-12-04 | Choosing the Right Spread | Summary and Reference
© d-fine — All rights reserved | 35
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