CSA Pricing - quantlib.org · CSA Pricing Roland Lichters QuantLib User Meeting, 30 November - 1...
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CSA Pricing
Roland Lichters
QuantLib User Meeting, 30 November - 1 December 2015
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Agenda
About us
Introduction
CSA Pricing
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Agenda
About us
Introduction
CSA Pricing
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About us
www.quaternion.com
News:
I Increasing client base in EuropeI Added Quaternion Risk Management US Inc.I 25 staff and growing in US, IE, UK and DEI Hybrid offering: Software solutions rooted in
quant consulting services for Tier 1 banks
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Agenda
About us
Introduction
CSA Pricing
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Introduction
Exciting things to do in quant finance
... and using QuantLib
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Introduction
Tenor and cross currency basis:
Multi curve pricing, OIS discounting
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Introduction
Negative rates:
Review and revise your pricing models
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Introduction
Derivatives catch up with loans:
Valuation Adjustments for Credit, Funding,Capital
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Introduction
Margin requirements for centrally cleared andOTC derivatives:
More VAs - Margin Value Adjustment
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Introduction
Tighter supervision of Internal Model banks:
Credit Exposure simulation for derivatives
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Introduction
New standard approach for credit risk capital(SA-CCR):
Impact analysis, comparison to CEM Add-On,and internal model EAD
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Introduction
Optimization of capital and funding cost:
A combination of all of the above
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Introduction
Stress tests and sensitivity analysis on top ofxVA and exposures
⇒ It is has got more complicated and morecomputationally demanding
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Introduction
So there is a need for efficient, clever andtransparent tools to cover all this.
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Agenda
About us
Introduction
CSA Pricing
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CSA Pricing and yet another VA
Ideal CSA
I SymmetricI Cash collateral in single currencyI Daily margining
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CSA Pricing and yet another VA
Real CSAs are often more complex
I One-sided thresholdsI Optional bond collateralI Rating triggersI Collateral currency choiceI Cash collateral compounding rate (Eonia)
shifted and floored at zero
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CSA Pricing and yet another VA
New regulations: Increased capital charges forresidual risks e.g. due to asymmetric CSAs
Increased cost of risk mitigation measures
Trend to simplify CSAs (ISDA)
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CSA Pricing and yet another VA
A real case:
I Harmonize CSAs for adequatecompensation, i.e. price the features
I Portfolio: A few thousand Swaps, FX Swaps,Bermudan Swaptions, Inflation Swaps, BMASwaps, CDS, and a bunch of structuredproducts
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What does this involve?
At the high level
I One-sided thresholds⇒ CVA, DVA, FVA
I Bond collateral and rating triggers⇒ Credit modelling
I Collateral currency choice⇒ Cross currency basis modelling
I Cash collateral compounding rate (Eonia)shifted and floored at zero⇒ See next slides
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Market
Eonia curves as of 30/09/2014 and 23/10/2015
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Methodology
The Eonia floor feature in a CSA has twoeffects:
1. It affects the fair value of each derivative inthe netting set
2. It affects the fair value of future interest cashflows paid/received in the margining process
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Discounting
’Ordinary’ OIS Discounting
Discount(T) = E[e−
∫ T0 r(s) ds
]’Floored’ OIS Discounting
Discount(T) = E[e−
∫ T0 r+(s) ds
]does not have closed form, but approximatesolutions.
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Discounting
0.00
0.50
1.00
1.50
2.00
0 2 4 6 8 10
For
war
d R
ate
/ Per
cent
Maturity / Years
EoniaEonia - 10bp
Figure: EONIA forward curve as of 30/09/2014 with negative rates up to 2years. Under the CSA collateral is paid in EUR and based on EONIA - 10 bp.
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Discounting
0.00
0.50
1.00
1.50
2.00
0 2 4 6 8 10
For
war
d R
ate
/ Per
cent
Maturity / Years
Eonia - 10bpEonia - 10bp, floored
Figure: Shifted EONIA forward curve compared to the forward curve withcollateral floor; Hull-White parameters are λ = 0.05 and σ = 0.004.
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Swap Pricing
Needs more than discounting, floating leg:
ΠFloat = E
[n∑
i=1
L(ti−1, ti)× δ(ti−1, ti)× D(ti)
]
L(ti−1, ti) =1
δ(ti−1, ti)
(e∫ ti
ti−1r(s) ds − 1
)
D(ti) = e−∫ ti
0 r+(s) ds
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Swap Pricing
It is tempting to build the ’floored’ discountcurve, keep the forward curves unchanged andto do curve pricing as usual.
Unfortunately, this does not yield the ’exact’Swap value.
Only full MC pricing yields the ’exact’ price, evenfor a vanilla Swap.
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Swap Pricing
The error is noticeable but quite small.
Term no floor with floor diff approx. error2 -665.21 -663.02 2.19 -663.26 -0.243 -1135.64 -1129.83 5.81 -1130.54 -0.714 -1459.73 -1450.40 9.34 -1451.84 -1.445 -1743.69 -1730.53 13.16 -1732.88 -2.357 -2216.14 -2195.89 20.24 -2200.47 -4.58
10 -2757.42 -2728.46 28.96 -2735.91 -7.4512 -3056.87 -3022.23 34.64 -3031.87 -9.6415 -3485.65 -3442.96 42.69 -3455.76 -12.8020 -4204.30 -4148.54 55.77 -4165.78 -17.24
Table: Vanilla swaps, 4% fixed vs Euribor 6m flat, Hull White model withλ = 0.01 and σ = 0.005, market data as of 30/06/2015. Approx: Onlydiscount curve replaced, forward curve unchanged
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Margin Effect, Eonia Floor Value
Move on to the second effect:
Impact on interest on collateral
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Margin Effect, Eonia Floor Value
Without Eonia floor, the value of collateral interest cashflows is
ΠNotFloored = E
[∑i
C(ti) · r(ti) · δi · D(ti+1)
]with
I C(ti): posted collateralI r(ti): overnight rate applicable to period (ti, ti+1)
I δi: day count fraction for period (ti, ti+1)
I D(ti+1): stochastic discount factor
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Margin Effect, Eonia Floor Value
With floored Eonia rates, the value of collateral interestcash flows is
ΠFloored = E
[∑i
C̃(ti) · r+(ti) · δi · D̃(ti+1)
]
where C̃ and D̃ denote floor-induced modified collateralamounts and stochastic discount factors.
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Margin Effect, Eonia Floor Value, COLVA
In summary
ΠFloor = ΠFloored − ΠNotFloored
= E
[∑i
(C̃(ti) D̃(ti+1) δi (r(ti))
+ − C(ti) D(ti+1) δi r(ti))]
≈ E
[∑i
C(ti) D(ti+1) δi (−r(ti))+
].
So-called COLVA, see Burgard-Kjaer, or ...
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Methodology
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Margin Effect, Eonia Floor Value, COLVA
ΠFloor ≈ E
[∑i
C(ti) D(ti+1) δi (−r(ti))+
]is the price of a floor
I paying off when overnight rates are negative, i.e.currently in the money
I with stochastic notional given by the amount of postedcollateral
I potentially with significant correlation between notionaland rate, depending on netting set composition
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Implementation
Quantify floor effects by means of
1. Floor-induced single-trade pricing for interestrate, FX and inflation derivatives usingbespoke pricing engines
2. Monte Carlo simulation of the netting setcollateral in conjunction with simulation of thecompounding rate
QuantLib applied in Quaternion Risk Engine
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Implementation
Monte Carlo Simulation Framework
I IR: Linear Gauss Markov models
I FX: Geometric Brownian Motion driven bystoch. IR differential
I INF: Jarrow Yildirim
I CR: Cox Ingersoll Ross and Black Karasinski
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Example Portfolio as of 23/10/2015
Exposure evolution
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Example Portfolio
Floorlets, intrinsic values
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Example Portfolio
Floorlets
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Example Portfolio
Floor, intrinsic
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Example Portfolio
Floor
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Summary
CSA ’Eonia Floor’ valueI can be seen as COLVAI should be computed by full MC simulation
due to the correlation between postedcollateral and compounding rate
I has significant time valueI is exposed to significant model risk (rate
distribution for negative rates)
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Questions
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