Capital requirements, market, credit, and liquidity risk · Introduction Acceptability Bid and Ask...
Transcript of Capital requirements, market, credit, and liquidity risk · Introduction Acceptability Bid and Ask...
Introduction
Acceptability
Bid and Ask
Modeling
EmpiricalResults
Accounting
References
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Capital requirements, market, credit,and liquidity risk
Ernst Eberlein
Department of Mathematical Stochasticsand
Center for Data Analysis and Modeling (FDM)University of Freiburg
Joint work with Dilip Madan and Wim Schoutens.
Croatian Quants DayUniversity of Zagreb, Croatia, May 11, 2012
Introduction
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Modeling
EmpiricalResults
Accounting
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c©Eberlein, Uni Freiburg, 1
Law of one Price
In complete markets and for liquid assets
EQ[X ]
Reality however is incomplete: no perfect hedges
(quick seller)bid price ask price
(quick buyer)
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Acceptability of Cashflows
X random variable: outcome (cashflow) of a risky position
In complete markets: unique pricing kernel given by a probabilitymeasure Q
value of the position: EQ[X ]
position is acceptable if: EQ[X ] ≥ 0
company’s objective is: maximize EQ[X ]
Real markets: incomplete
Instead of a unique probability measure Q we have to consider a set ofprobability measures Q ∈M
EQ[X ] ≥ 0 for all Q ∈M or infQ∈M
EQ[X ] ≥ 0
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Coherent Risk MeasuresSpecification ofM (test measures, generalized scenarios)
Axiomatic theory of risk measures: desirable properties
Monotonicity: X ≥ Y =⇒ %(X ) ≤ %(Y )
Cash invariance: %(X + c) = %(X )− c
Scale invariance: %(λX ) = λ%(X ), λ ≥ 0
Subadditivity: %(X + Y ) ≤ %(X ) + %(Y )
Examples: Value at Risk (VaR)Tail-VaR (expected shortfall)
General risk measure: %m(X ) = −∫ 1
0qu(X )m(du)
Any coherent risk measure has a representation
%(X ) = − infQ∈M
EQ[X ]
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Operationalization
Link between acceptability and concave distortions(Cherny and Madan (2009))
→ Concave distortions
Assume acceptability is completely defined by the distribution functionof the risk
Ψ(u): concave distribution function on [0, 1]
⇒M the set of supporting measures is given by all measures Qwith density Z = dQ
dP s.t.
EP [(Z − a)+] ≤ supu∈[0,1]
(Ψ(u)− ua) for all a ≥ 0
Acceptability of X with distribution function F (x)∫ +∞
−∞xdΨ(F (x)) ≥ 0
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Distortion
x
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Ψ
(
x)
γ = 2γ = 10γ = 20γ =100
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Families of Distortions (1)
Consider families of distortions (Ψγ)γ≥0
γ stress level
Example: MIN VaR
Ψγ(x) = 1− (1− x)1+γ (0 ≤ x ≤ 1, γ ≥ 0)
Statistical interpretation:
Let γ be an integer, then %γ(X ) = −E(Y ) where
Y law= min{X1, . . . ,Xγ+1}
and X1, . . . ,Xγ+1 are independent draws of X
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Families of Distortions (2)
Further examples: MAX VaR
Ψγ(x) = x1
1+γ (0 ≤ x ≤ 1, γ ≥ 0)
Statistical interpretation: %γ(X ) = −E [Y ]
where Y is a random variable s.t.
max{Y1, . . . ,Yγ+1}law= X
and Y1, . . . ,Yγ+1 are independent draws of Y .
Combining MIN VaR and MAX VaR: MAX MIN VaR
Ψγ(x) = (1− (1− x)1+γ)1
1+γ (0 ≤ x ≤ 1, γ ≥ 0)
Interpretation: %γ(X ) = −E [Y ] with Y s.t.
max{Y1, . . . ,Yγ+1}law= min{X1, . . . ,Xγ+1}
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Families of Distortions (3)
Distortion used: MIN MAX VaR
Ψγ(x) = 1−(
1− x1
1+γ
)1+γ
(0 ≤ x ≤ 1, γ ≥ 0)
%γ(X ) = −E [Y ] with Y s.t. Y law= min{Z1, . . . ,Zγ+1},
max{Z1, . . . ,Zγ+1}law= X
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Families of Distortions (4)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
Ψγ (x
)
γ = 0.50γ = 0.75γ = 1.0γ = 5.0
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Marking Assets and Liabilities
Assets: Cash flow to be received A ≥ 0
Largest value A s.t. A− A is acceptable
⇒ A = infQ∈M
EQ[A]
Bid Price
Liabilities: Cash flow to be paid out L ≥ 0
Smallest value L s.t. L− L is acceptable
⇒ L = supQ∈M
EQ[L]
Ask Price
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Two Price Economics
Range of application: Markets which are not perfectly liquid
Bid and ask prices of a two price economy: not to be confused with bidand ask prices of relatively liquid markets like stock markets
Markets for OTC structured products or structured investments
Both parties typically hold a position out to contract maturity
Liquid markets: one price prevails
Nevertheless liquidity providers will need a bid-ask spread
Bid-ask spreads reflect
• the cost of inventory management
• transaction costs (commissions)
• asymmetric information cost, etc.
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Directional Prices in a TwoPrice Economy
The goal is not to get a single risk neutral price which could beinterpreted as a midpoint between bid and ask
Instead modeling two separate prices at which transactions occur−→ directional prices
Bid price: Minimal conservative valuation s.t. the expectedoutcome will safely exceed this price
Ask price: Maximal valuation s.t. the expected payout will fall belowthis price
−→ specification of the set of valuation possibilities
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Directional Prices in a TwoPrice Economy
Midquote in such two price markets is in general not the risk neutralprice (Carr, Madan, Vicente Alvarez (2011))
Midquotes would generate arbitrage opportunities(Madan, Schoutens (2011))
Pricing of liquidity: nonlinear (infimum and supremum of a set ofvaluations)
Spread: capital reserve
No complete replication: spread is a charge for the need to holdresidual risk
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Relating Bid and Ask Prices
Consider real-valued cashflows X , e.g. swaps
X = X + − X−
⇒ b(X ) = b(X +)− a(X−)
and a(X ) = a(X +)− b(X−)
Valuation as asset: X + is an asset and priced at the bidX− is a liability and priced at the ask
Valuation as liability: X− is an asset and priced at the bidX + is a liability and priced at the ask
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Explicit Bid and Ask PricingBid Price of a cash flow X : Acceptability of X − b(X )
b(X ) =
∫ ∞−∞
xdΨ(FX (x))
Ask Price of a cash flow X : Acceptability of a(X )− X
a(X ) = −∫ ∞−∞
xdΨ(1− FX (−x))
Examples: Calls and Puts
bC(K , t) =
∫ ∞K
(1−Ψ(FSt (x))
)dx
aC(K , t) =
∫ ∞K
Ψ(1− FSt (x))dx
bP(K , t) =
∫ K
0
(1−Ψ(1− FSt (x))
)dx
aP(K , t) =
∫ K
0Ψ(FSt (x))dx
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Modeling of Stock Prices
X self decomposable:for every c, 0 < c < 1: X = cX + X (c) X (c) independent of X
subclass of infinitely divisible prob. distributions
Sato (1991): process (X (t))t≥0 with independent increments
X (t) L= tγX (t ≥ 0)
Write E [exp(X (t))] = exp(−ω(t))
Define the stock price process
S(t) = S(0) exp((r – q)t + X (t) + ω(t))
with rate of return r – q for interest rate r and dividend yield q
Discounted stock price: martingale
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Choice of the Generating Distribution
Variance Gamma: difference of two Gamma distributions
fGamma(x ; a, b) =ba
Γ(a)xa−1 exp(−xb) (x > 0)
Take X = Gamma(a = C, b = M)Y = Gamma(a = C, b = G)
X ,Y independent → X − Y ∼ VG
Alternatively: G = Gamma(a = 1ν, b = 1
ν)
Define X = Normal(θG, σ2G) → X = VG(σ, ν, θ)
Characteristic function
E [exp(iuX )] =
(1− iuθν +
σ2νu2
2
)− 1ν
→ four parameter process
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Accomodating Default in this Model
(∆(t))t≥0 process which starts at one and jumps to 0 atrandom time T
Survival probability given by a Weibull distribution
p(t) = exp(−(
tc
)a)c = characteristic life timea = shape parameter
Define the defaultable stock price
S(t) = S(t)∆(t)p(t)
If Ft (s) = P[S(t) ≤ s], then Ft (s) = 1− p(t) + p(t)Ft (sp(t))
→ 6 parameter model so far
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Liquidity
Refined distortion: MIN MAX VaR 2
Ψ(x) = 1−(
1− x1
1+λ
)1+η
λ: rate at which Ψ′ goes to infinity at 0(coefficient of loss aversion)
η: rate at which Ψ′ goes to 0 at unity(degree of the absence of gain enticement)
λ, η liquidity parameters
λ, η increased: bid prices fall, ask prices rise(acceptable risks are reduced)
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Capital Requirements (Risk)
Reserves expressed as difference between ask and bid prices
For a liability to be acceptable: ask pricecapital or cost of unwinding the position
One gets credit for the bid priceonly excess needs to be held in reserve
Reserves are then responsive to movements of
option surface parameters: σ, ν, θ, γcredit parameters: c, aliquidity parameters: λ, η
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1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
Credit Life Parameter
Opt
ion
Pric
e
Effect of Credit Life Parameter on bid and ask prices of puts and calls
80 one year put ask
120 one year call ask
120 one year call bid
80 one year put bid
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0.05 0.1 0.15 0.28
10
12
14
16
18
20
22
Liquidity parameter
Opt
ion
Pric
e
Effect on bid and ask prices of varying the symmetric Liquidity parameter
120 one year call ask
80 one year put ask
80 one year put bid
120 one year call bid
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Empirical Results
Four banks: BAC, GS, JPM, WFC
Data from 3 years ending Sept. 22, 2010 → 237 calibrations
Movements around Lehman bankruptcy
Hypothetical options portfolio: spot 100; strikes 80, 90, 100, 110, 120;maturities 3 and 6 months
parameters: Aug. 26, 2008; Oct. 8, 2008
Sum over the spreads of the ten options
Pre and post Lehman capital needs on the hypothetical portfolio
BAC GS JPM WFCPre Lehman 2.3684 1.1851 2.0325 4.5648Post Lehman 5.2694 3.8898 4.4995 8.3947
Percentage increase 122.48 228.22 121.38 83.89
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Decomposition in Option,Credit and Liquidity Parameters
Reserve c = g(θ)
∆c ≈(∂g∂Θ
∣∣∣Θ0
)∆Θ
gradient vector at Θ0 i.e. for Aug. 26, 2008
∆Θ: change in parameter value from Aug. 26 to Oct. 8, 2008
Relative parameter contributions to capital requirements (risk sources)from pre to post Lehman bankruptcy
BAC GS JPM WFCσ 0.0406 −0.0657 0.0136 0.1003ν 0.0254 −0.0026 0.0002 0.0192θ 0.3476 0.0526 0.0307 −0.0264γ 0.0409 0.0672 0.0750 0.1077λ 0.0374 0.8513 0.3998 0.0673η 0.4854 0.0972 0.4808 0.7318c −0.0073 0.0 0.0 0.0a 0.0299 0.0 0.0 0.0
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0 50 100 150 200 2500
0.2
0.4
0.6
0.8sigma
0 50 100 150 200 2500
0.5
1
1.5
2
2.5nu
0 50 100 150 200 250−2
−1
0
1
2theta
0 50 100 150 200 2500
0.2
0.4
0.6
0.8gamma
BACGS
JPM WFC
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20 40 60 80 100 120 140 160 180 200 220
0.5
1
1.5
2
2.5
3
3.5
Capital Activity for BAC
20 40 60 80 100 120 140 160 180 200 220
0.2
0.4
0.6
0.8
1Risk Contributions for BAC Capital Activity
Liquidity
Option Surface
Credit
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20 40 60 80 100 120 140 160 180 200 220
0.5
1
1.5
2
Capital Activity for JPM
20 40 60 80 100 120 140 160 180 200 220
0.2
0.4
0.6
0.8
Risk Contributions for JPM Capital Activity
Liquidity
Option Surface
Credit
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Bonds on a Balance Sheet
Balance sheet: investor
assets liabilities
cash equity
bonds...
...
Balance sheet: issuer (bank, corporate)
assets liabilities
cash equity... bonds
...
Rating of the bonds deteriorates: → losses for investor, gains for issuer
Rating of the bonds improves: → gains for investor, losses for issuer
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Financial Accounting Standards
FASB: Financial Accounting Standard Board
US-GAAP: United States General Accepted Accounting Principles
IFRS: International Financial Reporting Standards
IASB: International Accounting Standards Board
Question: What is the correct value of a position?
Answer according to the current standards
Mark to market
Consequences: Volatile behavior in times of crises
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Marking Assets and Liabilities
Assets: Cash flow to be received A ≥ 0
Largest value A s.t. A− A is acceptable
⇒ A = infQ∈M
EQ[A]
Bid Price
Liabilities: Cash flow to be paid out L ≥ 0
Smallest value L s.t. L− L is acceptable
⇒ L = supQ∈M
EQ[L]
Ask Price
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1 1.5 2 2.5 3 3.5 4−1500
−1000
−500
0
500
1000
1500Profits Due to Credit Deterioration
Quarters
PnL
impa
ct
Marked as an asset
Marked as a Liability
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References
Eberlein, E., Madan, D., Schoutens, W.:Capital requirements, the option surface, market, credit, and liquidityrisk. Preprint, University of Freiburg, 2010.
Eberlein, E., Madan, D.:Unbounded liabilities, capital reserve requirements and the taxpayer putoption. Quantitative Finance (2012), to appear.
Eberlein, E., Gehrig, T., Madan, D.:Pricing to acceptability: With applications to valuing one’s own creditrisk. The Journal of Risk (2012), to appear.