This document discusses XVA, which represents the cost of running an over-the-counter derivatives operation. It includes credit valuation adjustment (CVA) for counterparty risk, debit valuation adjustment (DVA) for one's own default risk, and funding valuation adjustment (FVA) for the cost of funding collateral in pricing portfolios. The document provides examples of interest rate swaps and uses the LIBOR market model and Monte Carlo simulation to model term structures and calculate XVA.

1. XVA
(CREDIT, DEBIT, FUNDING VALUE ADJUSTMENT)
FOR FIXED INCOME PRODUCTS (INTEREST RATE SWAPS)

2. PURPOSE OF XVA
XVA REPRESENTS THE COST OF RUNNING AN OTC DERIVATIVES OPERATIONS
XVA DETERMINE THE AMOUNT OF CAPITAL REQUIRED UNDER BASEL III
RISK FACTORS
COUNTERPARTY RISK (CREDIT VALUATION ADJUSTMENT)
RISK OF ONE’S OWN DEFAULT(DEBIT VALUATION ADJUSTMENT)
FUNDING COLLATERAL IN PRICING TRADING PORTFOLIO(FUNDING
VALUATION ADJUSTMENT)
EXAMPLE - BANK A TRADES WITH BANK B OR INTEL OR MICROSOFT

3. FINANCIAL CRISIS 2008
SUBPRIME
MORTGAGES
MORTGAGES WHERE RISK OF DEFAULT IS HIGHER DUE TO POOR CREDIT HISTORY OR THE RATIO OF LOAN TO VALUE IS HIGH OR BOTH (Hull, 2012)
DEMISE OF BEAR
STEARNS (GLOBAL
INVESTMENT
BANK)
MARCH 12, 2008 – STOCK CLOSES AT $61.58, AVERAGE TARGET PRICE : $98.87
MARCH 14, 2008 – STOCK CLOSES AT $30.85, AVERAGE TARGET PRICE : $93.62
MARCH 16 , 2008 – JP MORGAN AGREES TO BUY BEAR STEARNS $2 A SHARE
MARCH 17, 2008 – STOCK CLOSES AT $4.81, AVERAGE TARGET PRICE : $2
(https://www.reuters.com)
LEHMAN
BROTHERS
SEPTEMBER 15, 2008 –FILED FOR BANKRUPTCY (https://www.financialexpress.com)

4. COMPONENTS OF DERIVATIVES PRICES
(BEFORE AND AFTER THE FINANCIAL CRISIS OF 2007-2009) (Andrew Green, 2015)
PRE-CRISIS
 RISK-NEUTRAL PRICE(LIBOR DISCOUNTING)
 HEDGING COSTS
 CVA
 PROFIT
POST-CRISIS
 RISK-NEUTRAL(OIS DISCOUNTING)
 HEDGING COSTS
 CVA & DVA
 PROFIT
 FVA(INCLUDING COST OF LIQUIDITY BUFFER)
 KVA(LIFETIME COST OF CAPITAL)
 MVA(MARGIN COST)
 TVA(TAX ON PROFITS/LOSSES)

5. INTEREST RATE SWAP (HULL, 2012)
INTEL BANK
(N=1000,000.00)
MICROSOFT
LIBOR + 20 BPS LIBOR + 60BPS
4.15%3.35%
Tenor Libor Rates Zero Curve Discount
Factor
Forward
Rates
Swap Rates Fixed
Cashflows
Floating
Cashflows
Net
Cashflows
Fixed PV Floating PV Net Value
1 0.01 0.01 0.990099 0.01 0.0338756 33,500.00 12,000.00 -21,500.00 1,016,190.00 1,028,790.00 12,600.00
2 0.018 0.018 0.964949 0.0260634 0.0356391 33,500.00 28,063.40 -5,436.60 981,532.00 1,016,910.00 35,378.00
3 0.024 0.024 0.931323 0.0361063 0.0363819 33,500.00 38,106.30 4,606.30 947,759.00 989,828.00 42,069.00
4 0.029 0.029 0.891946 0.044147 0.0364042 33,500.00 46,147.00 12,647.00 915,163.00 954,339.00 39,176.00
5 0.033 0.033 0.850156 0.0491561 0.0357537 33,500.00 51,156.10 17,656.10 883,944.00 913,178.00 29,234.00
6 0.0335 0.820611 0.0360036 0.034587 33,500.00 38,003.60 4,503.60 854,189.00 869,688.00 15,499.00
7 0.034 0.034 0.791327 0.0370051 0.034457 33,500.00 39,005.10 5,505.10 825,468.00 838,502.00 13,034.00
Zero Curve
𝑅 𝑡 = 𝑅 𝑡𝑖 +
𝑅 𝑡 𝑖+1 −𝑅(𝑡 𝑖)
𝑡 𝑖+1 − 𝑡 𝑖
× 𝑡 − 𝑡𝑖
Discount Factors
𝐷𝐹 𝑡𝑖 =
1
(1+𝑅 𝑡 𝑖 ) 𝑡 𝑖
Forward Rates
𝐿 𝐹 𝑡𝑖 =
𝐷𝐹 𝑡 𝑖 −1
𝐷𝐹 𝑡 𝑖
- 1
Swap Rate
𝑆(𝑡𝑖) =
𝐷𝐹 𝑡 𝑖 −𝐷𝐹(𝑡 𝑛)
𝑗=𝑖+1
𝑗=𝑛
𝐷𝐹 𝑡 𝑗

6. LIBOR MARKET MODEL
LFM – lognormal Forward-LIBOR model
LSM – lognormal Forward-SWAP model

7. LFM – LOGNORMAL FORWARD-LIBOR MODEL
 𝑑𝐿 𝑛 𝑡 = 𝜇 𝑛 𝑡 𝐿 𝑛 𝑡 𝑑𝑡 + 𝜎 𝑛
⊺ 𝑡 𝐿 𝑛(𝑡)𝑑𝑊(𝑡) - generic SDE for LFM
 Solution to SDE 𝐿 𝑛(𝑡𝑖+1) = 𝐿 𝑛(𝑡𝑖)exp 𝜇 𝑛
𝑄 𝑁
(𝑡𝑖) −
1
2
𝜎 𝑛
⊺ 𝑡𝑖 𝜎 𝑛(𝑡𝑖) 𝛿𝑖 + 𝜎 𝑛
⊺ 𝑡 𝛿𝑖 𝑍𝑡 𝑖
 Where
 𝜇 𝑛
𝑄 𝑁
𝑡𝑖 = 𝑗=𝑙
𝑛 𝛿 𝑗 𝐿 𝑗 𝑡
1+𝛿 𝑗 𝐿 𝑗 𝑡
𝜎 𝑛
⊺ 𝑡𝑖 𝜎𝑗 𝑡𝑖 𝜌 𝑛,𝑗 𝑡𝑖 (Drift Term under Spot measure)
 𝜎𝑗 𝑡𝑖 = 𝑎 + 𝑏𝛿𝑖 . exp −𝑐. 𝛿𝑖 + 𝑑 (Functional Form)
 𝜌 𝑛,𝑗 𝑡 = exp(−𝛼. 𝑛 − 𝑗 ) (Functional Form)
 𝑄 𝑁
- Spot measure (EMM)
 𝛿𝑖 = t 𝑖+1 − 𝑡𝑖
(Fries, 2007),(Glasserman,2004)

8. MONTE CARLO SIMULATION LFM TERM STRUCTURE
𝒕𝒊 𝑳 𝟎 𝑳 𝟏 𝑳 𝟐 𝑳 𝟑 𝑳 𝟒 𝑳 𝟓 𝑳 𝟔 𝑳 𝟕 𝑳 𝟖 𝑳 𝟗 𝑳 𝟏𝟎
1 0.01 0.009136 0.012514 0.006653 0.007444 0.010372 0.009118 0.009549 0.008374 0.010654 0.008675
2 0.026063 0.032592 0.017328 0.019387 0.027013 0.023748 0.024871 0.021809 0.027749 0.022593 0.016244
3 0.036106 0.023961 0.026808 0.037354 0.032839 0.034391 0.030157 0.03837 0.031242 0.022461 0.041197
4 0.044147 0.032695 0.045557 0.040051 0.041944 0.036779 0.046797 0.038103 0.027394 0.050244 0.052152
5 0.049156 0.050571 0.044459 0.04656 0.040827 0.051948 0.042297 0.030409 0.055774 0.057892 0.041901
6 0.036004 0.032453 0.033987 0.029802 0.03792 0.030875 0.022198 0.040713 0.042259 0.030586 0.036767
7 0.037005 0.034845 0.030554 0.038877 0.031654 0.022758 0.04174 0.043325 0.031358 0.037694 0.050317
8 0.03667 0.030199 0.038425 0.031286 0.022493 0.041255 0.042822 0.030993 0.037256 0.049732 0.058036
9 0.037337 0.039024 0.031774 0.022844 0.041899 0.04349 0.031477 0.037838 0.050508 0.058941 0.06181

9. XVA
CVA – Credit Valuation Adjustment
DVA – Debit Valuation Adjustment
FVA – Funding Valuation Adjustment

10. CVA – CREDIT VALUATION ADJUSTMENT
 𝐶𝑉𝐴 𝑎𝑝𝑝𝑟𝑜𝑥 = 1 − 𝑅 𝑖=2
𝑛
[ Φ 𝑡𝑖 − Φ 𝑡𝑖−1 𝐸 𝑒 𝑜
𝑡 𝑖 𝑟 𝑢 𝑑𝑢
𝑉𝑡𝑖
+
]
 Where
 𝑉𝑡 𝑖
+
= max 𝑉𝑡 𝑖
, 0 = 𝐸𝑥𝑝𝑜𝑠𝑢𝑟𝑒 = 𝑁𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑤𝑎𝑝 = 𝐹𝑙𝑜𝑎𝑡𝑖𝑛𝑔𝑃𝑉 − 𝐹𝑖𝑥𝑒𝑑 𝑃𝑉
 𝐸 𝑒 𝑜
𝑡 𝑖 𝑟 𝑢 𝑑𝑢
𝑉𝑡 𝑖
+
− 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑒𝑑 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝐸𝑥𝑝𝑜𝑠𝑢𝑟𝑒
 Φ 𝑡𝑖 − 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 = 1 − e
−𝐶×𝑡 𝑖
1−𝑅 𝑤ℎ𝑒𝑟𝑒 𝑐 = 𝑐𝑑𝑠 𝑠𝑝𝑟𝑒𝑎𝑑
 𝑅 − 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑅𝑎𝑡𝑒
(Andrew Green, 2015)

11. CVA - MONTE CARLO SIMULATION

12. DVA – DEBIT
VALUATION
ADJUSTMENT
 𝐵𝐶𝑉𝐴 = 𝐶𝑉𝐴 + 𝐷𝑉𝐴
 𝐵𝐶𝑉𝐴 = 1 − 𝑅 𝐶 𝑖=2
𝑛
Φ 𝑡𝑖−1 < 𝜏 𝐶 < 𝑡𝑖 ∩ ( 𝜏 𝐷 >

13. FVA – FUNDING VALUATION ADJUSTMENT
 𝐹𝑉𝐴 = − 1 − 𝑅 𝐵 𝑡
𝑇
𝜆 𝐵 𝑠 𝑒− 𝑡
𝑠
𝑟 𝑢 +𝜆 𝐵 𝑢 +𝜆 𝐶 𝑢 𝑑𝑢
𝐸 𝑉 𝑠 𝑑𝑠
 Where
 𝑉 𝑠 − 𝑁𝑒𝑡 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒
 𝑅 𝐵 = 𝐵𝑜𝑛𝑑𝑠 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑅𝑎𝑡𝑒 𝑤ℎ𝑒𝑛 𝑓𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝐵𝑜𝑛𝑑𝑠
 𝜆 𝐵 = 𝐻𝑎𝑧𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 𝑓𝑜𝑟 𝐵𝑎𝑛𝑘′
𝑠 𝑑𝑒𝑓𝑎𝑢𝑙𝑡
 𝑆𝑝𝑟𝑒𝑎𝑑 𝑜𝑓 𝑧𝑒𝑟𝑜 − 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑍𝐶 𝑏𝑜𝑛𝑑𝑠
 𝜆 𝐶 = 𝐻𝑎𝑧𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 𝑓𝑜𝑟 𝐶𝑜𝑢𝑛𝑡𝑒𝑟𝑝𝑎𝑟𝑡𝑦′
𝑠 𝑑𝑒𝑓𝑎𝑢𝑙𝑡
 𝑦𝑖𝑒𝑙𝑑 𝑜𝑛 𝐶𝑃 𝑏𝑜𝑛𝑑 − 𝐶𝑃 𝑏𝑜𝑛𝑑 𝑟𝑒𝑝𝑜 𝑟𝑎𝑡𝑒
 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑓𝑖𝑛𝑎𝑛𝑐𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑜𝑓 𝐶𝑃 𝑏𝑜𝑛𝑑
(Andrew Green, 2015)

14. THE FUTURE (ANDREW GREEN, 2015)
XVA IS GROWING WITH
MORE ADJUSMENTS BEING
ADDED TO THE FRAMEWORK
XVA IS ABOUT EXPLAINING
COSTS OF DOING
DERIVATIVES BUSINESS
THESE COSTS WERE ALWAYS
PRESENT BUT IGNORED
SOME COSTS ARE NEW AND
ASSOCIATED WITH CHANGES
IN THE MARKET

15. Q & A