FVA for General Instruments: Theory and Practice

FVA for General Instruments:

Theory and Practice

Alexandre Antonov Numerix

Outline

◮ FVA theory framework ◮ Single rate classical models

◮ Multi-rate models for general accounts

◮ Theoretical and Numerical Issues

◮ Approximation for general instruments (including payments

and exercises)

◮ Numerical experiments for a Bermudan option: comparison of

Literature

◮ Fujii et al (2009)

pioneered the funding business for swap prices

◮ Vladimir Piterbarg (2010)

was first to extend the Black-Scholes logic to take into account that the derivative, the hedging instruments, the funding and the possible collateral have different grow rates

◮ Pallavicini et al (2011)

considered a joint framework including credit and funding under risk-neutral pricing with additional costs

◮ Christoph Burgard and Mats Kjaer (2012)

dealt with a joint framework by replication using bonds emitted by the two defaultable Counterparties in the funding strategy

FVA basics: standard theory

Origins of Funding Valuation Adjustment (FVA):

existence of multiple rates corresponding to different possibilities to borrow/lend money

→ the classical arbitrage-free theory should be modified

The classical theory postulates existence of a single rate r(t): all tradable assets grow with this rate (under risk-neutral measure). A portfolio value v(t) satisfied the standard PDE

L(t)v(t) =r(t)v(t)

where L(t) is the evolution operator of the model drivers.

FVA basics: modern theory

The modern market has multiple rates, for example:

◮ raterC – collateral rate (almost risk-free rate) ◮ raterR for asset secured borrowing (“repo”) ◮ raterF for unsecured funding

For example, a fully collateralized payment is discounted with rC

rate, but:

→ What happens if the collateralization is only partial?

Input and assumptions

◮ Postulate evolution of our rates, rC, rR,rF (maybe correlated)

stochastic processes

◮ Our portfolio may contain general instruments (both vanillas

and exotics)

◮ Instrument payments/indexes are considered to be known

functions of our rates.

◮ Portfolio collateral is known function of the portfolio price Definition. The FVA is the difference between the true (modern) price and the base one. The base price is often related with fully collateralized portfolio– this is the only difference between the true and the base models: the payments/indexes coincide.

FVA theory

Piterbarg (2010) calculated a true portfolio value by replication ◮ The replication gives a modified pricing PDE (w.r.t. classical

theory)

◮ The replication is a unique way to determine the price ◮ The default risk (of the bank or counterparty) was not taken

into account while replication

Let V(t) be the portfolio price andC(t) be the collateral, a known function of the portfolio (e.g. C(t) = (V(t))+₎

then by replication →

L(t)V(t) =rCC(t) +rF(V(t)−C(t))

where L(t) is the evolution operator corresponding to rates processes.

It also assumes that the equity grows with the repo rate rR(t)

minus dividend.

The r.h.s. intuition:

◮ The portfolio value is divided in two parts V =C + (V −C) ◮ Part under collateral C ”discounted” with collateral rate rC ◮ Residual part not covered by collateral, V −C, ”discounted”

Solution between payment/exercise dates is obtained via the

Feynman-Kac theorem

L(t)V(t) =rCC(t) +rF(V(t)−C(t))

⇓

V(t) = E

h

e−

RT

t rF(s)dsV(T) Ft i + E Z T t e− Rτ

t rF(s)ds_{(rF(τ)−rC(τ))C(τ)dτ} Ft or, equivalently,

V(t) = Ehe−

RT

t rC(s)ds_V(T) Ft i + E Z T t e− Rτ

t rC(s)ds

(rC(τ)−rF(τ)) (V(τ)−C(τ))dτ

Remark 1. In ABM (2013) we brought the reader’s attention to the inclusion into the replicating self-financing portfolio of risky assets which are related to collateralized and uncollateralized zero coupon bonds. However, the result, written in probabilistic terms, coincides with Piterbarg (2010).

Remark 2. We call the pricing probability measure a

FVA theory: special cases

This PDE is non-linear (except when collateral is a linear function of the price V)

Special cases:

A solution for a non-collateralized deal (C = 0)

V(t) =Ehe−

RT

t rF(s)ds_V(T) Ft

i

A solution for a fully collateralized situation (C =V)

V(t) =Ehe−

RT

t rC(s)ds_V(T) Ft

FVA theory: measure changes

It is important to notice that the numeraire is dissociated from our multi-risk-neutral measure. Indeed, a fully-collateralized security

VC(t) is discounted with the collateralized savings account

VC(t) =E

h

e−

RT t rC(s)ds

VC(T) Ft

i

,

while the uncollateralized security is discounted with the pure funding account

VF(t) =E

h

e−

RT t rF(s)ds

VF(T) Ft

i

.

Note that both conditional expectations are done in the same

Of course, one can change our pricing measure using, for example, the Radon-Nikodym derivative

M(t) =PC(t,T)/BC(t),

where PC(t,T) is a collateralized zero bond

PC(t,T) =E

h

e−

RT t rC(s)ds

| Ft

i

The collateralized deal propagation leads to a standard formula

VC(t) =E

h

e−

RT t rC(s)ds

VC(T) Ft

i

=M(t)ECT

e−

RT

t rC(s)ds_VC(T) 1

M(T) Ft

=PC(t,T)_ECT

VC(T)

PC(T,T) Ft

However, for the pure funding security, such a measure change gives

VF(t) =E

h

e−

RT t rF(s)ds

VF(T) Ft

i

=M(t)ECT

e−

RT t rF(s)ds

VF(T) 1

M(T) Ft

Different rates for borrowing and lending

One can generalize the theory to different rates for posivite/negative parts (Pallavicini et al.). This reflects a distinction between borrowing and lending

◮ Positive part of collateralC

+ growing with raterC+

◮ Negative part of collateral C

− growing with rate rC−

◮ Positive part of funding amount (V −C)+ growing with rate

rF+

◮ Negative part of funding amount (V −C)− growing with rate

rF

−

L(t)V(t) = rC₊C+(t) +rF₊(V(t)−C(t))+ + rC

−C

−_{(t) +rF}

−(V(t)−C(t))

Collateralization of domestic deals in foreign currency

We cover this case in details:

◮ these results are new and the derivation is didactic

Let V(t) be a domestic deal value (expressed in domestic ccy) partially collateralized in the foreign ccy.

Denote foreign attributes using tilde˜and introduce the rates

◮ raterC (˜rC) for fully collateralized borrowing in domestic

(resp., foreign ccy)

◮ raterF (˜rF) for unsecured borrowing in domestic (resp.,

foreign ccy)

For simplicity, suppose these rates being deterministic and the FX-rate X(t) having the following diffusion part

Collateral C(t) (expressed in the foreign ccy) being converted to the domestic ccy is a certain function of the derivative, e.g.

X(t)C(t) = max(V(t),0)

Denote the following money accounts

◮ foreign collateralized money account ˜BC(t) growing with rate

˜

rC(t),

dBC˜ (t) = ˜BC(t) ˜rC(t)dt,

◮ domestic uncollateralized money account BF(t) growing with

a funding rate rF(t),

Han et al. (2013) tried to replicate (erroneously) the derivative price V(t) using

◮ ψC˜ (t) units of money account ˜BC(t) ◮ ψF(t) units of money account BF(t)

We will follow their logic (supposing, for simplicity, that all assets are deterministic), show the contradiction and correct the

Replication portfolio (Han et al. (2013))

V(t) = ˜ψC(t)X(t) ˜BC(t) +ψF(t)BF(t)

Distribution of the money between accounts ˜

ψC(t) ˜BC(t) =C(t)

ψF(t)BF(t) =V(t)−C(t)X(t)

The self-financing property implies

dV(t) = ψC˜ (t)d(X(t) ˜BC(t)) +ψF(t)dBF(t)

= ψC˜ (t) ˜BC(t)dX(t) + ˜ψC(t)X(t) ˜BC(t) ˜rC(t)dt

+ ψF(t)BF(t)rF(t)dt

On the other hand, by Ito,

dV(t) =∂X V(t)dX +

∂tV(t) +1 2X

2_σ2_(t,X)∂2

X V(t)

◮ Diffusion

˜

ψC(t) ˜BC(t) =∂XV(t) ⇔ ∂XV(t) =C(t) (1)

◮ Drift

∂t+1 2X

2_σ2_(t,X)∂2

X

V(t) = ˜ψC(t)X(t) ˜BC(t) ˜rC(t)

+ψF(t)BF(t)rF(t)

∂t+1 2X

2_σ2_(t,X)∂2

X

V(t) =X(t)∂XV(t) ˜rC(t) (2)

+ (V(t)−C(t)X(t))rF(t)

Problem → the priceV(t,x) satisfies two PDEat the same time

(with a terminal condition): equation (1) and (2)

Solution → add to the replication set an account in domestic ccy fully collateralized in the foreign ccy, BC˜, with an interest rate rC˜,

dBC˜ =rC˜BC˜dt

and the foreign collateralization rate ˜rC.

Such account does not exits but can be synthesised as o/n rolling FX-swap: the account interest is related with the o/n FX-swap rate rd,f

Indeed, the o/n FX-swap rate rd,f is an interest rate which makes

the following cashflow exchange having zero price att

◮ at time t: exchange 1 unit of foreign ccy vs. X(t) units of

domestic ccy

◮ at time t+dt: exchangeX(t) (1 +rd

,f dt) of domestic ccy

units vs 1 unit of foreign ccy This means that a cashflow of

X(t) (1 +rd,f dt)−X(t+dt)

Now consider cashflows generated by our account in domestic ccy

BC˜ (with the interestrC˜) fully collateralized in the foreign ccy

◮ at time t: ◮ payB_˜

C(t) in domestic ccy (open account) ◮ receive

B_C˜(t)

X(t) in foreign ccy (receive collateral)

◮ at time t+dt: ◮ receiveB_˜

C(t) (1 +rC˜dt) in domestic ccy (close account

receiving the interest)

◮ pay B_C˜(t)

X(t) (1 + ˜rCdt) in foreign ccy (return collateral with the

collateral interest)

The cashflows at time t+dt in the domestic currency will look like

BC˜(t) (1 +rC˜dt)−X(t+dt) BC˜(t)

X(t) (1 + ˜rCdt)

= BC˜(t)

X(t)

Thus, the o/n FX-swap with rate rd,f and notional B_˜

C(t)

X₍t₎ will

synthesize our account BC˜ with the interest rate

rC˜ =rd,f + ˜rC

Complete the hedging set with amount of money ψC˜(t) in the the

account BC˜ to obtain

V(t) = ˜ψC(t)X(t) ˜BC(t) +ψF(t)BF(t) +ψC˜(t)BC˜

with two constraints

X(t)C(t) = X(t) ˜ψC(t) ˜BC(t) +ψC˜(t)BC˜ V(t)−C(t)X(t) = ψF(t)BF(t)

The self-financing property

Match the diffusion

∂XV(t) = ˜ψC(t) ˜BC(t)

and the drift

∂t+1 2X

2_σ2_(t,X)∂2

X

V(t) =X(t) ˜ψC(t) ˜BC(t) ˜rC(t)

+ψF(t)BF(t)rF(t) +ψC˜(t)BC˜rC˜

∂t+1 2X

2_σ2_(t,X)∂2

X

V(t) =X(t) ˜ψC(t) ˜BC(t) ˜rC(t)

+ (V(t)−C(t)X(t))rF(t)

Finally

∂t+X(rC˜ −˜rC(t))∂X +

1 2X

2_σ2_(t,X)∂2

X

V(t)

= (V(t)−C(t)X(t))rF(t) +C(t)X(t)rC˜(t)

Remark 1. We see that the equivalent FX-process drift (in sort of risk-neutral measure) equals to

rd,f =rC˜ −˜rC(t)

as deduced in Piterbarg (2010). Note that the FX-drift in this replication scheme was obtained ”automatically”.

Remark 2. A special case of a fully foreign-collateralized

(domestic) deal (V(t) =C(t)X(t)) naturally corresponds to the rate rC˜. This was derived in Piterbarg (2012) for domestic zero

FVA theory: general r.h.s.

Further generalization→ arbitrary portfolio parts Φi(V) funded

with interest rates ri give rise to the PDE,

main evolution equation presented in a compact and general form,

L(t)V(t) =X

i

Φi(V(t))ri(t)

with the obvious ”conservation” condition P

iΦi(V) =V Example.

◮ Different rates for borrowing and lending

Φ1(V) =C+(V),r1 =rC+; Φ2(V) =C−(V),r2 =rC−, etc. ◮ Domestic deals with the foreign collateral

funding part Φ1(V) =V −C X,r1=rF

Difficulties with the general approach

◮ The pricing is portfolio-wide because the replication

arguments are applied to the whole portfolio (if the collateral is defined on the portfolio level)

→ it is not clear how we fund an individual instrument:

◮ notion of individual instrument is lost (impossible to write a

pricing PDE on an instrument: its r.h.s. is unknown)

◮ callable instruments price/exercise becomes vague: exercise

condition requires global optimization (very complicated numerically)

◮ Certain numerical difficulties with non-linear r.h.s., e.g. fine

Single-rate model vs. true one

Introduce a single-rate model,our main calculation tool L(t)v(t) =r(t)v(t)

with portfolio price v(t) and rater(t), a deterministic function of the general rates ri(t) (e.g. the collateralized rate).

The difference between the true

L(t)V(t) =X

i

Φi(V(t))ri(t)

and the single rate solution has order of spreads si =ri−r

V(t) =v(t) +O(spread)

Approximation

◮ Exercises:

→ Approximate exercise decisions in the true model by those coming from the single-rate model

◮ Non-linear r.h.s. of the true model: → Linearize the non-linear r.h.s.

→ Approximate the obtained solution by a single adjustment

Approximation workflow

Start with the true PDE (between payment/exercise dates)

L(t)V(t) =X

i

Φi(V(t))ri(t) =r_Φ(V(t))V(t)

where rΦ(V(t)) iseffective non-linear rate

◮ Approximate the rate with corresponding single-rate model,

rΦ(V(t)) =rΦ(v(t)) +O(spread)

◮ Approximate all exercises of the multi-rate model by the

Individual single-rate prices v(k)_{(t) satisfy}

L(t)v(k)(t) =r(t)v(k)(t)

We approximate the true prices as

L(t)Vi(t) =rΦ(v(t))Vi(t)

To continue we need a result from ABM (2013): the exact price of an automatically exercisable instrument (e.g. European-style or Barrier) having different stochastic discount rates before and after exercises.

Suppose an instrument V(t) can be exercised into ˜V(t)

Lx(t)V(t) =h(t) +R(t)V(t)

Lx(t) ˜V(t) =h(t) + ˜R(t) ˜V(t)

Remark 1. We note different discounting rates (R(t) and ˜R(t)) before and and after exercise.

Remark 2. The free rhs term h(t) is interpreted as instantaneous payment.

⇓

The price of the (automatically) callable instrument V(t)can be exactly calculated given instrument prices in the single-rate environment, v(t) and ˜v(t)

Lx(t)v(t) =r(t)V(t)

Lx(t) ˜v(t) =r(t) ˜v(t)

Intuition

Let us emphasize that the effective raterΦ(v(t)) in

L(t)Vi(t) =rΦ(v(t))Vi(t) +O(spread)

is exercise-dependent.

Suppose the portfolio contains only one instrument k, which can be exercised into the ℓk-th one.

If we stay in instrument k, i.e. haven’t exercise prior to t, then the portfolio value v(t) will be the instrumentk value:

v(t) =v(k)(t).

Otherwise, if we already exercised into instrument ℓk, then

v(t) =v(ℓk)_(t)

We obtain

Lx(t)V( k₎

(t) =rΦ(t,v(k)(t))V(k)(t),

Lx(t)V(ℓk)(t) =r_Φ(t,v(ℓk)(t))V(ℓk)(t).

We see that the instrument discount rate before exercise coincides withrΦ(t,v(k)(t)), while the rate after the exercise is

rΦ(t,v(ℓk)(t)).

Remark 1. The rates depend on the single-rate pricesv and are independent of the multi-rate prices V.

Remark 2. The situation corresponds to an option where the discount rate changes after exercise.

We can unify values v(k₎

Future and continuation values

For instrument with exercises one should distinguish future and continuation values (see also AIM (2011))

◮ Continuation valuev(k)(t)

Participates in the PDE and represents the k-th deal value provided that it was not exercises before time t, usually a function of model driving factors at time t

◮ Future value v(k)

0 (t)

Represents the true price of k-th deal taking into account exercises before t, usually a path-dependent function (marked with the subscript (v₀(k)(t)))

Remark 1. Future and continuation values coincide for non-callable deals.

Correct and Naive approximations

Application of the ABM (2013) result leads to the adjustment

∆V(k)(0) =V(k)(0)−v(k)(0)

= −E

Z T

0

du rΦ

v₀(k)(u)−r(u)e−

Ru 0ds rΦ(v0(

k)_(s))

v₀(k)(u)

It is important to stress that that the formula uses the futurevalue

v₀(k)(u) in the effective rate, reflecting the fact that possible exercises change the rate.

We warn against the use of the continuationvalue of the instrument v(k)(u) for the effective rate1

∆V(k)₍₀₎₆₌₋

E

Z T

0

du rΦ(v(k)(u))−r(u)

e−

Ru 0 ds rΦ(v(

k₎

(s))_v(k) 0 (u)

Adjustment on the portfolio level

∆V(0) =X

k

∆V(k)(0)

≃ −E

Z T

0

du FΦ(u,v0(u))e−

Ru

0 ds rΦ(s,v0(s))

≃ −E

Z T

0

du FΦ(u,v0(u))e−

Ru 0 ds

F

Φ(s,v₀₍s))−FΦ(s,0)

v

0(s) e−

Ru 0ds r(s)

where FΦ(t,v) is the “perturbation” of the true PDE’s right-hand

side over the single-rate one

FΦ(t,v)≡

X

i

Φi(v)ri(t)−v r(t) = (r_Φ(v)−r(t))v

Remark. For rare cases the FΦ(t,v) can be nonzero for v = 0

Approximation features

◮ The adjustment is exact for linear r.h.s. (e.g. fully

collateralized deal) and non-callable instruments

◮ The adjustment has the second order error in spreads for

other cases, however, according to our experiments the accuracy is much higher

◮ For non-callable instruments or for a portfolio containing a

single callable instrument, the prices can be obtained numerically even for non-linear r.h.s. (for multiple exotics in portfolio the exercise optimization is very complicated numerically)

◮ The adjustment does not take into account counterparty

Funding and Credit

Certain authors consider the funding and credit issues at the same time

◮ either introducing default processes into the expectations

(Pallavicini et al. (2011))

◮ or considering an explicit portfolio replication by self and

counterparty risky bonds to hedge the corresponding default risks (Burgard-Kjaer (2012))

Given absence of consensus we exclude the default risk from the

funding replication → instead

◮ We consider CVA and DVA as separate adjustments

◮ We leave possibility to add the default indicators into the FVA

average

Remark. Introduced ”inaccuracies” are of second or third orders in spreads, thus, can be ignored due to theoretical/practical

Implementation

◮ Simulate the model ratesri and all payment indexes ◮ Build the single-rate pricing model equipped with Least

Square Monte Carlo (proxy model for the FVA calculations)

◮ Calculate future values v

0(t) for all instruments in portfolio on

this model (can be done independently

”instrument-by-instrument” using the Algorithmic Exposure methods (AIM (2011))– important for parallel computation)

◮ Aggregate the instrument future prices into the portfolio ones

v0(t) =Pkv

(k) 0 (t)

Numerical experiments: model

◮ Yield curves (log-linear interpolation for discount factors) are

set using the rates below (DF(t) =e−R(t)t)

Curve Maturity_{1Y 20Y} model curve 1.50% 2.00% collateral curve 1.50% 2.00% funding curve 2.50% 2.50%

◮ HW model, 1% of vol and 5% of mean-reversion, is written

for the model yield curve.

◮ Spreads between the model short rate and collateral/funding

Numerical experiments: instrument

10Y Bermudan swaption giving annually the right to enter into a 10,000 notional swap where we receive annually the fixed rate and pay semi-annually the floating rate

Float leg fixing dates: 1Y 1.5Y 2Y . . . 9.5Y Float leg payment dates: 1.5Y 2Y 2.5Y . . . 10Y

Fixed leg start dates: 1Y 2Y 3Y . . . 9Y

Fixed leg payment dates: 2Y 3Y 4Y . . . 10Y

Exercise dates: 1Y 2Y 3Y . . . 9Y

Output: prices

We calculate prices

◮ Single-rate price →

non-perturbed classical version for given deterministic model rateL(t)v(t) =r(t)v(t)

◮ Exact price →

Exact Numerical pricing

Formally the exact PDE L(t)V(t) =rΦ(V(t))V(t) has solution

V(t) =E

h

e−

RT

t rΦ(V(s))dsV(T) Ft

i

Choosing small spacing ∆t we approximate

V(t) = _Ehe−

Rt+∆t

t rΦ(V(s))ds_{V(t+ ∆t)} Ft

i

≃ Ehe−

Rt_+∆t

t r(s)ds e−∆t(rΦ(V(t+∆t)−r(t+∆t))_{V(t+ ∆t)} Ft

i

Thus, the exact calculation can be done using a backward discounted propagation of the single-rate model provided that we modify the underlying as follows

V(t+ ∆t)→e−∆t(rΦ(V(t+∆t)−r(t+∆t))V(t+ ∆t)

The non-linear effective rate makes the instrument pricing different from the usual single-rate habits.

Example. A payment atT of an index I fixed at a previous timet

cannot be simply added to the instrument leg L at timet,

Lnew(t)6=L(t) +P(t,T)I(t)

where P(t,T) is a zero bond. Instead, non-linear effects force us to add the payments to legs when they are really paid, i.e.

Remark 1. The subsequent backwards propagation should take into account the partial ”path-dependence” of the new value of the leg Lnew(T). Indeed, it will depend on the model states at timeT

as well as on the model states at time t via the index I(t). This means that we should augment the space of regression variables. In our concrete case, for each observation date t in a period of the floating leg [Tn,Tn+1], we add the Libor rate observed atTnas an

extra regression variable at t.

Collateral

The collateral, as a function of portfolio value, is defined as

C(V) = (V −H)+

where the threshold H has been set at H= 500, corresponding to an asymmetric CSA (only our counterparty posts collateral, to cover any exposure beyond the threshold).

The uncollateralized part

V −C = min(V,H),

gives rise to the PDE

LV = (V −H)+rC+ min(V,H)rF =rΦ(V)V,

where the effective non-linear rate rΦ(V) is defined implicitly by

Swap and bermudan prices

Strike _{Single-rate price}Swap prices_{Exact price Single-rate price}Bermudan prices_{Exact price}

0.05% -1,604.54 -1,554.05 85.21 82.18

1.05% -802.27 -776.71 210.82 204.14

2.05% 0.00 3.20 469.89 458.04

3.05% 802.27 790.23 941.75 925.68

4.05% 1,604.54 1,585.77 1,625.61 1,606.25

5.05% 2,406.82 2,385.07 2,408.26 2,386.47

6.05% 3,209.09 3,185.99 3,209.10 3,186.01

7.05% 4,011.36 3,987.66 4,011.36 3,987.66

8.05% 4,813.63 4,789.68 4,813.63 4,789.68

9.05% 5,615.90 5,591.84 5,615.90 5,591.84

10.05% 6,418.17 6,394.06 6,418.17 6,394.06

Output: FVA

FVAs (=corrections over the single-rate price)

◮ ”True” (difference between exact and single-rate price) ◮ ”Approx” (rates are evaluated for future valuesv

0)

−E

Z T

0

du(rΦ(v0(u))−r(u))e−

Ru

0ds rΦ(v0(s))_v(k)

0 (u)

◮ ”Naive” (rates are evaluated for continuation values v)

−E

Z T

0

du(rΦ(v(u))−r(u))e−

Ru

0 ds rΦ(v(s))_v(k)

0 (u)

FVA table

Strike _TrueSwap FVA_Approx. Bermudan swaption FVA_{True Approx. Naive}

0.05% 50.49 50.49 -3.02 -3.03 -3.07

1.05% 25.56 25.57 -6.67 -6.66 -6.80

2.05% 3.20 3.24 -11.85 -11.77 -12.27

3.05% -12.04 -11.93 -16.07 -15.93 -17.21

4.05% -18.77 -18.63 -19.36 -19.21 -21.52

5.05% -21.75 -21.62 -21.79 -21.65 -24.79

6.05% -23.10 -22.98 -23.10 -22.98 -26.66

7.05% -23.70 -23.59 -23.70 -23.59 -27.62

8.05% -23.95 -23.87 -23.95 -23.87 -28.11

9.05% -24.06 -23.99 -24.06 -23.99 -28.35

10.05% -24.11 -24.05 -24.11 -24.05 -28.44

0% 2% 4% 6% 8% 10%

−30 −20 −10

0 10 20 30 40 50

Strike

F

V

A

Swap FVA Bermudan FVA

0% 2% 4% 6% 8% 10%

−30 −25 −20 −15 −10 −5

Strike

P

ri

ce

True FVA Approx. FVA Naive FVA

Behavior explanation

FVA for our concrete case (C(V) = (V −H)+ _{andr =rC)}

−E

Z T

0

du sF(u) min(H,v0(u))e−

Ru 0 dt

min(H_,v 0(t)) v₀₍t) sF(t)

e−

Ru 0 dt rC(t)

.

where sF(u) =rF(u)−rC(u) is the funding spread.

◮ The swaption future valuev

0(t) is very likely to be positive

→ the FVA negative (due to min(H,v0(u)))

◮ For large and positive future values (far in-the-money options)

the FVA saturates on the level

J+(0) =−H

Z T

0

duEhsF(u) e−

Ru

0 dt rC(t)i

◮ For highly negative values, say for out-of-money swap, the

Conclusion

We presented:

◮ Replication for domestic deals collateralized with foreign

curency

◮ a generalized funding theory

◮ a practical and very accurate approximation for a portfolio

containing both vanilla and exotic instruments

◮ an understanding of theory tolerances

◮ numerical experiments for a Bermudan swaption

Alexandre Antonov, Serguei Issakov and Serguei Mechkov (2011) ”Algorithmic Exposure and CVA for Exotic

Derivatives”, Available at SSRN

Alexandre Antonov, Marco Bianchetti and Ion Mihai (2013) ”FVA for General Instruments: Theory and Practice”, Available at SSRN

Christoph Burgard and Mats Kjaer (2012), ”Generalised CVA with Funding and Collateral via Semi-Replication”, Available at SSRN

Masaaki Fujii, Yasufumi Shimada and Akihiko Takahashi (2009), ”A Note on Construction of Multiple Swap Curves with and without Collateral.” Available at SSRN

Meng Han, Yeqi He and Hu Zhang (2013), ”A Note on Discounting and Funding Value Adjustments for Derivatives.” Available at SSRN

Andrea Pallavicini, Daniele Perini and Damiano Brigo (2011), ”Funding Valuation Adjustment: A Consistent Framework Including CVA, DVA, Collateral, Netting Rules and Re-Hypothecation.” Available at SSRN.

Contact information

www.numerix.com