A Bond Consistent Derivative Fair Value
C. Johan Gunnesson∗ Alberto Fernández Muñoz de Morales†
BBVA - Risk Methodologies‡
September 22, 2014
arXiv:1406.5755v2 [q-fin.PR] 19 Sep 2014
Abstract
In this paper we present a rigorously motivated pricing equation for derivatives,
including cash collateralization schemes, which is consistent with quoted market bond
prices. Traditionally, there have been differences in how instruments with similar
cash flow structures have been priced if their definition falls under that of a financial
derivative versus if they correspond to bonds, leading to possibilities such as funding
through derivatives transactions. Furthermore, the problem has not been solved with
the recent introduction of Funding Valuation Adjustments in derivatives pricing, and
in some cases has even been made worse.
In contrast, our proposed equation is not only consistent with fixed income assets
and liabilities, but is also symmetric, implying a well-defined exit price, independent
of the entity performing the valuation. Also, we provide some practical proxies, such
as first-order approximations or basing calculations of CVA and DVA on bond curves,
rather than Credit Default Swaps.
∗
†
‡
The opinions of this article are those of the authors and do not reflect in any way the views or business
of their employer.
1
,1 Introduction and Final Pricing Formula
Ever since Black, Scholes and Merton’s seminal works [1, 2], and until recently, financial
derivatives products have been priced without taking into consideration credit- or funding
spreads of either counterparty in the transaction. A frequently stated reason for this
approach (eg, [3]) was that financial institutions could, in the pre-financial crisis world,
borrow funds at the prevailing Libor rate, and any funding considerations could therefore
be taken into account by discounting cash flows accordingly.
This did, however, not correctly reflect the counterparty credit risk inherent in any
given derivative. A digital option which is far in the money behaves similarly to a zero-
coupon bond, yet traditionally its cash flows were discounted at Libor, or a similar rate,
instead of applying the corresponding bond curve. The inconsistency in the way these
two, functionally similar, deals were treated led some market participants to securing
funding through derivatives transactions. Derivatives desks are of course aware of cur-
rent bond prices, and it is therefore expected that they should charge the counterparties
accordingly, reflecting the value of the liquidity provided to the counterparty. But the
accounting mismatch still provided incentives for closing deals entailing funding, thus ren-
dering an upfront profit for the dealer, while at the same time reducing funding costs for
the counterparty. It is true that, from the risk management side, some sophisticated banks
were provisioning expected counterparty credit losses based on market-implied estimates.
However, many were basing such provisions on historical data.
A step in the direction of reconciling bond- and derivative valuations was taken with
the entry into force of the accounting standard IFRS 13. This standard defines fair value
as an exit price, further stressing the use of market-implied (or at least market-adjusted)
valuations, and including a bank’s own non-performance risk, ie, the possibility that the
bank may not fulfill all of its obligations. The fair value should not be entity-specific,
in the sense that other market participants should arrive at the same valuation. The
standard interpretation of IFRS 13 is to include in the derivative price a Credit Valua-
tion Adjustment (CVA), representing the market value of the deal’s counterparty credit
risk1 , together with a Debit Valuation Adjustment (DVA), representing the bank’s non-
performance risk and based on its credit spread. The fair value obtained in this way is
symmetric, in the sense that two counterparties will arrive at the same value if they use
the same calculation methodology and market inputs. For a technical account of these
subjects see [4, 5], or the textbooks [6, 7, 8].
After including CVA and DVA, deals with risky counterparties that once seemed ar-
tificially appealing will not produce as large an accounting profit upfront, thus reflecting
the true nature of these transactions. The most frequent way to quantify CVA and DVA
is to estimate market-implied default probabilities using Credit Default Swaps (CDS), in-
1
The market value of a given risk can be defined as the cost of buying protection against it in the market,
ie, the cost of hedging it. It is true that CVA is often expressed as the expected value of discounted losses
due to counterparty defaults, but it should be born in mind that these expectations are based on market
inputs, and not on historical (or real-world) losses due to counterparty defaults.
2
,struments in which an insurance premium, the CDS spread, is exchanged for protection
against losses stemming from a given bond issuer’s default. With equal recovery rates, a
higher CDS spread entails a greater probability of default. Credit risk is not the entire
story however. A persistent property of bond markets is the existence of a difference be-
tween the excess rates of return of bonds over the risk-free rate2 plus the CDS spread.
Nevertheless, a bond and a CDS on the same reference basically refer to the same type of
risk, so in a frictionless market, arguments of arbitrage should end up driving such differ-
ence, called the bond-CDS basis, to zero. There are a number of reasons that explain why
this gap fails to disappear completely, besides the classical argument of capital constraints
preventing arbitrage opportunities (see [9]). Certain frictions, like the Cheapest-to-deliver
option embedded in CDS contracts ([10, 11]) or haircuts that the arbitrageur encounters
when financing bond purchases in the repo market ([12]), explain the rationale behind the
basis. For further details on the origin of the basis see, for example, [13]. For simplicity,
the many reasons underlying the basis are commonly referred to as liquidity risk.
In the past couple of years, an additional step has been taken by some sophisticated
banks, with the inclusion of a Funding Valuation Adjustment3 (FVA). The aim of such an
adjustment is to take into consideration the funding costs associated to the ”production” of
a derivative’s transaction, ie, the cost of funding the hedging of its risks during the lifetime
of the deal. This introduces the bank’s complete bond spread into the derivatives price.
However, rather than solving the discrepancy between bond- and derivatives prices, many
approaches to FVA actually make it larger. Consider, for example, the in-the-money digital
option mentioned above, and suppose that the bank has bought the option, analogously to
a bond purchase. Any approach to pricing it consistently with bonds should thus contain
a CVA, reflecting the counterparty’s credit risk, together with an additional term governed
by the counterparty’s bond-CDS basis to reflect the bond’s liquidity premium. In contrast,
one approach to FVA adds the bank’s complete funding cost (proportional to its funding
spread) to the calculated CVA. The counterparty’s bond-CDS basis is therefore not part
of the price and furthermore, as we will explain later, its CVA contributes implicitly to
the bank’s funding spread, and is therefore double counted. Needless to say, the obtained
valuation will not be symmetric, and the counterparty will calculate a different derivatives
price. In Section 2 we will explain the limitations of current FVA frameworks in more
detail.
In this paper we provide a solution in the form of a rigorously motivated derivative
pricing equation that is completely consistent with market bond prices. We will be con-
cerned here with uncollateralized derivatives, since the direct exposure that they generate
to the counterparty is analogous to bond exposure, although we will briefly comment on
the partially collateralized case in Section 4. At a given time t, the pricing equation takes
2
It is, of course, doubtful that any truly risk-free interest rate can be said to exist, but for practical (and
theoretical, as will be discussed below) purposes an Overnight Indexed Swap (OIS) rate is often employed
(see [3]). It has become a standard to pay such rates for held collateral, and they are therefore often
referred to as collateral rates.
3
See, for example, the aforementioned textbooks or [14, 15, 16].
3
, the form
Vt = Vtc − CV At + DV At + BF V At , (1)
where Vt is the fair value at t, and its components are
• Vtc : the fair value that would be obtained at t if the derivative were perfectly col-
lateralized, meaning that collateral is posted in a continuous fashion by the bank or
counterparty in response to changes in the derivative valuation. In [17] it was shown
that in such idealized cases the derivative value is simply obtained by discounting
all future cash flows using the rate paid on the collateral accounts.
• CV At : The CVA calculated to adjust for the counterparty’s credit risk.
• DV At : The DVA reflecting own credit risk, and which equals the CVA that would
be calculated by the counterparty.
• BF V At : The new term in our approach, which we call a Bilateral Funding
Valuation Adjustment, and that incorporates the effects of both the bank’s and
counterparty’s bond-CDS bases. In turn, we separate it as
BF V At = −CF V At + DF V At , (2)
where CF V At stands for Credit Funding Valuation Adjustment, and is gov-
erned by the counterparty’s bond-CDS basis, while DF V At means Debit Funding
Valuation Adjustment, and depends on the bank’s bond-CDS basis. They can
be thought of as correcting CVA and DVA, respectively, extending them to a full
funding adjustment. In particular, positive exposure to the counterparty, given by
Vt+ ≡ max(Vt , 0), (3)
and which arises when the derivative can be considered an asset, generates CF V At ,
while negative exposure (the derivative is a liability)
Vt− ≡ max(−Vt , 0) (4)
gives rise to a DF V At . In more detail,
Z T
C +
CF V At = E 1alive (s) D(t, s) γs Vs ds , (5)
t
where E[·] stands for ”Expected Value4 ”, t is the current valuation time, T is ma-
turity of the deal, s is an integration variable representing all intermediate times
between the present (t) and maturity (T ), 1alive (s) means that the deal should be
alive at s (it is a variable that is equal to 1 if the deal is alive at s, and zero otherwise),
D(t, s) is the discount factor between t and s, γsC is the counterparty’s bond-CDS
basis at s, and Vs+ is the aforementioned positive exposure. In the same way,
Z T
B −
DF V At = E 1alive (s) D(t, s) γs Vs ds , (6)
t
where γsB is the bank’s bond-CDS basis.
4
Under the risk-neutral, or market-implied, measure.
4
C. Johan Gunnesson∗ Alberto Fernández Muñoz de Morales†
BBVA - Risk Methodologies‡
September 22, 2014
arXiv:1406.5755v2 [q-fin.PR] 19 Sep 2014
Abstract
In this paper we present a rigorously motivated pricing equation for derivatives,
including cash collateralization schemes, which is consistent with quoted market bond
prices. Traditionally, there have been differences in how instruments with similar
cash flow structures have been priced if their definition falls under that of a financial
derivative versus if they correspond to bonds, leading to possibilities such as funding
through derivatives transactions. Furthermore, the problem has not been solved with
the recent introduction of Funding Valuation Adjustments in derivatives pricing, and
in some cases has even been made worse.
In contrast, our proposed equation is not only consistent with fixed income assets
and liabilities, but is also symmetric, implying a well-defined exit price, independent
of the entity performing the valuation. Also, we provide some practical proxies, such
as first-order approximations or basing calculations of CVA and DVA on bond curves,
rather than Credit Default Swaps.
∗
†
‡
The opinions of this article are those of the authors and do not reflect in any way the views or business
of their employer.
1
,1 Introduction and Final Pricing Formula
Ever since Black, Scholes and Merton’s seminal works [1, 2], and until recently, financial
derivatives products have been priced without taking into consideration credit- or funding
spreads of either counterparty in the transaction. A frequently stated reason for this
approach (eg, [3]) was that financial institutions could, in the pre-financial crisis world,
borrow funds at the prevailing Libor rate, and any funding considerations could therefore
be taken into account by discounting cash flows accordingly.
This did, however, not correctly reflect the counterparty credit risk inherent in any
given derivative. A digital option which is far in the money behaves similarly to a zero-
coupon bond, yet traditionally its cash flows were discounted at Libor, or a similar rate,
instead of applying the corresponding bond curve. The inconsistency in the way these
two, functionally similar, deals were treated led some market participants to securing
funding through derivatives transactions. Derivatives desks are of course aware of cur-
rent bond prices, and it is therefore expected that they should charge the counterparties
accordingly, reflecting the value of the liquidity provided to the counterparty. But the
accounting mismatch still provided incentives for closing deals entailing funding, thus ren-
dering an upfront profit for the dealer, while at the same time reducing funding costs for
the counterparty. It is true that, from the risk management side, some sophisticated banks
were provisioning expected counterparty credit losses based on market-implied estimates.
However, many were basing such provisions on historical data.
A step in the direction of reconciling bond- and derivative valuations was taken with
the entry into force of the accounting standard IFRS 13. This standard defines fair value
as an exit price, further stressing the use of market-implied (or at least market-adjusted)
valuations, and including a bank’s own non-performance risk, ie, the possibility that the
bank may not fulfill all of its obligations. The fair value should not be entity-specific,
in the sense that other market participants should arrive at the same valuation. The
standard interpretation of IFRS 13 is to include in the derivative price a Credit Valua-
tion Adjustment (CVA), representing the market value of the deal’s counterparty credit
risk1 , together with a Debit Valuation Adjustment (DVA), representing the bank’s non-
performance risk and based on its credit spread. The fair value obtained in this way is
symmetric, in the sense that two counterparties will arrive at the same value if they use
the same calculation methodology and market inputs. For a technical account of these
subjects see [4, 5], or the textbooks [6, 7, 8].
After including CVA and DVA, deals with risky counterparties that once seemed ar-
tificially appealing will not produce as large an accounting profit upfront, thus reflecting
the true nature of these transactions. The most frequent way to quantify CVA and DVA
is to estimate market-implied default probabilities using Credit Default Swaps (CDS), in-
1
The market value of a given risk can be defined as the cost of buying protection against it in the market,
ie, the cost of hedging it. It is true that CVA is often expressed as the expected value of discounted losses
due to counterparty defaults, but it should be born in mind that these expectations are based on market
inputs, and not on historical (or real-world) losses due to counterparty defaults.
2
,struments in which an insurance premium, the CDS spread, is exchanged for protection
against losses stemming from a given bond issuer’s default. With equal recovery rates, a
higher CDS spread entails a greater probability of default. Credit risk is not the entire
story however. A persistent property of bond markets is the existence of a difference be-
tween the excess rates of return of bonds over the risk-free rate2 plus the CDS spread.
Nevertheless, a bond and a CDS on the same reference basically refer to the same type of
risk, so in a frictionless market, arguments of arbitrage should end up driving such differ-
ence, called the bond-CDS basis, to zero. There are a number of reasons that explain why
this gap fails to disappear completely, besides the classical argument of capital constraints
preventing arbitrage opportunities (see [9]). Certain frictions, like the Cheapest-to-deliver
option embedded in CDS contracts ([10, 11]) or haircuts that the arbitrageur encounters
when financing bond purchases in the repo market ([12]), explain the rationale behind the
basis. For further details on the origin of the basis see, for example, [13]. For simplicity,
the many reasons underlying the basis are commonly referred to as liquidity risk.
In the past couple of years, an additional step has been taken by some sophisticated
banks, with the inclusion of a Funding Valuation Adjustment3 (FVA). The aim of such an
adjustment is to take into consideration the funding costs associated to the ”production” of
a derivative’s transaction, ie, the cost of funding the hedging of its risks during the lifetime
of the deal. This introduces the bank’s complete bond spread into the derivatives price.
However, rather than solving the discrepancy between bond- and derivatives prices, many
approaches to FVA actually make it larger. Consider, for example, the in-the-money digital
option mentioned above, and suppose that the bank has bought the option, analogously to
a bond purchase. Any approach to pricing it consistently with bonds should thus contain
a CVA, reflecting the counterparty’s credit risk, together with an additional term governed
by the counterparty’s bond-CDS basis to reflect the bond’s liquidity premium. In contrast,
one approach to FVA adds the bank’s complete funding cost (proportional to its funding
spread) to the calculated CVA. The counterparty’s bond-CDS basis is therefore not part
of the price and furthermore, as we will explain later, its CVA contributes implicitly to
the bank’s funding spread, and is therefore double counted. Needless to say, the obtained
valuation will not be symmetric, and the counterparty will calculate a different derivatives
price. In Section 2 we will explain the limitations of current FVA frameworks in more
detail.
In this paper we provide a solution in the form of a rigorously motivated derivative
pricing equation that is completely consistent with market bond prices. We will be con-
cerned here with uncollateralized derivatives, since the direct exposure that they generate
to the counterparty is analogous to bond exposure, although we will briefly comment on
the partially collateralized case in Section 4. At a given time t, the pricing equation takes
2
It is, of course, doubtful that any truly risk-free interest rate can be said to exist, but for practical (and
theoretical, as will be discussed below) purposes an Overnight Indexed Swap (OIS) rate is often employed
(see [3]). It has become a standard to pay such rates for held collateral, and they are therefore often
referred to as collateral rates.
3
See, for example, the aforementioned textbooks or [14, 15, 16].
3
, the form
Vt = Vtc − CV At + DV At + BF V At , (1)
where Vt is the fair value at t, and its components are
• Vtc : the fair value that would be obtained at t if the derivative were perfectly col-
lateralized, meaning that collateral is posted in a continuous fashion by the bank or
counterparty in response to changes in the derivative valuation. In [17] it was shown
that in such idealized cases the derivative value is simply obtained by discounting
all future cash flows using the rate paid on the collateral accounts.
• CV At : The CVA calculated to adjust for the counterparty’s credit risk.
• DV At : The DVA reflecting own credit risk, and which equals the CVA that would
be calculated by the counterparty.
• BF V At : The new term in our approach, which we call a Bilateral Funding
Valuation Adjustment, and that incorporates the effects of both the bank’s and
counterparty’s bond-CDS bases. In turn, we separate it as
BF V At = −CF V At + DF V At , (2)
where CF V At stands for Credit Funding Valuation Adjustment, and is gov-
erned by the counterparty’s bond-CDS basis, while DF V At means Debit Funding
Valuation Adjustment, and depends on the bank’s bond-CDS basis. They can
be thought of as correcting CVA and DVA, respectively, extending them to a full
funding adjustment. In particular, positive exposure to the counterparty, given by
Vt+ ≡ max(Vt , 0), (3)
and which arises when the derivative can be considered an asset, generates CF V At ,
while negative exposure (the derivative is a liability)
Vt− ≡ max(−Vt , 0) (4)
gives rise to a DF V At . In more detail,
Z T
C +
CF V At = E 1alive (s) D(t, s) γs Vs ds , (5)
t
where E[·] stands for ”Expected Value4 ”, t is the current valuation time, T is ma-
turity of the deal, s is an integration variable representing all intermediate times
between the present (t) and maturity (T ), 1alive (s) means that the deal should be
alive at s (it is a variable that is equal to 1 if the deal is alive at s, and zero otherwise),
D(t, s) is the discount factor between t and s, γsC is the counterparty’s bond-CDS
basis at s, and Vs+ is the aforementioned positive exposure. In the same way,
Z T
B −
DF V At = E 1alive (s) D(t, s) γs Vs ds , (6)
t
where γsB is the bank’s bond-CDS basis.
4
Under the risk-neutral, or market-implied, measure.
4
