Derivatives pricing when one cannot
borrow at the risk free rate
S. Benaim and D. Kainth, QuaRC, RBS
Derivatives pricing when one cannot borrow at the risk
free rate
Disclaimer:
The views expressed here are those of the presenters and not necessarily those of RBS.
2
Derivatives pricing when one cannot borrow at the risk
free rate
• Conventionally, derivatives pricing models have assumed that market participants can “borrow at the risk free
rate” whenever they need to do so to replicate derivatives payoffs or to carry out arbitrages.
• Post-crisis, credit spreads and the differences between different funding rates have increased significantly. This
has meant that these assumptions are no longer even approximately valid, and this has a significant impact on
derivatives pricing, even for products conventionally thought of as “vanilla”.
• The way that derivatives are valued and risk managed is often strongly dependent on what credit risks and
funding commitments are embedded in the payoffs.
• We will explore these issues for uncollateralised derivatives concentrating on valuation and risk management,
and show how conventional pricing theory needs to be adapted, why assumptions like “price=expected payoff”,
that pricing for derivatives must be symmetric (i.e. both sides of a swap use the same methodology to price the
derivative) and that payoffs can easily be replicated, often no longer apply and need to be adapted.
3
Derivatives pricing when one cannot borrow at the risk
free rate
•For collateralised derivatives, it turns out that the assumption that we can borrow freely is still (mostly) a
reasonable assumption, effectively because the counterparty is contractually obliged to lend us all the money we
need to fund the position at a rate specified in the CSA (the contract governing the collateralisation).
•This rate is (usually) OIS, and there is now a consensus in the market that assuming that we can borrow (or lend)
at OIS gives us the right price for collateralised derivatives (more or less).
However, for uncollateralised derivatives, it is less clear how we should price derivatives now that borrowing is
expensive and constrained, for all market participants (at least for all institutions who make markets or look for
arbitrages).
•It is simplest to split uncollateralised exposures into two categories (from the perspective of a given bank or other
institution):
−Derivatives whose payoffs are contingent on the bank surviving.
−Derivatives whose payoffs are not contingent on the bank surviving.
•In practice (under certain assumptions) much of banks’ derivatives portfolios does not fall neatly into one of these
two categories. However, these positions can be thought of as a combination of these two cases, and therefore
such positions should be dealt with using a combination of the ways each of the two cases above are dealt with.
While technically difficult, the principles involved will be the same.
4
The value of term funding commitments
• Term funding is currently expensive, and it would be cheaper to fund all positions at short term rates.
• It is worth recalling why banks do not do that, as this will provide insight into how we should quantify the costs
and benefits of contingent funding commitments embedded in derivatives and structured notes, and show why it
is important to incorporate these costs properly.
• We use term funding because:
− If we fund long term cashflows by borrowing short, and the short term funding becomes more expensive
than the return we receive on the asset we are funding, we will lose money.
− If short term funding becomes too expensive or disappears completely, and we are dependent on it, we will
be forced into bankruptcy (or a government bailout).
− In these stressed markets, the higher our funding requirements are, the more our cost of funding will go up,
so there will be feedback effects that will exacerbate these issues.
• Thus, term funding is effectively insurance against having to borrow in scenarios where markets are stressed
and funding is expensive.
5
Uncollateralised derivatives contingent on bank survival
• We first analyse derivatives whose cashflows are contingent on the bank surviving. For simplicity, we assume 0
recovery (the issues are generally similar if recovery is non-zero). In effect we are assuming that cashflows
cease after the bank defaults, and that there are no cashflows at the point of default.
• This category will include:
–Any derivative liabilities that cannot be netted against assets.
–Any swaps that are embedded in liabilities (such as structured notes).
–In many jurisdictions, due to the ability to net against bonds, derivative assets (see below).
The arguments in this section are similar to those made in Burgard and Kjaer (2010).
• If RBS holds an uncollateralised derivative asset traded with a counterparty, and the counterparty holds an RBS bond with the
same notional as the MTM of the asset, if RBS defaults on the bond, the counterparty can net the bond liability against the
derivative asset and so he would not have to make any payment to RBS. This means that, if we assume that the counterparty
behaves rationally, he will, when he sees that RBS is close to default, buy RBS bonds cheaply and avoid having to pay out the MtM
of the derivative.
6
Uncollateralised derivatives contingent on bank survival
• Typically positions are hedged using collateralised derivatives traded on the inter-bank market, so we look at
pricing the uncollateralised position by replicating with the collateralised version. Note that our replicating
portfolio should match all cashflows, including those occurring if the bank defaults.
• Consider the situation where we have sold an uncollateralised option to a client, and then bought a collateralised
version of the same option from another bank. (Note that the argument will be the same for a swap or derivative
asset in this category).
• We will receive collateral from the collateralised counterparty immediately, but we will only need to make a
payment to the uncollateralised counterparty at maturity. What should we do with the cash until then?
• It turns out that in order to hedge the payoff perfectly, we should use the cash to buy an instrument that, like the
uncollateralised derivative, is contingent on the bank’s survival, such as a bond issued by the bank or a term
deposit with bank’s treasury function. The instrument should have the same maturity as the uncollateralised
derivative.
• In practice, the derivatives desk would not go out and buy bonds in the market, they would deposit the money
with the treasury function who would manage it as part of the bank’s overall funding requirements. The more
funding derivatives desks bring in, the less term debt the treasury function will need to issue, and vice versa.
Thus, “buying back bonds” is not as unrealistic as it might seem at first sight. We will refer to these instruments
as “risky deposits”.
• Of course, this needs to be done dynamically, so whenever the price of the collateralised derivative in the hedge
portfolio moves, the desk replicating the payoff will need to buy or unwind some term deposits to match the
change in the amount of collateral posted.
7
Uncollateralised derivatives contingent on bank survival
• If the bank survives until the maturity, cashflows are matched. Treasury return the deposit (equal to the option
payoff) and this is passed on to the client. We keep the collateral posted by the other bank, now equal to the
option payoff, so there are no cashflows there.
Option Payoff
Option Payoff
Option
Desk
Other
Bank
Collateral
returned
Deposit Returned
Group
Treasury
8
Client
Uncollateralised derivatives contingent on bank survival
•If the bank defaults, there are no cashflows at all. Treasury default on the term deposit, the option desk default on
the uncollateralised option, and the collateralised position is unwound at the MtM (which is equal to the collateral
already posted by the counterparty).
Default on
uncollateralised
derivative
Option MTM
Option
Desk
Other
Bank
Collateral
returned
Client
Default on
Deposit
Group
Treasury
9
Uncollateralised derivatives contingent on bank survival
• The interest we receive on the risky deposit/bond will be based on the term funding rate, and it is easy to show
that this is the rate we should use to discount the uncollateralised derivative.
• Thus, if a derivative pays a cashflow of CT at T and the risky cost of funding is rf :
Uncollateralised Option Price =
where r is the risk free rate, R is the recovery (assuming that the payoff on recovery is R*MTM) and tau is the
default time..
• If we make the risky bond the numeraire (and we assume that the R>0 a.s), we can write this as:
Uncollateralised Option Price= Bank Issued Zero Coupon Bond Price x ER[CT]
• Clearly, this argument can be extended to more complex payoffs and assets (in which case the deposits would
be replaced by loans).
There is therefore a clear replication strategy and therefore a clear, unique, price for this derivative (assuming of
course a relatively simple model for the cost of funding which has no unhedgeable stochastic factors like
stochastic correlation).
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Uncollateralised derivatives contingent on bank survival
Correlation
• Note that the uncollateralised derivative will be discounted more heavily than the collateralised derivative, so:
–On day 1 we will need to buy a smaller notional on the collateralised hedge than for the uncollateralised
position.
–We will be paying OIS on the collateral and receiving term funding on the deposit. As we move forward in
time we the discounting differential will narrow, so we will use the carry on the collateral to increase the
notional of the collateralised hedge, in such a way that by maturity the notional of the collateralised and
uncollateralised positions is the same.
–As the funding spread fluctuates we will need to increase or decrease our position in the collateralised
derivative. Similarly, as the derivative price fluctuates then we will need to buy or sell risky
bonds/deposits.
–The cost of re-hedging will thus depend on the way funding spreads and defaults and the payoff are
related. Whenever spreads go down or stay the same, we will need to buy more of the collateralised
hedge, and if the price of the hedge is, on average, high in those scenarios, and low when we need to
reduce the hedge, we will bleed P&L. Conversely, if the correlation is in the opposite direction, we will
accrue a profit from the cross gamma.
–Thus, the correlation between RBS funding/credit and the payoff will have a significant impact on the
valuation.
–In other words, like any other derivative that requires dynamic replication, the pricing model needs to
model the future costs of the hedging instruments well.
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Uncollateralised derivatives contingent on bank survival
DVA and funding
Note that in the discussion above we showed that we can perfectly replicated the payoff, even when the bank
defaults. Therefore if the trade is priced and risk managed in that way there is no need to make a separate
adjustment for own credit risk (i.e. DVA), as that is already incorporated when we discount off the term funding
spread (which includes own credit risk).
Thus, DVA can be thought of as a measure of the expected funding benefit from a derivative, as we can use the
cash generated by the trade to reduce our term funding requirements. This is a real benefit that could be
monetised. Therefore, attempting to monetise by, say, selling protection on other banks will be counterproductive
and force us to sacrifice these benefits
Downgrade contingent CSAs
If the derivative liability is subject to one of these, we are effectively borrowing money that we need to pay back
when we get downgraded, when the replacement cost is likely to be high. We pay term funding spreads precisely
to avoid having to replace the funding in stressed markets, so we should be paying a very low spreads, even
though the expected term of the funding may be relatively high.
Note that pricing the credit risk explicitly leads to the same conclusion, the counterparty is taking very little bank
credit risk, so there should only a small DVA added to the risk free price.
Recovery
Up to now, we have assumed that recovery is 0. However, if we assume non-zero recovery, it is easy to check
that, as long as the counterparty’s claim on default is on the MTM of the derivative, the same logic applies. This is
close to true in most cases, but there are some exceptions.
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Uncollateralised derivatives contingent on bank survival
Structured Notes
• A structured note is, in theory, no different from any other uncollateralised liability. It is however particularly
important to incorporate the term funding spread as this is often one of the key drivers of the valuation.
• Typically the valuation of these notes is split into:
–A term deposit with the treasury function that pays Libor+term funding spread and is effectively
discounted off term funding rates.
–A swap that swaps the interest and principal of the deposit for the exotic cashflows from the note.
• The swap component would be hedged in the market using a collateralised swap (or equivalently replicated
using simpler collateralised derivatives).
Discounting
• In the previous slides we have argued that the appropriate discount rate for the swap component (as for the rest
of the note) is the term funding rate, as the entire structure is a liability contingent on the bank’s survival and this
needs to be incorporated.
• Discounting off a lower rate than this can make a significant difference to the valuation, particularly if the
durations of the different legs of the swap are different or if the swap is no longer ATM.
13
Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Correlation
• If, as is often the case, the payoff of the note is positively correlated with the bank’s survival (i.e. the noteholder,
on average, receives a higher payoff in scenarios where the bank survives), then this will reduce the value of the
note (from the bank’s perspective). Examples of this type of note would be notes where we promise to pay
interest and principal as long as a reference credit correlated to the bank does not default, or as long as an
equity index does not go down substantially.
• This reduction in value occurs because, on average, when the market’s perception of the bank’s probability of
default increases (i.e. when the bank’s funding spreads increase), it is likely that we will need to post collateral
on our hedges. We would need to borrow money/cancel some of the term deposit to finance this collateral
posting, which will be expensive. Conversely, the opposite will happen in scenarios where funding is cheap, so
the benefit from the collateral we receive in that case is smaller.
• The contingent funding we receive from this type of note does a relatively poor job of insuring against the
stressed scenarios that term funding is supposed to insure us against. This is because, on average, in these
stressed scenarios, we will need to use some of the money the client has deposited with us to post collateral on
our hedges, so the amount of money available to reduce our funding requirements in these markets is reduced.
Effectively, while we are borrowing 100 on average, say, we would be borrowing only a small amount in
scenarios where having locked in term funding is valuable, and a large amount where it is not valuable.
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Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Correlation modelling
The simplest way to model correlation is to make the funding spread stochastic using a Hull White model (as this
allows for easy analytic pricing), correlate this with the underlying (whether a credit spread, a stock or a rate), and
to incorporate this spread into the discounting as is done, for example in Piterbarg (2010).
However, we would argue that this substantially underestimates the impact of correlation, particularly for shorter
dated trades.
Note that the spread is essentially a credit spread, so we should treat it like one. In particular, we know that this
spread can go to infinity with a known and significant probability so it does not behave like a (risk free) interest
rate.
The model essentially assumes that credit is modelled as follows:
•The hazard rate (which should be proportional to the spread) is a Hull White process. This process is correlated
with the underlying.
•A default process which is driven by a Poisson process, with intensity determined by the hazard rate. Conditional
on the hazard rate this process is independent of everything else.
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Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Correlation modelling
This approach has a number of shortcomings.
•The model is also very dependent on the default process to generate defaults. If we look at a two year bond, with
hazard rate at 200bps and HW volatility also at 200 bps (which is relatively high), then we know that there is a 4%
chance of the bond going to 40 (if that is the recovery). However, even a move from 100 to 94 caused by the
hazard rate would be very unlikely (3 standard deviations) so almost all the movement from 94 to 40 must come
from the default process. However, only the hazard rate process is correlated to the underlying, so we are
essentially assuming that “most” of the movement to default is independent of everything else.
•The spread is normally distributed, which means that it is much more negatively skewed than the real world
distribution of spreads, so it significantly underestimates the possibility of large moves upwards, while overestimating the possibility of large moves downwards (even to negative levels). This means that large correlated
moves upwards in spreads and the underlying have a very low probability, but it is precisely these moves that are
important.
Note that, even though the natural reaction of many people will be “who cares what happens when we default?”, it
is important to model this well, as understanding the (risk neutral) distribution conditional on survival is
indisputably vital. Given that all we can observe directly is the unconditional distribution, understanding how the
distribution conditional on survival differs from the unconditional distribution is equivalent to understanding the
distribution conditional on default so if the latter is badly modelled, the former will be too.
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Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Correlation modelling
The following graph shows the empirical volatility of credit spreads of financial companies over the last five years
as a function of the level of credit spreads, and for comparison shows the distribution predicted by a Hull White
model (with the vol chosen so that the vols are the same as the empirical distribution at 100 bps).
5 year CDS daily volatility
Log-normal volatility
(annualised)
1
0.8
0.6
0.4
0.2
0
0
0.01
0.02
0.03
0.04
Spread level
Empirical vol
Hull White implied vol
17
0.05
0.06
Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Correlation modelling
These shortcomings can be remedied either by:
•Using a more realistic spread model (e.g. log-normal) and to correlate jumps to default with the underlying,
perhaps with a jump in the underlying on default of the credit.
•Using a copula to correlate defaults and the underlying. This approach is well established for credit correlation
products (note that a credit linked note is essentially identical to a first to default note). This automatically ensures
that the entire distribution is correlated with the underlying, and allows us to avoid having to build and calibrate a
realistic spread model, as the terminal distribution of bond prices is essentially known (modulo stochastic
recovery), bonds either default (with the default distribution given by the spreads) or mature. The approach is
therefore more robust and parsimonious, though imperfect.
•Neither of these is perfect as the former can be computationally intensive and the latter cannot be used to price
path dependent products. In both cases parameter estimation is difficult.
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Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Correlation impact (credit linked notes)
The following are estimates of how prices of CLNs issued by a bank with the same spreads as RBS change when
we incorporate correlations (market data based on Autumn 2010 with RBS spread c. 250 bps, correlations based
loosely on FTD prices and modelled with a Gaussian copula). We assume for simplicity that recovery is fixed at
40% of notional if issuer defaults, although this depends on the terms of the note.
Credit
Maturity
Correlation
CDS spread
Impact of correlation (% of
notional)
Impact of
correlation
(bps running)
AIG
5
70.00%
236
3.90%
92
Korea
5
50.00%
89
1.40%
32
Korea
5
77.50%
89
2.60%
59
BT
5
42.00%
134
1.40%
33
AIG
10
70.00%
236
7.30%
103
Korea
10
50.00%
109
3.30%
44
Korea
10
77.50%
109
5.70%
74
BT
10
42.00%
137
3.00%
41
19
Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Correlation impact (equity linked notes)
The following are correlation impacts (relative to simply discounting at cost of funding) calculated using a
Gaussian copula as a percentage of notional for a knock in reverse exchangeable on the FTSE, (European
barrier, coupons assumed to be independent of equity performance). While the correlation choice is somewhat
subjective, it should clearly be significantly above 0.
Correlations
Maturity
Barrier
30%
50%
80%
100%
3
50%
0.70%
1.40%
2.80%
4.30%
5
50%
1.30%
2.40%
4.50%
6.80%
10
50%
2.00%
3.50%
6.30%
9.10%
Note that these differences are significantly larger than many more conventional sources of model risk like, say,
the impact of stochastic volatility on barrier pricing, or stochastic rates on autocallable pricing.
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Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Early Termination
• Similarly, the value of a structured note that terminates early in some scenarios will be dependent on the type of
scenario where the early termination takes place. If it is more likely to terminate early when markets are stressed
and the funding is difficult to replace, it will be considerably less valuable than if the early termination is
uncorrelated with cost of funding. Similarly, if early termination is more likely in good economic scenarios when it
is likely that we will be able to replace the funding cheaply, that will make the funding more valuable.
• All other things being equal, this early termination should be more valuable when we have the option to call than
when the early termination is automatic (“autocallable”), as we will be able to call when doing so is in our favour,
and not when the funding replacement cost is high.
• Furthermore, our realised profit or loss in the event of early termination will depend on the replacement cost of
the funding, which may be significantly higher than when the trade was done, so it is important to incorporate
that into the valuation. If we do not, i.e. we assume that the deposit component is unwound at par, we risk misvaluing the trade and distorting call decisions.
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Uncollateralised derivatives contingent on bank survival
Structured Notes Cont…
Recovery
• The recovery that the noteholder is entitled to varies depending on the legal terms of the note.
• In particular, the noteholders’ claim may be on the notional (so they would receive Recovery x Notional) or on the
MTM of the note without the bank credit risk (i.e. they receive Recovery x MTM).
• The latter case is the same as for other derivatives, but the former can lead to significant differences in the
expected payoff.
• It effectively means that the swap component has 0 recovery, as the MTM of the swap component at default
does not affect the recovery.
• In practice, this means that the swap component should be discounted by the hazard rate/survival probability,
which will be higher than the credit/funding spread (as the latter incorporates the expectation of non-zero
recovery).
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Uncollateralised derivatives not contingent on bank
survival
We now move on to derivatives where the cashflow is not contingent on the bank’s survival.
• This will include all derivative assets, as long as we assume that, because of legal or operational reasons, the
counterparty will not net his liability against the bank’s bonds or other liabilities in the event of a default by the
bank.
• Effectively we are assuming if a derivative has a positive MtM at the time of a default by the bank, the
counterparty will be required pay the bank the MtM.
• As before, we will generally hedge these exposures using collateralised interbank trades. In this case, we will
need to post collateral to the collateralised counterparty before we receive the cashflows from the
uncollateralised counterparty. We therefore have a borrowing requirement .
• Ideally, we would want to structure this borrowing so that the cashflows of the borrowing match the cashflows of
the derivative. This is only possible if we can fund at risk free rates (i.e. promise to repay the borrowing even if
the bank defaults).
• This is not possible unless we can use the uncollateralised derivative as collateral (which is generally impossible
in practice), so in practice we can only fund at unsecured rates.
• It is therefore not possible to perfectly replicate the payoff of these positions; the market is incomplete.
• There is a range of possible “arbitrage free” prices, any price between risk free discounting and discounting at
the bank’s credit spread (or at least the credit spread of the potential arbitrageur with the lowest credit spread) is
arbitrage free.
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Uncollateralised derivatives not contingent on bank
survival
What should the price of these derivatives be? To determine this we will need to use equilibrium arguments and
analyse where the market clearing price should be.
•We will assume that there are two sides to the market, banks who attempt to replicate the derivatives’ payoffs
and clients (e.g. corporates) who want to hedge some risk or to express a view on the real world distribution of
market prices.
•We will discuss at what price banks should be willing to take on trades, and assuming enough corporates etc. are
willing to pay this price for the hedging or other benefits they want, this should be where the market will clear.
•We will not require that if both counterparties price the trade based on replication costs they should have the
same price. In practice uncollateralised trades where both counterparties are banks are very rare, so there is no
reason to dismiss a model that predicts they do not happen.
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Uncollateralised derivatives not contingent on bank
survival
• There are three ways in which we can approach the hedging and valuation in this context, and the choice
depends on the liquidity assumptions we want to make.
1. Super-replication
• The first possibility is that we use term funding as before, and discount the uncollateralised derivative off the term
funding curve.
• Following the same hedging strategy as before, all cashflows are matched except that the bank will make a
windfall profit if the bank defaults (there are no cashflows to match the MtM of the uncollateralised position we
receive on default).
Derivative MTM
Derivative MTM
Option
Desk
Other
Bank
Collateral
returned
Client
Default on
Borrowing
Funding
Market
• However, there is no way the bank can monetise this windfall. Even if the bank could sell CDS protection on
itself, that would not help much, as the cost of funding collateral calls (on the CDS) when funding spreads blow
out would be punitive. However, as will be discussed on a later slide, there may be some indirect ways in which
the bank benefits.
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Uncollateralised derivatives not contingent on bank
survival
2. Perfect liquidity
• The second possibility is to fund using short term borrowing and make some assumptions about liquidity. This
short term cost of funding is low today (around Libor) for a typical bank. However, the funding curve is usually
steeply upward sloping, implying that there is a significant possibility that the cost of short term funding will
increase significantly in the future. Thus, the risk adjusted expected cost of rolling over the funding will be
equivalent to what the term funding curve implies and will therefore be high.
• If, however, we assume that we can sell the derivative in the market (or unwind the derivative at the price the
client would be able to replace it with another bank) as soon as our funding costs spike, we can avoid paying
these high costs.
• It is not clear at what price we could sell it at. Because perfect replication is impossible, we cannot use the
standard arbitrage arguments. There is a self consistent replication based argument that can be made that the
equilibrium price would correspond to Libor discounting (or a little above Libor). This would be based on the
following assumptions:
–We assume that other banks will fund these derivatives using short term borrowing.
–Whenever a given bank’s funding costs spike, they sell the derivative to another bank, who are funding at
Libor, at a price based on Libor discounting. This bank will be willing to pay this price because it will be
making the same assumptions that it will fund at Libor and sell the trade on if its funding costs spike.
Libor is (in theory) banks’ average cost of funding, so there will always be some bank funding short term
at Libor.
• However, these assumptions are very strong. We are effectively assuming that the bank will be able to sell or
unwind all of its uncollateralised derivative assets and swaps without any significant discount as soon as funding
costs spike. Furthermore, we are assuming it will be able to do so in stressed markets (like in Autumn 2008).
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Uncollateralised derivatives not contingent on bank
survival
3. Offsetting funding costs against other benefits
The third possibility is to value the derivative position using short term rates (a small spread above Libor) or
somewhere in between these rates and the term funding spread, and accept that we will bleed money at least in
some scenarios if we look at the derivative trade in isolation, but argue that these losses will be offset by other
benefits. We will see in the following slides how this could come about and what assumptions are needed to
justify it.
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Uncollateralised derivatives not contingent on bank
survival
Corporate finance approach
We have seen that there is a strong case for arguing that the value of a derivative whose payoff does not depend
on the bank’s credit, should incorporate the bank’s funding spread, which does depend heavily of the bank’s credit
spread.
This seems counterintuitive, particularly if one is used to thinking of the price of a derivative as its risk neutral
expected payoff (although of course there exists a probability measure where the expected payoff is equal to this
price).
To better understand if this is the right price, and if so to better understand why this is, we will take a step back
and explicitly model the bank’s balance sheet.
We will relax a key assumption that derivatives pricing models usually make, and that we have been implicitly
assuming up to now, that market prices, in particular the cost of borrowing for the bank, is independent of the
actions of the derivatives desk which trades and hedges the derivative. Instead, we will explicitly price the
liabilities of the bank as contingent claims on the bank’s assets.
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Uncollateralised derivatives not contingent on bank
survival
We will analyse what happens when the bank purchases an illiquid risk free asset. This will be our proxy for an
uncollateralised derivative with all the market and counterparty risk hedged with collateralised instruments.
We assume that:
•The bank has assets worth A at the start. The ultimate payoff of these assets is random.
•To finance these it issues notional D of debt (yielding s0 above the risk free rate r in the base case and s1+r after
the risk free asset is purchased), and equity with a value of E.
•At the end of the period, the bank’s assets are used to pay off creditors first, and then shareholders if the value of
the asset is sufficient. The payoff of the debtholders is therefore equivalent to a forward minus a call on the bank’s
assets, and the shareholders’ payoff is a call option on the bank’s assets.
•We assume perfect elasticity of the bank’s debt markets, so that the price of debt instruments does not depend
on the amount of debt issued (as long as the payoff of the debt does not change).
•We analyse what happens when the bank buys notional of this risk free asset, and finances this by issuing
additional debt. In particular we calculate the price at which the bank’s shareholders are indifferent to the
transaction (because their payoff does not change).
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Uncollateralised derivatives not contingent on bank
survival
In the base case, without the risk free asset, the payoff to bondholders is:
And to shareholders is :
After it purchases the risk free asset, the bondholder’s payoff is
And to shareholders is
Now, it should be clear from these equations that adding the risk free asset increases the bondholders’ payoff
(before we consider any change in bond spreads). This is because it is adding an extra component that is
available to increase the bondholders’ recovery in the scenarios where they are not paid back in full. In other
words, the total amount of risk the bondholders are exposed to is essentially still the same, but the risk is now
divided among a slightly larger number of bondholders.
In particular, the payoff is now equivalent to a linear combination of a risk free asset and the payoff in the base
case.
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Uncollateralised derivatives not contingent on bank
survival
In order to do this we need to make some assumptions about the cost of the debt. We analyse three different sets
of assumptions, although the reality could be in between the three extremes.
1. Efficient funding markets, no existing long-term debt. We assume that the price of the bonds incorporates
all information about the ultimate payoff of the bonds, and that all the bank’s debt will mature imminently and will
need to be rolled over.
Now, adding the risk free asset makes the debtholders’ payoff slightly less risky, and therefore, with our
assumptions, this will mean that the debt will be slightly cheaper than before for the bank.
It is straightforward to show that, if the risk free asset is purchased at the risk free yield, the shareholders break
even overall, as the negative carry on the risk free asset is exactly offset by the rest of the balance sheet will be
cheaper to fund.
Thus, with these set of assumptions, we get the conventional answer, the derivative should be discounted at the
risk free rate.
2. Efficient funding markets, all existing debt is long-term. Here, as in the previous case, trading the risk free
asset makes the bank’s debt less risky. However, in this case, this only serves to benefit the existing debtholders,
and shareholders do not benefit (except that funding the risk free asset is slightly cheaper). In this case, the
shareholders only break even at the cost of funding.
3. Inefficient funding markets. In this case the market cost of funding does not change even if the payoff of the
debt changes. This means that the shareholders break even only if the risk free asset yields the same as the
bank’s debt.
31
Uncollateralised derivatives not contingent on bank
survival
Thus, we see that the assumption that the cost of funding is independent of the derivative desk’s actions is
equivalent to assuming that the market is not efficient, and it is these inefficiencies that mean that the derivative
asset is worth significantly less than its “expected payoff” (to shareholders, ultimately the difference accrues to the
bank’s bondholders).
Which assumption is the right one? Ultimately it is an empirical question, but it seems unlikely that the “perfect
efficiency” extreme is the right one, given that
•Much of banks’ (at least universal banks) unsecured funding comes from sources that are relatively insensitive to
the bank’s credit risk, e.g. insured deposits.
•For it to be true, we would need to assume that when the bank increases the size of its balance sheet without
increasing the amount of capital it holds, its cost of funding will come down. This seems unlikely, not least
because it is difficult for outsiders to ascertain whether the assets genuinely are risk free (and of course they are
unlikely to be completely risk free in practice), even though in theory (if there is perfect elasticity and the assets
are genuinely risk free) this is what should happen.
•Debt markets are not perfectly elastic which adds a further reason why the marginal cost of taking on a derivative
position is higher than a simple model predicts.
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Uncollateralised swaps
As mentioned earlier, most derivative portfolios are composed of swaps that are a combination of the two types of
derivatives we have discussed (at least under the conventional assumptions), they are contingent on the bank’s
survival when they have a negative MTM and not when they have a positive MTM.
We can write the payoff as contingent on defaults that affect the payoff, and we will call this default time τ*, and
we will denote the bank’s default time by τ. We assume the derivative has a series of cashflows C. and that the
risk free discount factors are D..
If we believe that assets should be priced using risk free discounting, then only defaults that take place when the
swap has a negative (risk free) MTM are relevant.
τ*= τ if MTM(τ)<0
= infinity if MTM(τ)>0
If we believe that assets should also be priced using the cost of unsecured funding, then all defaults are relevant
τ*= τ
33
Uncollateralised swaps
If the counterparty is also risky, we can add the dependence on the counterparty
where again we exclude certain defaults of the counterparty, in particular defaults that occur when the MTM is
negative (and, depending on the legal technicalities of the counterparties’ claims when one of them defaults, we
may exclude second defaults i.e. defaults of a counterparty that take place after the default of the other
counterparty ).
What is clear from these equations, particularly if we conclude that unsecured funding costs should be priced for
assets, is that the payoff depends strongly on the probability of both credits surviving, so it is very sensitive to the
correlation between the credits, which is almost certainly positive and significant.
This means that incorporating the full funding costs into valuations is not as punitive as it may seem, as we end
up discounting by significantly less than the sum of the two spreads.
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Conclusions
•There is a clear arbitrage free price for derivative liabilities (and similar products). Once we price in
the funding benefits to the bank we get to this price, and we have shown why it is important to price
these benefits well. Once we include those funding benefits we are already including DVA, and
therefore DVA represents genuine economic value to the bank and is not just an accounting charge
with no economic relevance.
•There is no clear arbitrage free price for derivative assets (under the standard assumptions), and
the value should depend on how we price the funding costs involved in replicating these payoffs.
We have discussed what empirical assumptions about the market are necessary to justify the
different possibilities.
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Bibliography
Bibliography
C. Burgard and M Kjaer. PDE Representations of Options with Bilateral Counterparty Risk and Funding Costs,
2010.http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1605307&download=yes
V. Piterbarg. Funding beyond discounting: collateral agreements and derivatives pricing. Risk, February:97{102,
2010.
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