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 rateS. Benaim and D. Kainth, QuaRC, RBS

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Derivatives pricing when one cannot borrow at the risk free rateDisclaimer:

The views expressed here are those of the presenters and not necessarily those of RBS.

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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.

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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.

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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.

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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.

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•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.

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• 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 Desk

Other Bank

Client

Group Treasury

Deposit ReturnedCollateral returned

Option Payoff Option Payoff

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•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).

Option Desk

Other Bank

Client

Group Treasury

Default on Deposit

Collateral returned

Option MTM

Default on uncollateralised

derivative

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•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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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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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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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.

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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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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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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 over-estimating 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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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

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06

Spread level

Lo

g-n

orm

al v

ola

tilit

y (a

nn

ual

ised

)

Empirical vol Hull White implied vol

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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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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 spreadImpact 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

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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.

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.

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%

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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 mis-valuing the trade and distorting call decisions.

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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 survivalWe 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 survivalWhat 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).

•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.

Option Desk

Other Bank

Client

Funding Market

Default on Borrowing

Collateral returned

Derivative MTM Derivative MTM

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Uncollateralised derivatives not contingent on bank survival2. 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 survival3. 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 survivalCorporate 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 survivalWe 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 survivalIn 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 survivalIn 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.

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Uncollateralised derivatives not contingent on bank survivalThus, 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

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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.

Documents

Derivatives pricing when one cannot borrow at the risk free rate S. Benaim and D. Kainth, QuaRC, RBS