E4731: IEOR Credit Risk and Credit Derivatives
Lecture 03: Risk Neutral Pricing. Basic arbitrage relations.
Notes originally written by Prof. Rama Cont
Instructor: Xuedong He
Spring, 2012
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Single-name credit derivatives
Single-name credit derivatives are contracts whose payoff depends on a default event of a single entity. The payoffs of these contracts involve expressions of the form F (T, i , Li ) where
T : maturity i : time of default Li : loss given default
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Multi-name credit derivatives
Multi-name credit derivatives are contracts whose payoff depends on default events of several entities. The payoffs of these contracts involve expressions of the form F (T, 1 , . . . , N , L1 , . . . , LN ) where
T : maturity i : time of default for obligor i Li : loss given default for obligor i
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Pricing of credit derivatives
Goal of credit derivative pricing models: assign values to various credit-risky payoffs in a manner which is
arbitrage-free: consistent across payoffs consistent with observed market prices of benchmark instruments
How can we build pricing models verifying these requirements? Problem: payoffs are uncertain/ random notion of value is not straightforward.
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Stochastic modeling of market prices
We model the uncertainty in the market by a probability space (, F, P) equipped with a filtration (Ft )t[0,T ] where
is the set of market scenarios Ft represents the information available at time t, i.e., the market history on [0, t]
A financial contract (with maturity/horizon T ) can then be represented by a payoff function H : R which defines the payoff H() of the contract in each scenario . H is thus a random variable. Denote by H the set of bounded random variables, representing the payoffs at T of financial contracts.
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Pricing rules
A market is a mechanism for associating a price t (H) to each payoff H at each point in time t: in short, it is a pricing rule, : H Y H (t (H))t[0,T ] which assigns a value process (t (H))t[0,T ] to each contract H. Here Y is the set of the non-anticipative processes Y : [0, T ] R i.e. such that for each t, Yt only depends on information available up to t. Two approaches:
Actuarial pricing Arbitrage pricing
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Actuarial pricing
Principle: contract value = sum of discounted value of expected cash flows. Using past information/history Ft , estimate objective probability distribution P of risk factors involved in contract. Then t (H) = B(t, T )EP [H | Ft ] Used a lot for pricing of insurance contracts. Rationale: insurance policies are sold in large numbers to many policyholders for which loss events are approximately independent. Example: life insurance and mortality curves.
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Actuarial pricing (Cont'd)
Consider a defaultable AA-rated zero coupon bond, with probability of default at maturity p(T ), expected recovery rate R and nominal N Expected payoff: = N [p(T )R + (1 - p(T ))] Actuarial value = N B(0, T )[p(T )R + (1 - p(T ))] Assume the bonds are priced in this way by the market. Then the market yield spread s(T ) is given by 1 - e-T s(T ) 1 ln(p(T )R + 1 - p(T )) p(T ) = T 1-R Five year default probability of AA-rated obligors, estimated by Standard and Poors: = 18 bps Assuming a recovery rate R = 40% we get a 5-yr yield spread of 1 s(T ) = - ln(p(T )R + 1 - p(T )) = 0.0002 = 2bp T for AA bonds s(T ) = -
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Actuarial pricing (Cont'd)
Jan 31 2007: 5 yr Treasury rate = 4.78%, 5 yr rate on AA bond= 5.34% 5 year yield spread: 56 bps Applying the above formula yields an implied 5-year default probability of p(5) = 1 - e-T s(T ) = 4.6% = 460bps! 1-R
So: the market does not seem to be pricing default risk using historically estimated default probabilities... Market-implied default probabilities are systematically higher than historical default probabilities. Need to take into account a premium for default risk.
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Arbitrage-free pricing rules
Instead of choosing an ad-hoc pricing rule, we simply assume the market to be arbitrage-free. The first fundamental theorem of asset pricing: no arbitrage is equivalent to the existence of an equivalent martingale measure. The price of any asset is equal to its expected discounted payoff under this measure. Recall that f0 (t, T ) is the forward rate at time t with a forward time T . Let r(t) be the short rate at time t, i.e., r(t) := f0 (t, t). Short rate r(t) represents the instantaneous rate of borrowing and lending at time t. The short rate process r(t), t 0 can be a random process.
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Arbitrage-free pricing rules (Cont'd)
Denote Q the equivalent martingale measure, also named as risk-neutral measure, which is equivalent to the objective measure P. Then, for any asset that pays H at time T , its time t-price is given by t (H) = EQ e-
T t
r(u)du
H | Ft
The price of the asset is the expected discounted future payoff under Q, the risk-neutral measure. The process e- 0 r(u)du t (H) is a martingale under Q, the equivalent martingale measure.
t
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Arbitrage-free pricing rules (Cont'd)
Nt := e 0 r(u)du can be considered to be the price of a particular asset in the market. This asset can be constructed once borrowing and lending are available at any time t at the short rate r(t). By using this asset as a numeraire, there exists the equivalent martingale measure Q such that the price of any asset, when discounted using the numeraire, is a martingale under Q. One can choose any assert as a numeraire, and there exists an equivalent martingale measure such that the price of any asset, when discounted using the numeraire, is a martingale under this measure. The equivalent martingale measure depends on the choice of the numeraire. If we use a T -matured zero coupon bond as the numeraire, we may have an equivalent martingale measure QT that is different from Q.
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Arbitrage-free pricing rules (Cont'd)
In this course, we always use the numeraire with price t e 0 r(u)du , t 0 and the resulting equivalent martingale measure Q. As a result, the discount factor B(t, T ), which is the time-t price of the T -matured zero coupon bond, is B(t, T ) = EQ e-
T t
r(u)du
| Ft
When the short rate is independent of the asset payoff H at time T , then the asset price can be written as t (H) = EQ e-
T t
r(u)du
H | Ft = B(t, T )EQ [H | Ft ]
In this case, Q is same as QT .
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Arbitrage-free pricing rules (Cont'd)
If asset an pays HTi at time Ti > t, i = 1, . . . , m, then the time-t price of the asset is
m
E
Q i=1
e-
Ti t
r(u)du
HTi | Ft
If an asset pays hs ds in the time period [s, s + ds), for any s (t, T ], then its price is EQ
(t,T ]
e-
s t
r(u)du
hs ds | Ft
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Calibrate Q to market data
The representation of the prices as conditional expectations with respect to some probability measure Q are derived under very weak and general conditions--no aribtrage, usually satisfied in all liquid markets. Q can be constructed from observed market prices: Q(A) is, up to a discount factor, the price of a binary/digital option: Q(A) = 0 (1A ) , B(0, T )
if the short rate is independent of A. So: the right way to construct Q is to extract information on Q from market prices of traded contracts. This is called calibrating the model to market prices.
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Calibrate Q to market data (Cont'd)
However, in order to fully calibrate Q, we need all the contracts 1A to be tradable in the market. If this is the case, we call the market is complete. Under market completeness, the pricing measure Q is uniquely determined, i.e., uniquely calibrated. It is more reasonable to assume that the market is incomplete, i.e., not all the contracts 1A are tradeable. In this case, the existence of the pricing measure Q is still valid. However, it is not unique by only calibrating to market data. Further criteria might be needed to calibrate a unique pricing measure.
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A Proof of Arbitrage-free pricing rules
We give a proof of the first fundamental theorem of asset pricing: no arbitrage is equivalent to the existence of equivalent martingale measures In the proof we use the T -matured bond as the numeraire. However, one will see that one can use any other denominators. We start from four axioms that are consequences of no-arbitrage assumption.
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Four Axioms
A1 positivity. For any H H, if H 0, then (H) 0. Positivity ensures that the pricing rule verifies static arbitrage A2 Ft -linearity: For any H1 , H2 H and any bounded Ft -measurable random variable (i.e. whose value is known at t), t (H1 + H2 ) = t (H1 ) + t (H2 ) inequalities. Ft -linearity rules out arbitrage opportunities by constructing portfolios at any time t
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Four Axioms (Cont'd)
A3 Time consistency. H H, t s (H) s (1) = t (H), 0tsT
This is from the observation that the two following strategies lead to the same terminal payoff H:
entering at t a long position in a contract with T -payoff H and holding until maturity: needs initial investment t (H). entering at time s, into a long position in a contract with T -payoff H, at cost s (H) and holding until maturity. The T -value of s (H) paid at time s is: s (H)/s (1). So, its value at t is t (s (H)/s (1)).
Time consistency rules out cash and carry arbitrage strategies for traded assets.
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Four Axioms (Cont'd)
A4 Continuity. If (Hn )n1 is an increasing sequence of payoffs with Hn H, then 0 (Hn ) 0 (H) Continuity rules out strange pricing rules which would attribute very different prices to very similar portfolios. t (1) is the value at t of $1 paid at T = default-free discount factor.
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Representation of pricing rules as conditional expectations
Theorem
For any pricing rule verifying the above properties, there exists a probability measure QT defined on the set of market scenarios (, FT ) such that 1 the value at time t T of any terminal payoff H H is equal to its discounted conditional expectation computed with respect to QT : t (H) = t (1)EQT [H | Ft ] = B(t, T )EQT [H | Ft ] , QT -a.s.
2 The discounted prices t (H)/t (1), 0 t T of all traded contracts are QT -martingales. QT is called the pricing measure (or risk-neutral measure) associated to the market .
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Proof of the theorem
Define QT on FT by A FT , QT (A) = 0 (1A ) 0 (1)
Step 1 show that QT is a probability measure QT is positive by positivity of , additive by linearity of and normalized. In order to show -additivity of QT , consider a sequence (Ai )i1 of disjoint events. Then
n
1n Ai = i=1
i=1
1A i
i=1
1A i = 1 A i . i=1
The continuity property (A4) of 0 implies
n
0
i=1
1A i
0
i=1
1A i
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Proof of the theorem (Cont'd)
But by linearity of 0
n n
0
i=1
1A i
=
i=1
0 (1Ai ) .
Dividing by 0 (1) we obtain that
n
QT (Ai ) QT ( Ai ) . i=1
i=1
So QT is positive, normalized and -additive, therefore a probability measure.
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Proof of the theorem (Cont'd)
Step 2 Show that 0 (H) = 0 (1)EQT [H] for simple payoffs H: Define a simple payoff as a sum of binary options:
n
H=
i=1
ci 1Ai ,
Ai FT , ci R.
Since 0 is linear, for any simple payoff H we have
n n
0 (H) =
i=1
ci 0 (1Ai ) =
i=1
ci 0 (1)QT (Ai ) = 0 (1)EQT [H].
Any positive payoff H L , H 0 can be approximated from below by a monotone sequence (Hn )n1 of simple payoffs: Hn H.
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Proof of the theorem (Cont'd)
Using the monotone convergence theorem for QT -expectation and the continuity property for , we can pass to the limit in EQT [Hn ] and 0 (Hn ) and we thus obtain 0 (H) = 0 (1)EQT [H]. Consider now a general payoff, not necessarily positive. Since H is bounded, both its positive and negative part H+ , H- have finite QT -expectation and by additivity of QT and we get 0 (H) = 0 (1)EQT [H].
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Proof of the theorem (Cont'd)
Step 3 Show t (H) = t (1)EQT [H | Ft ]. We use the following characterization of the conditional expectation: X = E [Y | A] A A, E [1A X] = E [1A Y ] . Fix t [0, T ]. Applying Ft -linearity and time consistency, for any A Ft we have that 1A t (H) 0 (1A H) = 0 . t (1) Hence, for any A Ft EQT [1A H] = EQT 1A t (H) t (1)
which characterizes tt(H) as a version of the QT -conditional (1) expectation of H with respect to Ft : t (H) = EQT [H | Ft ] t (1)
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Proof of the theorem (Cont'd)
Step 4 Show the discounted prices of all trade contracts are QT -martingales. For any 0 t s T , by time-consistency and s (H) = s (1)EQT [H | Fs ], we immediately have EQT t (H) s (H) | Ft = s (1) t (1)
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Two Approaches of Modeling Default Times
In order to apply the general no arbitrage pricing rule to credit derivative pricing, we need to model the default times Two approaches
Structural approach: economically meaningful, but complicated Reduced-form approach: with no economic interpretation, but simple
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