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Unformatted text preview: IEOR E4731: 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 1 / 28 Singlename credit derivatives I Singlename credit derivatives are contracts whose payoff depends on a default event of a single entity. I The payoffs of these contracts involve expressions of the form F ( T,τ i ,L i ) where I T : maturity I τ i : time of default I L i : loss given default 2 / 28 Multiname credit derivatives I Multiname credit derivatives are contracts whose payoff depends on default events of several entities. I The payoffs of these contracts involve expressions of the form F ( T,τ 1 ,...,τ N ,L 1 ,...,L N ) where I T : maturity I τ i : time of default for obligor i I L i : loss given default for obligor i 3 / 28 Pricing of credit derivatives I Goal of credit derivative pricing models: assign values to various creditrisky payoffs in a manner which is I arbitragefree: consistent across payoffs I consistent with observed market prices of benchmark instruments I How can we build pricing models verifying these requirements? I Problem: payoffs are uncertain/ random → notion of value is not straightforward. 4 / 28 Stochastic modeling of market prices I We model the uncertainty in the market by a probability space (Ω , F , P ) equipped with a filtration ( F t ) t ∈ [0 ,T ] where I Ω is the set of market scenarios I F t represents the information available at time t , i.e., the market history on [0 ,t ] I A financial contract (with maturity/horizon T ) can then be represented by a payoff function H : Ω 7→ R which defines the payoff H ( ω ) of the contract in each scenario ω ∈ Ω . H is thus a random variable . I Denote by H the set of bounded random variables, representing the payoffs at T of financial contracts. 5 / 28 Pricing rules I A market is a mechanism for associating a price Π t ( H ) to each payoff H at each point in time t : in short, it is a pricing rule , Π : H → Y H 7→ (Π t ( H )) t ∈ [0 ,T ] which assigns a value process (Π t ( H )) t ∈ [0 ,T ] to each contract H . I Here Y is the set of the nonanticipative processes Y : Ω × [0 ,T ] → R i.e. such that for each t , Y t only depends on information available up to t . I Two approaches: I Actuarial pricing I Arbitrage pricing 6 / 28 Actuarial pricing I Principle: contract value = sum of discounted value of expected cash flows. I Using past information/history F t , estimate objective probability distribution P of risk factors involved in contract. Then Π t ( H ) = B ( t,T ) E P [ H  F t ] I Used a lot for pricing of insurance contracts. I Rationale: insurance policies are sold in large numbers to many policyholders for which loss events are approximately independent....
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 Spring '12
 HeXuedong
 Probability theory, equivalent martingale measure, Arbitragefree pricing rules

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