In this region when X is large and the pull back term is strongly negative we

# In this region when x is large and the pull back term

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In this region when X is large and the pull-back term is strongly negative, we use a branching in which the possibilities are stay-the-same, move down by Δ X , or move down even further by 2Δ X . The probabilities on the three branches top to bottom are q u = 7 6 + 1 2 ( a 2 j 2 t ) 2 3 aj Δ t ) (51) q m = 1 3 a 2 j 2 t ) 2 + 2 aj Δ t q d = 1 6 + 1 2 ( a 2 j 2 t ) 2 aj Δ t ) For X small enough, we switch to branching (b) which allows for two upward movements. We do this for j j min := j max . The probabilities in this case are q u = 1 6 + 1 2 ( a 2 j 2 t ) 2 + aj Δ t ) (52) q m = 1 3 a 2 j 2 t ) 2 2 aj Δ t q d = 7 6 + 1 2 ( a 2 j 2 t ) 2 + 3 aj Δ t ) 78

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Finally the Y values for the tree corresponding to the OU process with mean-level m and non-zero starting point Y 0 can be found by reversing the transformation (48). At the node ( n, j ), this gives Y n,j = X n,j + m + ( Y 0 m ) e - at n . 17 Credit Risk Part II The market in credit-linked derivative products grew astonishingly, from \$631 . 5 billion global volume in the first half of 2001, to above \$12 trillion through the first half of 2005 3 . The growth from mid-2004 through mid-2005 alone was 128%. Up till the 2008 crisis, they accounted for approximately 10% of the total OTC derivatives market. Many of the products that traded in large number up till the “credit crunch” are complex in their dependence on numerous underlying risks, while the quantitative technology for their valuation (and hedging) lags far behind. This is particularly true with respect of mortgage-backed securities (MBSs). The main difficulty is the high-dimensionality of the basket derivatives, discussed in the next section, which are typically written on hundreds of underlying names, such that computational efficiency severely limits model choice. 17.1 Single-name Instruments and Intensity-Based Models We discussed the pricing of defaultable bonds in Section 10 using structural models where default occurs when the value of the firm’s assets drops below some debt threshold. Such models have the benefit of providing economic intuition for the cause of default, but, as we have seen, they predict essentially zero yield spreads at short maturities, because defaults are “predictable” in the sense that we can see them coming. A popular alternative is to model default events as pure surprises using a jump process. The simplest example is a Poisson process with parameter ( a.k.a. intensity or hazard rate) λ > 0. The Poisson process starts at zero and jumps to one at some random time τ , which signals the firm defaulting. We do not need any specific details about Poisson processes, only that the distribution of τ is exponential with parameter λ : IP Q { τ T } = 1 e - λT . Note that everything here is under the risk-neutral probabilities Q (chosen by the market) that we use for pricing. Then the price of a corporate bond paying \$1 at time T if it does not default, and nothing otherwise, is simply: P D ( T ) = e - rT IP Q { τ > T } = e - ( r + λ ) T . Note that the effect of this simple intensity model is to effectively increase the discount rate from r to r + λ . The yield spread is given by Y ( T ) = 1 T log P D ( T ) parenleftbigg 1 T log( e - rT ) parenrightbigg = λ 3 Source: ISDA data reported at 79
is constant as a function of maturity T . However, market data usually reveals an upward sloping concave yield spread curve, which motivates generalizing these models to allow for stochastic intensity .
• Fall '11
• COULON
• Variance, Probability theory, Trigraph, Credit default swap, Wiener process

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