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This article discusses the Forward LIBOR Model and its application in valuating knockout cap securities. It covers topics such as lattice calculation, transitional probability, martingale measures, and valuation techniques. The impact of different measures and lattice generation methods on the pricing accuracy is examined. The article concludes with a discussion on the future research directions in incorporating transitional probabilities of other periods for valuing various derivatives.
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Lattice Calculation for Forward LIBOR Model Tadashi Uratani Hosei University uratani@k.hosei.ac.jp and Makoto Utsunomiya Bank of Tokyo-Mitsubishi Jafee 99
Outline • Interest rate sensitive security (e.g. Cap) • Definitions: Bond Price, LIBOR • Forward LIBOR Model • Martingale, No Arbitrage, and • Backward, Forward Induction Method • Lattice Calculation • Transitional Probability among Measures • Valuation of Knockout Cap Jafee 99
Short term interest（LIBOR） difference Interest rate derivatives E.g. Cap rate Company premium difference Exercise rate Bank date Jafee 99
３ month Returning date Borrowing date LIBORLondon Interbank Offer Rate Typical short-term interest rate in international capital market 3 month LIBOR 6 month LIBOR Jafee 99
Forward LIBORand Discount Bond Price Forward LIBOR Jafee 99
Forward LIBOR and Bond Price Jafee 99
Bond process Forward LIBOR Model Ito’s Lemma Valuation of derivatives by Forward LIBOR Jafee 99
Change of Probability Measure Girsanov Theorem Jafee 99
Market value of risk Martingale No Arbitrage Condition Jafee 99
Pricing by Martingale Martingale under Jafee 99
Risk Neutral Measure Jafee 99
Risk Adjusted Measure Jafee 99
Valuation of cap Generation :F. LIBOR Choice :Meas. Forward Backward Lattice martingale Jafee 99
Lattice Calculation Martingale Measures Jafee 99
Transition Probability Jafee 99
Relation of Transition Probabilities Jafee 99
One measure Jafee 99
Cap contract is knockout Knockout Cap rate LIBOR Year Jafee 99
Binominal tree under each measure Calculate the knocknout Unify one measure Valuation of derivative Jafee 99
Knockout Cap Underlying security:3 month LIBOR maturity 3 years knockout cap Jafee 99
fd La fd La fd La fd La Error Knockout rate Jafee 99
Why lambda affect? Large λ Lattice generation Jafee 99
End Jafee 99
単純モンテカルロ法 試行回数１００００回 時点分割１０ 時点分割１０ Knockout Cap 前進法、後退法 多重格子法 ３ヶ月LIBORを対象とした３年満期の０時点における価格 Jafee 99
Conclusion • 異なった測度による推移確率関数を関係付けることによりいくつかの金利派生証券を評価 • 推移確率関数を格子法に利用 • Knockout Capletにおいてλが大きいとき他の方法に比べ誤差が大きい • 各格子の作成方法を検討 • 推移確率関数の検討 • 他の期間同士のForward LIBORの推移確率の導入し、他の派生証券を評価 Jafee 99
