99/08/29Jafee 99 Lattice Calculation for Forward LIBOR Model Tadashi Uratani Hosei University and Makoto Utsunomiya Bank of Tokyo-Mitsubishi
99/08/29Jafee 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
99/08/29Jafee 99 Interest rate derivatives E.g. Cap rate date Short term interest （ LIBOR ） Exercise rate Company Bank difference premium difference
99/08/29Jafee 99 LIBOR London Interbank Offer Rate Typical short-term interest rate in international capital market 3 month LIBOR 6 month LIBOR Borrowing date Returning date ３ month
99/08/29Jafee 99 Forward LIBOR and Discount Bond Price Forward LIBOR
99/08/29Jafee 99 Forward LIBOR and Bond Price
99/08/29Jafee 99 Forward LIBOR Model Bond process Ito’s Lemma Valuation of derivatives by Forward LIBOR
99/08/29Jafee 99 Change of Probability Measure Girsanov Theorem
99/08/29Jafee 99 Martingale No Arbitrage Condition Market value of risk
99/08/29Jafee 99 Pricing by Martingale Martingale under
99/08/29Jafee 99 Risk Neutral Measure
99/08/29Jafee 99 Risk Adjusted Measure
99/08/29Jafee 99 Valuation of cap Generation :F. LIBOR Choice :Meas. Backward Forward Lattice martingale
99/08/29Jafee 99 Lattice Calculation Martingale Measures
99/08/29Jafee 99 Transition Probability
99/08/29Jafee 99 Relation of Transition Probabilities
99/08/29Jafee 99 One measure
99/08/29Jafee 99 Knockout Cap rate Year LIBOR Cap contract is knockout
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99/08/29Jafee 99 Binominal tree under each measure Calculate the knocknoutUnify one measureValuation of derivative
99/08/29Jafee 99 Knockout Cap Underlying security:3 month LIBOR maturity 3 years knockout cap
99/08/29Jafee 99 fd La fd La fd La fd La Knockout rate Error
99/08/29Jafee 99 Why lambda affect? Large λ Lattice generation
99/08/29Jafee 99 End
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99/08/29Jafee 99 Knockout Cap 単純モンテカルロ法 試行回数１００００ 回 時点分割１０ 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
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99/08/29Jafee 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