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BDT Model Applications © SimCorp Financial Training A/S

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AFIX5-900©SimCorp Financial Training A/S2 af 54 Applications 1.Pricing bonds 2.American calls on bonds 3.Step up callables 4.Danish mortgage backed bonds 5.Caps (Floors) 6.Forward starting swap 7.Swaptions 8.CTD

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AFIX5-900©SimCorp Financial Training A/S3 af 54 Pricing Bonds 3 year 10% Bullet Binominal treeBond Prices Bond prices are found by discounting one period at a time, backwards, begin-ning at maturity.

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AFIX5-900©SimCorp Financial Training A/S4 af 54 Pricing Bond Options 2 year American call on 3 year 10% bullet, strike Binominal treeBond PricesAmerican Call * 1.13 * The option is exercised immediately Using the BDT model the price of the American call option can be found to be Value of Callable Bond is: NonCall-CallOption = – 1.08 = 98.51

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AFIX5-900©SimCorp Financial Training A/S5 af 54 Pricing step-up Callables 3Y step-up Callable Bond, strike at Binominal treeBond PricesAmerican Call Value of Callable Note is = 98.45

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AFIX5-900©SimCorp Financial Training A/S6 af 54 Pricing Mortgage Bonds Danish Mortgage-Backed Securities Bond Pool of underlying Loans Callable Bond Model Prepayment Risk of Call Option Debtors are not homogenous: Several Call options Other Features: –Cost of Prepaying –Premium required –Prepayment behaviour (first, optimal) –Prepayment Model –Tax –DK Cash flows Path Dependency? Debtor Model

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AFIX5-900©SimCorp Financial Training A/S7 af 54 Pricing Danish Mortgage-Backed Securities Zero Yields Volatility Short rate model, e.g. BDT Debtor Model Short Rates Price MBS Price/Risk/Return Rentability Calculations

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AFIX5-900©SimCorp Financial Training A/S8 af 54 Caps/Floors Product description Long term options based on a money market rate at future dates (often 3M or 6M LIBOR). Caps ensure a maximum funding rate compared to floors which ensure a minimum deposit rate. A purchased collar is a combination of a long cap and a short floor. Time (months) Strike Libor Compensation from purchased cap …. Strike Libor Compensation from purchased floor Time (months) ….

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AFIX5-900©SimCorp Financial Training A/S9 af 54 Pricing an Interest Rate Cap 3Y Cap on 1Y rate, strike 10% 3Y Cap (1Y) = 1Y Call IRG (1Y) + 2Y Call IRG (1Y) Binomial tree 1Y Call IRG 2Y Call IRG Value 3Y Cap = = 1.01 Tree is in Bond yields, strike is Money Market Rate (here is no difference) Also beware of Day Counts

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AFIX5-900©SimCorp Financial Training A/S10 af 54 Pricing a Forward Starting Swap 1Y forward, annual in 2Y, 10%, receive floating Term structure from Binomial tree Fixed leg Floating leg Diff = FloatPV - FixPV = 0.40

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AFIX5-900©SimCorp Financial Training A/S11 af 54 Can we construct a forward starting swap using swaptions? Swaption - product description A swaption is a right not a duty to buy/sell a forward starting IRS.

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AFIX5-900©SimCorp Financial Training A/S12 af 54 Synthetic Construction of Fwd Starting Swap Buy 1 Payers swaption (Strike = 10%) Sell 1 Receivers swaption (Strike = 10%) The swaptions are European Style The swaptions expire 1 year from today The swaptions have a 2-year swap as underlying instrument Swap-rate > 10% Bought Payers swaption Swap-rate < 10% Not exercised Pay 10% and receive floating Net Profile = forward starting swap Sold Receiver Swaption Pay 10% and receive floating Pay 10% and receive floating Not exercised Counterparty asks you to pay 10% and receive floating

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AFIX5-900©SimCorp Financial Training A/S13 af 54 Pricing a European Swaption 1Y swaption into 2Y swap, strike 10% Binomial tree2Y swap in 1Y UP-state2Y swap in 1Y DOWN-state FixPV = FloatPV = 100 Float – Fix = 2.01 Strike if PAY fix FloatPV = 100 Float – Fix = Strike if RECEIVE fix

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AFIX5-900©SimCorp Financial Training A/S14 af 54 Pricing a European Swaption Payer Swaption (Pay Fix)Receiver Swaption (Receive Fix) Swaption Put-Call parity Payer Swaption-Receiver Swaption=Forward Start Swap (Rounding error)

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AFIX5-900©SimCorp Financial Training A/S15 af 54 Pricing CTD Futures Future with delivery of 2Y bond (at contract maturity) with Notional Coupon of 10% in 1Y Conversion factors calculated as price at maturity with yield = 10%

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AFIX5-900©SimCorp Financial Training A/S16 af 54 Pricing CTD Futures Cost-of-Carry Find forward prices of Bonds: #1: #2: Futures prices are: #1: #2: CTD is Bond #2 F = Cheapest to deliver Adjust for Conversion Factors

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AFIX5-900©SimCorp Financial Training A/S17 af 54 Pricing CTD Futures The BDT Model Binomial treeBond # 1Bond # 2

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AFIX5-900©SimCorp Financial Training A/S18 af 54 Pricing CTD Futures The BDT Model Level 1 - Up State Futures Prices Level 1 - Down State Futures Prices #2 is CTD#1 is CTD The Futures price is then the expected Forward price = Delivery option value is 0.4 bp.

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AFIX5-900©SimCorp Financial Training A/S19 af 54 CTD Futures Pricing Determinants of Delivery Option Value Futures Contract Maturity( ) Bond Maturity( ) Bond Coupon Differential( ) Volatility of Zero Yields( ) Zero Coupon Yield Curve Shape( ) Number of Deliverables( )

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AFIX5-900©SimCorp Financial Training A/S20 af 54 Processes

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AFIX5-900©SimCorp Financial Training A/S21 af 54 The BDT Model Mean Reversion

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AFIX5-900©SimCorp Financial Training A/S22 af 54 The BDT Model Mean fleeing

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AFIX5-900©SimCorp Financial Training A/S23 af 54 The BDT Model Evolution of local volatility

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AFIX5-900©SimCorp Financial Training A/S24 af 54 The BDT Model Evolution of forward volatility

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AFIX5-900©SimCorp Financial Training A/S25 af 54 The BDT Model TSOI: 3->5%; TSOV: 25->11%;0-30Y 0% 5% 10% 15% 20% 25% 0%5%10%15%20%25%30%35% Log normality of short rates after 15 years

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AFIX5-900©SimCorp Financial Training A/S26 af 54 The BDT Model Advantages Consistency Term structure of volatility American options Mean reverting rates Log-normal (i.e. postive) rates Benchmark term structure model? Problems One-factor model Link between mean reversion and forward volatility No analytical prices (numerical solution) Constant steps Hull/White have provided an improved model, but this is more Complicated Products that depend on more factors, e.g.: –Quantos (diff. swaps) –Spread options –Convertibles Summary

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AFIX5-900©SimCorp Financial Training A/S27 af 54 The first method The calculations

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