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L.H.Z. de Miranda
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This thesis is devoted to option pricing on backward-looking rates. For the last decades, interest rate products were often linked to IBOR rates. IBORs are short-term borrowing rates charged between global banks in the unsecured interbank market. The purpose of this thesis is to compare the Hull-White model to the Black-Karasinski model for the pricing of caps/floors on compounded rates. Both models are so-called short-rate models, which are widely used for interest rate modelling. Due to the IBOR reform, new products are expected to appear in the market. One type of these products is caps/floors linked to the new Risk-Free Rate (RFR). The new RFRs will be in-arrears backward-looking rates and, as a consequence, have an impact on the choice of pricing models.
This thesis considers caps/floors on the new compounded RFR rates. For both models various pricing techniques for caps/floors on compounded rates are investigated. For the Hull-White model, the pricing kernel approach and a Monte Carlo simulation are explored. The pricing kernel approach yields an analytic formula for caps/floors on compounded rates. This formula is also used for the comparison. For the Black-Karasinski model, the pricing kernel approach, the trinomial tree method and the Monte Carlo simulation are considered. The pricing kernel approach yields a semi-analytic formula for caps/floors on compounded rates. However, in practice the computation time of this semi-analytic formula turned out to be substantial. Further, despite the fast computation time of the trinomial tree for LIBOR caps/floors, the trinomial tree method is rather slow for the caps/floors on compounded rates. As a result, the Monte Carlo simulation is the most suitable pricing technique, along the three
explored methods, for caps/floors on compounded rates under the Black-Karasinski model. Therefore, the Monte Carlo simulation is used for pricing caps on compounded rates in the model comparison.
Since there exists no liquid market yet for caps/floors linked to the new RFR, the models are calibrated to a proxy market. First, the Black-Karasinski model is calibrated to the proxy market using a heuristic approach. This heuristic approach is chosen as a compromise between computation time and accuracy. Then, the Hull-White model is calibrated to the Black-Karasinski model using the stripping method. Having both models calibrated, the price of caplets/floorlets on compounded rates are calculated. Thereafter, these prices are inverted to Bachelier implied volatilities for a uniform comparison. From the comparison of the two models, a difference in Bachelier implied volatility is observed in a range of 4 to 4 bps. This is of one order less than the volatility itself. ...
This thesis considers caps/floors on the new compounded RFR rates. For both models various pricing techniques for caps/floors on compounded rates are investigated. For the Hull-White model, the pricing kernel approach and a Monte Carlo simulation are explored. The pricing kernel approach yields an analytic formula for caps/floors on compounded rates. This formula is also used for the comparison. For the Black-Karasinski model, the pricing kernel approach, the trinomial tree method and the Monte Carlo simulation are considered. The pricing kernel approach yields a semi-analytic formula for caps/floors on compounded rates. However, in practice the computation time of this semi-analytic formula turned out to be substantial. Further, despite the fast computation time of the trinomial tree for LIBOR caps/floors, the trinomial tree method is rather slow for the caps/floors on compounded rates. As a result, the Monte Carlo simulation is the most suitable pricing technique, along the three
explored methods, for caps/floors on compounded rates under the Black-Karasinski model. Therefore, the Monte Carlo simulation is used for pricing caps on compounded rates in the model comparison.
Since there exists no liquid market yet for caps/floors linked to the new RFR, the models are calibrated to a proxy market. First, the Black-Karasinski model is calibrated to the proxy market using a heuristic approach. This heuristic approach is chosen as a compromise between computation time and accuracy. Then, the Hull-White model is calibrated to the Black-Karasinski model using the stripping method. Having both models calibrated, the price of caplets/floorlets on compounded rates are calculated. Thereafter, these prices are inverted to Bachelier implied volatilities for a uniform comparison. From the comparison of the two models, a difference in Bachelier implied volatility is observed in a range of 4 to 4 bps. This is of one order less than the volatility itself. ...
This thesis is devoted to option pricing on backward-looking rates. For the last decades, interest rate products were often linked to IBOR rates. IBORs are short-term borrowing rates charged between global banks in the unsecured interbank market. The purpose of this thesis is to compare the Hull-White model to the Black-Karasinski model for the pricing of caps/floors on compounded rates. Both models are so-called short-rate models, which are widely used for interest rate modelling. Due to the IBOR reform, new products are expected to appear in the market. One type of these products is caps/floors linked to the new Risk-Free Rate (RFR). The new RFRs will be in-arrears backward-looking rates and, as a consequence, have an impact on the choice of pricing models.
This thesis considers caps/floors on the new compounded RFR rates. For both models various pricing techniques for caps/floors on compounded rates are investigated. For the Hull-White model, the pricing kernel approach and a Monte Carlo simulation are explored. The pricing kernel approach yields an analytic formula for caps/floors on compounded rates. This formula is also used for the comparison. For the Black-Karasinski model, the pricing kernel approach, the trinomial tree method and the Monte Carlo simulation are considered. The pricing kernel approach yields a semi-analytic formula for caps/floors on compounded rates. However, in practice the computation time of this semi-analytic formula turned out to be substantial. Further, despite the fast computation time of the trinomial tree for LIBOR caps/floors, the trinomial tree method is rather slow for the caps/floors on compounded rates. As a result, the Monte Carlo simulation is the most suitable pricing technique, along the three
explored methods, for caps/floors on compounded rates under the Black-Karasinski model. Therefore, the Monte Carlo simulation is used for pricing caps on compounded rates in the model comparison.
Since there exists no liquid market yet for caps/floors linked to the new RFR, the models are calibrated to a proxy market. First, the Black-Karasinski model is calibrated to the proxy market using a heuristic approach. This heuristic approach is chosen as a compromise between computation time and accuracy. Then, the Hull-White model is calibrated to the Black-Karasinski model using the stripping method. Having both models calibrated, the price of caplets/floorlets on compounded rates are calculated. Thereafter, these prices are inverted to Bachelier implied volatilities for a uniform comparison. From the comparison of the two models, a difference in Bachelier implied volatility is observed in a range of 4 to 4 bps. This is of one order less than the volatility itself.
This thesis considers caps/floors on the new compounded RFR rates. For both models various pricing techniques for caps/floors on compounded rates are investigated. For the Hull-White model, the pricing kernel approach and a Monte Carlo simulation are explored. The pricing kernel approach yields an analytic formula for caps/floors on compounded rates. This formula is also used for the comparison. For the Black-Karasinski model, the pricing kernel approach, the trinomial tree method and the Monte Carlo simulation are considered. The pricing kernel approach yields a semi-analytic formula for caps/floors on compounded rates. However, in practice the computation time of this semi-analytic formula turned out to be substantial. Further, despite the fast computation time of the trinomial tree for LIBOR caps/floors, the trinomial tree method is rather slow for the caps/floors on compounded rates. As a result, the Monte Carlo simulation is the most suitable pricing technique, along the three
explored methods, for caps/floors on compounded rates under the Black-Karasinski model. Therefore, the Monte Carlo simulation is used for pricing caps on compounded rates in the model comparison.
Since there exists no liquid market yet for caps/floors linked to the new RFR, the models are calibrated to a proxy market. First, the Black-Karasinski model is calibrated to the proxy market using a heuristic approach. This heuristic approach is chosen as a compromise between computation time and accuracy. Then, the Hull-White model is calibrated to the Black-Karasinski model using the stripping method. Having both models calibrated, the price of caplets/floorlets on compounded rates are calculated. Thereafter, these prices are inverted to Bachelier implied volatilities for a uniform comparison. From the comparison of the two models, a difference in Bachelier implied volatility is observed in a range of 4 to 4 bps. This is of one order less than the volatility itself.
De locaties van ambulancestations zijn van belang bij het redden van levens. Dit onderzoek beschrijft een Mixed Integer Lineair Programming (MILP) model die de optimale locaties van ambulances bepaald zodat het dekkingsniveau maximaal is. Het dekkingsniveau is het percentage van het totale aantal oproepen waarbij een ambulance binnen een bepaalde vaste tijd op de locatie van het incident kan zijn. Het model houdt rekening met de kans dat een ambulance bezet kan zijn en de reistijden die afhankelijk zijn van de locatie van de ambulance en het incident. Er wordt ook rekening gehouden met de kans dat een ambulance op een oproep reageert. Deze kans wordt bepaald met een benadering van het zogeheten hypercube model. Hierbij wordt aangenomen dat de kans dat een ambulance reageert op een oproep onafhankelijk is van andere ambulances. In werkelijkheid is dit niet zo, daarom moet dit worden gecorrigeerd met correctiefactoren. Welke ook bepaald worden met het hypercube model. Het MILP-model is in dit onderzoek toegepast op twee regio’s in Nederland. Er is gekeken naar de optimale locaties om ambulances te plaatsen en het maximale dekkingsniveau. Het MILP-model is op een aantal regio’s toegepast met en zonder de correctiefactoren. Hieruit blijkt dat het dekkingsniveau hoger is bij het plaatsen van de ambulances met behulp van het MILP-model zonder de correctiefactoren en dat hetMILP-model niet geschikt is voor grote datasets.
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De locaties van ambulancestations zijn van belang bij het redden van levens. Dit onderzoek beschrijft een Mixed Integer Lineair Programming (MILP) model die de optimale locaties van ambulances bepaald zodat het dekkingsniveau maximaal is. Het dekkingsniveau is het percentage van het totale aantal oproepen waarbij een ambulance binnen een bepaalde vaste tijd op de locatie van het incident kan zijn. Het model houdt rekening met de kans dat een ambulance bezet kan zijn en de reistijden die afhankelijk zijn van de locatie van de ambulance en het incident. Er wordt ook rekening gehouden met de kans dat een ambulance op een oproep reageert. Deze kans wordt bepaald met een benadering van het zogeheten hypercube model. Hierbij wordt aangenomen dat de kans dat een ambulance reageert op een oproep onafhankelijk is van andere ambulances. In werkelijkheid is dit niet zo, daarom moet dit worden gecorrigeerd met correctiefactoren. Welke ook bepaald worden met het hypercube model. Het MILP-model is in dit onderzoek toegepast op twee regio’s in Nederland. Er is gekeken naar de optimale locaties om ambulances te plaatsen en het maximale dekkingsniveau. Het MILP-model is op een aantal regio’s toegepast met en zonder de correctiefactoren. Hieruit blijkt dat het dekkingsniveau hoger is bij het plaatsen van de ambulances met behulp van het MILP-model zonder de correctiefactoren en dat hetMILP-model niet geschikt is voor grote datasets.