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Pricing Interest Rate Derivative document sample
Remember the idea behind the BDT approach is that we match the model to the market prices of Z from exchange-traded options. These prices are used to build a binomial tree of the short-term such the model fits the current yield curve exactly. This approach is often called an arbitrage-fr The market data obviously includes risk premia associated with longer-term lending. Thus when of the short rate in the equivalent risk-neutral world. As such our tree can be used to price any Of course, at this point, this is a circular exercise (it would be reasonable to ask why we need a ra directly off Bloomberg). But, as noted, the model can deliver the price of any interest rate deriv Salomon Brothers seeking a quote on an interest rate collar. Salomon can use this model--calib fair value of such a derivative, and ascertain the risk of the position, and ultimately hedge the p Consider the following information obtained from Bloomberg. Maturity YTM (%) Volatility (Years) Of Yields (%) P0(T) Pricing 1 0.1 0.90909091 2 0.11 0.19 0.81162243 4.53697E-09 3 0.12 0.18 0.71178025 7.15713E-07 4 0.125 0.17 0.62429508 4.33964E-08 5 0.13 0.16 0.54275994 2.36621E-07 Short Rate Tree Zero price tree given Short Rate Tree 0.255245 0.217889 P11(2),P10(2) 0.194186 0.194767 0.837391 0.14318 0.160553 0.750704802 0.10 0.137668 0.148618 0.878991 0.097916 0.118304 0.815213318 0.097599 0.113404 0.911079 0.087173 0.086534 P11(4),P10(4) 0.588839 0.550976335 0.542760173 0.670891 0.643096045 0.741239 A Caplet is an interest-rate derivative that pays off the following for every dollar of notional princip The contract specifies the interest rate used, example 3-Month LIBOR, and its expiration date, alo with the difference between the yield setting and the expiration. Example: $10,000,000 caplet on 3-Month LIBOR struck at 3%, expiring on December 31, 2005. If on September 30, 2005, 3-Month LIBOR is 3.2%, then the holder of this caplet would receive {10,000,000 (.032 - .03) 92 } / 360, or $5,111.11 on December 31, 2005. (The strike rate is also called the cap rate.) A cap is a portfolio of caplets, the time between resets is called the tenor. (The compounding cor Note that a caplet is a call option on the interest rate. A floorlet is analogous to a caplet, but it pays off only if the prevaling rate is less than the floor rat A floor is a portfolio of floorlets. Floors, caps and swaps are connected to one-another through a put-call parity relationship. Consider a swap, cap and floor with the exact same terms (expiration date, tenor, strike rate). Value of Swap = Cap Price - Floor Price. (A long position in the cap and a short position in the The swap here is a "plain vanilla" (floating for fixed) s the swap (fixed) rate (using the same terms as the c Finally, a collar is a cap and a floor. Example: Consider a cap with a total life of 5 years, a tenor of 1 year, a notional principal of $10,000,000, and a cap the price of the cap given that the underlying LIBOR tree is the same as the one you obtained in (a). Cap Tree Caplet Prices 519777.6 458768.3 234331.6 39896.38 103401.3 105795.1 17188.53 48529.33082 49652.46 7554.283 0 23622.5523 7554.283 3440.284 0 3440.284244 0 0 0 0 0 95879.62 41935.47269 19061.5785 0 0 0 35050.65 15330.32166 6968.328027 0 0 0 0 0 0 Sum of Caplet prices= Cost of Tightening the cap rate to 18%: 38498.39 Consider a floor with a total life of 5 years, a tenor of 1 year, a notional principal of $10,000,000, and a flo the price of the floor given that the underlying 1-year LIBOR tree is the same as the one you obtained in ( Floor Tree Floorlet Prices 0 0 0 0 0 0 0 0 10167.97 0 0 3379.085198 0 22369.54 0 7433.987436 49119.73 0 16323.78 107827.5 77915.38 0 0 6788.886095 0 14935.54941 32795.95 0 0 0 0 0 0 0 0 0 Sum of Floorlet prices= Cost of Tightening the floor rate to 10%: 19490.13 10167.97 29658.1 he market prices of Zero-Coupon Treasury Securities and Implied Volatilities ree of the short-term interest rate. This tree is calibrated to these data--as called an arbitrage-free modeling approach. ending. Thus when we construct our tree, we have a picture of the dynamics be used to price any security. ask why we need a rather complicated model to produce prices that we can read ny interest rate derivative security. So, for example, a customer might call use this model--calibrated to prices on liquid securities--to determine the timately hedge the position. Vol Criterion -9E-08 8.09E-15 1.17E-07 5.26E-13 -3.4E-08 3.07E-15 -8.8E-08 6.37E-14 ee given Short Rate Tree P11(3),P10(3) 0.821093 0.70456 0.64568 0.861658 0.771697 0.727769 0.894211 0.826362 0.919817 0.796657 0.670685 0.836983 0.735682 0.870611 0.790821 0.898146 0.836345 0.920358 ar of notional principal: max( rt - r ,0) × d Notional Principal: s expiration date, along 360 Strike Prevaling LIBOR cember 31, 2005. Tenor this caplet would mber 31, 2005. he compounding corresponds to the tenor--note the last example has quarterly compounding.) ess than the floor rate. ty relationship. nor, strike rate). short position in the floor produces the same cash flows as a long position in the swap.) " (floating for fixed) swap--where the holder receives the prevaling (floating) rate and pays e same terms as the cap/floor). 10,000,000, and a cap rate of 19%. Find u obtained in (a). Princ 10000000 Cap 0.19 519777.6 229772.1 39896.38 17188.53 0 5-Year 0 0 0 0 228996.2 0 4-Year 0 0 3-Year 2-Year 49652 $10,000,000, and a floor rate of 9.5%. Find e one you obtained in (a). Princ 10000000 Floor 0.095 0 0 0 0 0 5-Year 0 0 35833.94 77915.38 0 0 4-Year 0 71993.6 3-Year 2-Year 10168 $10,000,000.00 0.03 0.032 92 pounding.) Remember the idea behind the BDT approach is that we match the model to the market prices of Z from exchange-traded options. These prices are used to build a binomial tree of the short-term such the model fits the current yield curve exactly. This approach is often called an arbitrage-fr The market data obviously includes risk premia associated with longer-term lending. Thus when of the short rate in the equivalent risk-neutral world. As such our tree can be used to price any Of course, at this point, this is a circular exercise (it would be reasonable to ask why we need a ra directly off Bloomberg). But, as noted, the model can deliver the price of any interest rate deriv Salomon Brothers seeking a quote on an interest rate collar. Salomon can use this model--calib fair value of such a derivative, and ascertain the risk of the position, and ultimately hedge the p Consider the following information obtained from Bloomberg. Maturity YTM (%) Volatility (Years) Of Yields (%) P0(T) Pricing 1 0.1 0.90909091 2 0.11 0.19 0.81162243 4.53697E-09 3 0.12 0.18 0.71178025 7.15713E-07 4 0.125 0.17 0.62429508 4.33964E-08 5 0.13 0.16 0.54275994 2.36621E-07 Short Rate Tree Zero price tree given Sho 0.255244816 0.217889 P11(2),P10(2) 0.194186 0.194766687 0.14318 0.160553 0.750704802 0.10 0.137668 0.148618346 0.097916 0.118304 0.815213318 0.097599 0.11340447 0.087173 0.086534227 P11(4),P10(4) 0.550976335 0.542760173 0.643096045 Here we can use the machinery of caps and floors to compute the swap rate. If you strike a cap a a swap. (Long one short other). So if use Solver to find the strike rate that makes the value of the floor equal to that of the cap, we Example: Cap Cap Tree Caplet Prices 1236769.225 1801369 1980817 793181.5322 1668123 1045844 396058.7056 1002506.774 969505.6 423276.7698 277964.1235 537391.8 406755.1 215462.3661 210516 120391.7358 55369.17 0 404569.7952 261689.0627 171146.1428 489772.5716 291161.7394 150783.2551 377722.0662 171691.8483 0 Sum of Caplet prices= Floor Tree Floorlet Prices 0 0 0 0 0 0 0 29658.10344 0 0 5374.818643 65247.83 0 11824.60101 101585.2 0 174982.4 123933.2596 0 11125.75149 24476.65327 0 4527.960371 9961.512817 0 8629.572939 18985.06047 Sum of Floorlet prices= Criterion: Difference between floor and cap: -972848.67 Squared Difference: 9.46435E+11 cap - floor 972848.7 Verification. Valuation of a fixed for floating swap at this swap rate: 1236769.225 1801369 1980817 793181.5322 1668123 1045844 $972,848.6704 969505.6 423276.7698 472143.9 406755.1 108930.7 120391.7358 -119613 -123933.26 Swaption Valuation of a swaption to enter a swap that pays fixed for floating swap at t Example: 2-Year option on 3-Year Swap, struck at 0.136159 Refer to the expiration date in the swap tree if the value in a node is positive, we would exercise the swapti That is, the swaption is an option to enter the swap tree at time 2. 1980817 1290401 809786.3951 969505.6 491129 108930.7 he market prices of Zero-Coupon Treasury Securities and Implied Volatilities ee of the short-term interest rate. This tree is calibrated to these data--as called an arbitrage-free modeling approach. ending. Thus when we construct our tree, we have a picture of the dynamics be used to price any security. sk why we need a rather complicated model to produce prices that we can read ny interest rate derivative security. So, for example, a customer might call use this model--calibrated to prices on liquid securities--to determine the imately hedge the position. Vol Criterion -8.98459E-08 8.09E-15 1.16887E-07 5.26E-13 -3.44947E-08 3.07E-15 -8.80206E-08 6.37E-14 ero price tree given Short Rate Tree 11(2),P10(2) P11(3),P10(3) 0.821093 0.837391 0.70456 0.64568 0.861658 0.878991 0.771697 0.727769 0.894211 0.911079 0.826362 0.919817 11(4),P10(4) 0.796657 0.670685333 0.588839 0.836983 0.73568161 0.670891 0.870611 0.790821252 0.741239 0.898146 0.836345132 0.920358 If you strike a cap and a floor at the same price, you have o that of the cap, we have the swap rate. Princ 10000000 Cap 0.1 1236769 833388.935 568368.3 793181.5 524085.7226 337164.8 423276.8 5-Year 243077.2055 135954.2 120391.7 55369.16861 0 967979.6378 623746.3 521758.224 301246.2 4-Year 163677.9291 74561.8 0 788702.3 331094.6 3-Year 0 2-Year let prices= 1002507 Princ 10000000 Floor 0.1 0 0 0 0 0 0 0 5-Year 0 25964.83 0 56997.9451 123933.3 0 0 0 0 4-Year 0 53746.6 117984.4409 0 0 3-Year 21873.8 2-Year rlet prices= 29658.1 r floating swap at this swap rate: would exercise the swaption in that node, otherwise we would not.