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					Derivatives - January 2009
            Question 1: Oil futures
            Spot price                    46.5
            IntRate                        2%
            Cost of storage                5%
            Forward price                  60

      (1.1) No arbitrage forward price   49.87 =SpotPrice*exp[(IntRate+CostOfStorage)]

      (1.3) Buy spot                     -46.50        ST
            Borrow                        46.50    -47.44
            Cost of storage                         -2.33
            Sell forward                            60.00 -ST
            Total                            0      10.24
Cost of storage is expressed as a percentage of the spot price.




Do not forget to include the cost of storage in the calculation of the arbitrage profit.
The no arbitrage forward price calculated in question (1.1) is based on the assumption
that the cost of storage is paid continuously.
The calculation here ignores this subtelty. Cost of storage is calculated as a percentage of the
final price.
Derivatives - January 2009
            Question 2 Euro-Bund Futures Contract

                                          Bond                A                   B
                     Maturity                             4/1/2018            4/7/2018
                     Coupon                                   4                  4.25
                     Quoted price                          111.69              112.89
                     Conversion factor                    0.865855            0.877459

          (2.1)      Cash forward price calculation
                     Month since last coupon                  0                  6
                     Accrued interest                         0                2.125
                     Cash spot price                        111.69             115.02

          (2.2)      Conversion factors are used to adjust the differences between the caracteristics of the
                     notional bond (coupon = 6%) and of the physical bonds.
                     They differ because of the different coupons (and also the different maturities).

          (2.3)      The cheapest to deliver bond minimizez the difference S-kF.
                     Here, F=                 124.75
                                          Bond               A                   B
                     s-kF =                                 3.67                3.43
                     => B is the CTD

          (2.4)                           F0                      124.75
                                          F1                      120.00
                                          n                        10.00
                                          Face value                100 m
                                                                  -47.50 m




          Data generation (the lines below are my calculations to generate the data)

                     Today                                    5/1/2009
                     Cont. comp. rate                            2.50% Continuous rate
                     An.comp.rate                                2.53% Annual compounding
                     Deliverable bonds                        A                B
                     Maturity                             4/1/2018         4/7/2018
                     Coupon                                   4               4.25
                     Months since last coupon today           0                 6
                     Nb coupons today                         9                10
                     Cash spot price                       111.69            115.02
                     Accrued interest                       0.00              2.13
                     Cash quoted price                     111.69            112.89

     (2.1) Forward contract maturity in                                3 months
                     Bond                                     A                   B
           Months to next coupon               9               3
           PV(Coupon)                        3.93            4.22
           Cash spot price - PV(Coupon)     107.76          110.79
           Cash forward price               107.76          111.49

Concordance Factors Calculation
          Coupon on notional bond                    6%
                                                      A               B
Months since last coupon                              3               9
Number coupon remaining                               9              10
Cash price                                        87.66           91.01
Concordance factor                               0.8666          0.8782

Si F =                             124.75
                               Bond                     A               B
                               S-kF                  3.57            3.33

Futures price calculation
Cash price                                       111.69          115.02
Forward price                                    107.76          111.49
Accrued interest                                   1.00            3.19
Clean forwardprice                               106.76          108.30
Futures price                                    123.19          123.31
                              C
                          4/1/2019
                             3.75
                           110.65
                          0.836179


                             0
                             0
                           110.65

s between the caracteristics of the

so the different maturities).




                             C
                            6.33




                                      This worsheet is more elaborate than the questions asked in the exam
                                      because it includes the calculation of the data.
Annual compounding
                              C
                          4/1/2019
                            3.75
                              0
                             10
                           110.65     Be sure to remember the difference between a clean price and a cash price!
                            0.00
                           110.65

                                      When calculating the cash forward price, be sure to keep track of coupon payment.
                                C
   9
 3.68
106.97
106.97




          C
          3
         10
      84.66
     0.8373


            C   The short will choose the physical bond that minimize S-kF.
         6.20   The cheapest to deliver bond is bond B.


     110.65
     106.97
       0.94
     106.03
     126.64
ns asked in the exam


                                        2009   2010   2011   2012   2013
                                    9             1      1      1      1
                                   10      1      1      1      1      1
                                   10             1      1      1      1

ean price and a cash price!




to keep track of coupon payment.
2014   2015   2016   2017   2018   2019
   1      1      1      1      1
   1      1      1      1      1
   1      1      1      1      1      1
Derivatives - January 2009
          Q3 Reverse Convertible

         Stock price         25       dt              0.25
         DivYield         0.00%       u              1.284
         Face value       1,000       p              0.448
         Coupon             10%
         n                40.00       PV(Principal amount)              980
         IntRate          2.00%       PV(Coupons)                        99
         Strike              25       Value of put options              174
         Volatility          0.5      Value of convertible              905
         Issue Price      1,010

                                               0      0.25      0.5     0.75       1
                       Coupon                          25       25       25       25
                       Principal                                                1,000
                       Dis. Factors                 0.9950   0.9900   0.9851   0.9802

         Stock price                        25.00    32.10    41.22    52.93    67.96
                                                     19.47    25.00    32.10    41.22
                                                              15.16    19.47    25.00
                                                                       11.81    15.16
                                                                                 9.20
         Reverse convertible               904.98   994.08 1039.68 1019.89      1,000
                                                    795.69  920.87 1019.89      1,000
                                                            656.16  803.68      1,000
                                                                    497.24        607
                                                                                  368
         Put option                          4.35     1.63     0.00     0.00     0.00
                                                      6.59     2.97     0.00     0.00
                                                               9.59     5.41     0.00
                                                                       13.07     9.84
                                                                                15.80
         Delta                              -0.39    -0.18     0.00     0.00
                                                     -0.67    -0.43     0.00
                                                              -1.00    -1.00
                                                                       -1.00

         # shares                           15.71     7.33     0.00     0.00    42.96
                                                     26.91    17.12     0.00    16.22
                                                              40.00    40.00     0.00
                                                                       40.00     0.00
                                                                                 0.00
Derivatives - January 2009
            Question 4 Using Black Scholes
                               1/1/2008 1/1/2009
            Index level          1,400      950             -450
            Dividend yield        2%        2%
            Striking price       1,400     1,400
            Maturity (years)      15        14
            Interest rate         3%        3%
            Volatility            20%       40%

          Put value               223.81     552.88
          Delta                   -0.208     -0.212
          Gamma                  0.00023    0.00018
          Theta                   0.601      5.677
          Elasticity               -1.30      -0.59
          Vega (per %)            13.54       9.04
          Rho (per %)             -77.23    -105.55
          N(d2)                    0.42       0.18


     (4.1) The delta of a put option is negative. The option value goes up if the level of the index goes down.
           The change of index was:         -450.00
           The delta approximation of the variation in the put value is:                         93.56
           The delta - gamma approximation is:                                                 116.86
           Compare this to the change in the value of the option:                              329.07

     (4.2) The other factor explaining the evolution of the put value is the sharp increase of the volatility.
           The impact can be measured using vega:                                            270.73

     (4.3) N(d2) is the risk-neutral probability of not exercizing the put option.
           As the index level drops, the RN probability of the option maturing in the money (ie being exercized) is higher.

     (4.4) Hedge short put
           Index                                 1.00
           Short put                             0.21
           Short -Delta shares                  -0.21
ng exercized) is higher.
Derivatives - January 2009
            Question 5
                    dt             0.25      Time interval                 Sigma       0.60%
            Time                           0       0.25        0.5           0.75          1
            Spot rates (cont.comp)               1.80%      2.20%          2.50%       2.70%
            Spot rate (simple)                   1.80%      2.21%          2.51%       2.71%
            Discount function         1.0000    0.9955     0.9891         0.9814      0.9734
            Fwd rate                             1.80%      2.61%          3.11%       3.31%

     (5.1) 6x9 FRA
                                   FRA = forward rate (si)                             3.11%
                                   Synthetic FRA
                                              Borrow for 6 m                          0.9891
                                              Invest for 9m FV =                      1.0078
                                              Forward rate                             3.11%

                                   Spot price ZC, 9m mat., F=100                       98.14
                                   Fwd price                                           99.23
                                   Forward rate                                        3.11%
     (5.2) LONG FRA

     (5.3) Current swap rate calculation   1=0.5*R*(SumDF)+DFT
                                 PV(Face)              97.34%
                                 Sum(DF)              393.94%
                                 Swap rate              2.70%

     (5.4)
             Short rate                  0.00         0.25       0.50       0.75
             (Simp.int.)       0       1.80%        2.91%      3.71%      4.21%
                               1                    2.31%      3.11%      3.61%
                               2                               2.51%      3.01%
                               3                                          2.41%
                               4

                                   6m Call on 3m rate     Strike          3.00%
                                              6-m rate Call on IR RN Proba
                                                  3.71%       0.176%        25%
                                                  3.11%       0.028%        50%
                                                  2.51%       0.000%        25%
                                              Value of call on IR
                                              Expected (risk neutral) future value              0.058%
                                              Present value                                     0.057%
                                              Binomial valuation              0.0         0.5       1.0
                                                                         0.057%       0.101%    0.176%
                                                                                      0.014%    0.028%
                                                                                                0.000%
                                                Valuation as a put on a zero coupon
          1-year Zero Coupon    0.00     0.25     0.50
                                0.987    0.991    0.998
                                         0.994    1.000
                                                  1.001

Put on Zero Coupon             0.057%   0.101%   0.176%
                                        0.014%   0.028%
                                                 0.000%
Long FRA
   0.75
 1.0075
100.75%
100.75%
100.75%

				
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