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# Risk Management and Derivatives Final Q1. You work for a UK building society (ie. mortgage bank) which is considering to launch a 10 year fixed mortgage based on the potential demand for such a...

Risk Management and Derivatives Final
Q1. You work for a UK building society (ie. mortgage bank) which is considering to launch a 10 year fixed mortgage based on the potential demand for such a product and similar products which are available from some of your competitors. but in the future it is expected that rates are going to rise (possible due to inflationary pressure). The Board of Directors knows that you have just completed a module in Derivatives and Risk Management and they want you to present to them the case of offering such a product with the potential risks to your institution highlighted and how you could deal with them. How would you launch a 10 year fixed rate mortgage product? [35 marks]
Q2. Use the BOPM to price a call and put option. Use Excel to build a 10 nodes tree. The time to expiry should be 1 year and interest rates are 1% per annum. You need to choose your own stock price, strike price and volatility. (Important : You need to choose different parameters than given in the Excel spreadsheet provided for the module). Price a European Call, European Put, American Call and American Put option. Compare your results against the Black-Scholes model and make suggestions how you could improve on the calculations undertaken in the BOPM. [30 marks]
Q3.You are a UK based asset management company and you hold the following international portfolio. On the 20th of December 2018 you set up the following international portfolio. You investment is just less than Â£10,000. Use data provided in â€˜cw Data 2020.xlsxâ€™.
CountryÂ Â
Company
Number of Shares
USÂ Â
Bank of AmericaÂ Â
85
GermanyÂ Â
BMW
30
Germany Â
Lufthansa
130
US
Amazon
3
UK
Tesco
273
(a.) Calculate the 10 day VaR of your portfolio at the 95% confidence level based on the variance-covariance method on the 10th June 2019, 24th June 2019, and 8th July 2019.
(b.) Repeat the calculations above, but use the exponential moving average for your volatility forecast in the calculation of the VaR using the variance-covariance method.
(c.) Repeat the calculations of the VaR of your portfolio using historic simulation.
(d.) Calculate the actual change in the value of your portfolio over the following 10 days. Compare your calculations with your VaR calculations performed in (a.) to (c.) and critically asses them. Comment critical on the practical implications. Explain what we understand by backtesting, how you would do it and what is the significance of it.
(Make any reasonable assumptions you need for the calculations.) [35 marks]
Hint : You have to draw standard normally distributed random variables. In Excel that can be done with the following command : NORMSINV(RAND())

Normal Distribution
Cumulative Normal Distribution : N(0,1)
z    0    0.01    0.02    0.03    0.04    0.05    0.06    0.07    0.08    0.09
0    0.5    0.504    0.508    0.512    0.516    0.5199    0.5239    0.5279    0.5319    0.5359
0.1    0.5398    0.5438    0.5478    0.5517    0.5557    0.5596    0.5636    0.5675    0.5714    0.5753
0.2    0.5793    0.5832    0.5871    0.591    0.5948    0.5987    0.6026    0.6064    0.6103    0.6141
0.3    0.6179    0.6217    0.6255    0.6293    0.6331    0.6368    0.6406    0.6443    0.648    0.6517
0.4    0.6554    0.6591    0.6628    0.6664    0.67    0.6736    0.6772    0.6808    0.6844    0.6879
0.5    0.6915    0.695    0.6985    0.7019    0.7054    0.7088    0.7123    0.7157    0.719    0.7224
0.6    0.7257    0.7291    0.7324    0.7357    0.7389    0.7422    0.7454    0.7486    0.7517    0.7549
0.7    0.758    0.7611    0.7642    0.7673    0.7704    0.7734    0.7764    0.7794    0.7823    0.7852
0.8    0.7881    0.791    0.7939    0.7967    0.7995    0.8023    0.8051    0.8078    0.8106    0.8133
0.9    0.8159    0.8186    0.8212    0.8238    0.8264    0.8289    0.8315    0.834    0.8365    0.8389
1    0.8413    0.8438    0.8461    0.8485    0.8508    0.8531    0.8554    0.8577    0.8599    0.8621
1.1    0.8643    0.8665    0.8686    0.8708    0.8729    0.8749    0.877    0.879    0.881    0.883
1.2    0.8849    0.8869    0.8888    0.8907    0.8925    0.8944    0.8962    0.898    0.8997    0.9015
1.3    0.9032    0.9049    0.9066    0.9082    0.9099    0.9115    0.9131    0.9147    0.9162    0.9177
1.4    0.9192    0.9207    0.9222    0.9236    0.9251    0.9265    0.9279    0.9292    0.9306    0.9319
1.5    0.9332    0.9345    0.9357    0.937    0.9382    0.9394    0.9406    0.9418    0.9429    0.9441
1.6    0.9452    0.9463    0.9474    0.9484    0.9495    0.9505    0.9515    0.9525    0.9535    0.9545
1.7    0.9554    0.9564    0.9573    0.9582    0.9591    0.9599    0.9608    0.9616    0.9625    0.9633
1.8    0.9641    0.9649    0.9656    0.9664    0.9671    0.9678    0.9686    0.9693    0.9699    0.9706
1.9    0.9713    0.9719    0.9726    0.9732    0.9738    0.9744    0.975    0.9756    0.9761    0.9767
2    0.9772    0.9778    0.9783    0.9788    0.9793    0.9798    0.9803    0.9808    0.9812    0.9817
2.1    0.9821    0.9826    0.983    0.9834    0.9838    0.9842    0.9846    0.985    0.9854    0.9857
2.2    0.9861    0.9864    0.9868    0.9871    0.9875    0.9878    0.9881    0.9884    0.9887    0.989
2.3    0.9893    0.9896    0.9898    0.9901    0.9904    0.9906    0.9909    0.9911    0.9913    0.9916
2.4    0.9918    0.992    0.9922    0.9925    0.9927    0.9929    0.9931    0.9932    0.9934    0.9936
2.5    0.9938    0.994    0.9941    0.9943    0.9945    0.9946    0.9948    0.9949    0.9951    0.9952
2.6    0.9953    0.9955    0.9956    0.9957    0.9959    0.996    0.9961    0.9962    0.9963    0.9964
2.7    0.9965    0.9966    0.9967    0.9968    0.9969    0.997    0.9971    0.9972    0.9973    0.9974
2.8    0.9974    0.9975    0.9976    0.9977    0.9977    0.9978    0.9979    0.9979    0.998    0.9981
2.9    0.9981    0.9982    0.9982    0.9983    0.9984    0.9984    0.9985    0.9985    0.9986    0.9986
3    0.9987    0.9987    0.9987    0.9988    0.9988    0.9989    0.9989    0.9989    0.999    0.999
3.1    0.999    0.9991    0.9991    0.9991    0.9992    0.9992    0.9992    0.9992    0.9993    0.9993
3.2    0.9993    0.9993    0.9994    0.9994    0.9994    0.9994    0.9994    0.9995    0.9995    0.9995
3.3    0.9995    0.9995    0.9995    0.9996    0.9996    0.9996    0.9996    0.9996    0.9996    0.9997
3.4    0.9997    0.9997    0.9997    0.9997    0.9997    0.9997    0.9997    0.9997    0.9997    0.9998
BS Option Pricing Model
Calculating the BS option price
Put-call parity concept is used here to calculate the Put premium.
only change the values in the yellow shaded area. Do not change anything else.
S        120
K        125
r        0.02
T         XXXXXXXXXX    165    days
d        0
Sigma        0.2
Auxillary inputs to calculate B-S premia
d1    d2    N -"dash"
(d1)
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
Imp. Vol. Trial and E
o
Calculating the Implied Volatility (using Trial and E
or)
Data Inputs
S        164
K        165
r        0.0521
T        0.0959
d        0
Trial and E
or calculations
Choose different values for "sigma" and calculate the "theoretical"
call premium using B-S until the latter equals the actual quoted call premium
Trial    d1    d2    N -"dash"    Call
Sigma                using B-S
0.281     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.282     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.283     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.284     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.285     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.286     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.287     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.288     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
0.289     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
Imp. Vol. Sove
Calculating the Implied Volatilities (Using "Solver")
Data Inputs
only change the values in the yellow shaded area. Do not change anything else.
S        164
K        165
r        0.0521
T        0.0959
d        0
Actual Call Premum        5.75
Cell to be minimised by SOLVER( Actual premium - B-S Premium)^2 =                             XXXXXXXXXX
Start value (and final value) for Sigma for use in SOLVER =                            0.2
Auxillary inputs to calculate B-S premia
d1    d2    N -"dash"
(d1)
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX

Daily ExR Returns
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Answered Same Day Jul 31, 2021

## Solution

Neenisha answered on Aug 06 2021
Question 1
Fixed rate mortgages is a loan for home which has fixed interest rate for the entire term. This means that the interest rate which is decided today will remain through the tenure of the mortgage. They usually have term of 10 â€“ 3- years.
These mortgages are prefe
ed because they are predictable and the person who has bo
owed the loan knows that what amount is to be paid in every instalment and thus there is no volatility. This is because there is no fluctuation in the interest rates and they remain same.
Advantages of 10 year fixed rate mortgages
The major advantage is that they are fixed for long term and there is no fluctuation. This means that the bo
ower can budget the loan and its payment very easily.
Disadvantage of 10 year fixed rate mortgages
The major disadvantage is that since the interest rate is fixed, therefore, there is a possibility that the rate decreases in future and thus this means that you took a loan at high interest rate.
In United Kingdom these loans or mortgages are widely used especially by building societies, lenders prefer different type of mortgages from variable interest rate to fixed interest rate.
Question 2
Binomial Pricing Model
Stock Price
810
Strike Price
800
Interest Rate
1%
Volatility
30%
Dividends
0%
Time to Maturity in Years
1
Binomial Steps
10
Delta T
0.1
Â
Â
Up Facto
1.10
Down Facto
0.91
Growth Factor (a)
1.00
p
0.48
q
0.52
European Call Option

2091.7

1291.6

1902.4

1103.2
1730.2

1730.2

930.2

931.8
1573.6

1573.6

774.4
1431.2

776.0
1431.2

631.2

1431.2

632.8
1301.6

634.4
1301.6

502.4
1183.8

1301.6

504.0
1183.8

383.8

505.6
1183.8

385.4
1076.7

1183.8

387.0
1076.7

277.5
979.2

391.1
1076.7

279.1
979.2

179.2

1076.7

285.5
979.2

180.8
890.6

293.2
979.2

191.8
890.6

91.4
810.0

979.2

202.7
890.6

111.1
810.0

10.0

213.1
890.6

126.3
810.0

46.5
736.7

890.6

139.1
810.0

65.6
736.7

4.8
670.0

150.5
810.0

80.3
736.7

23.6
670.0

0.0
810.0

92.6
736.7

37.7
670.0

2.3
609.4

103.5
736.7

49.6
670.0

11.9
609.4

0.0
554.2

60.1
670.0

21.3
609.4

1.1
554.2

0.0

30.0
609.4

6.0
554.2

0.0
504.1

11.8
554.2

0.5
504.1

0.0
458.4

3.0
504.1

0.0
458.4

0.0

0.3
458.4

0.0
416.9

0.0
416.9

0.0
379.2

0
379.2

0.0

0.0
344.9

0.0
313.7

0.0

European Put Option

2091.7

0.0

1902.4

0.0
1730.2

1730.2

0.0

0.0
1573.6

1573.6

0.0
1431.2

0.0
1431.2

0.0

1431.2

0.0
1301.6

0.0
1301.6

0.0
1183.8

1301.6

0.0
1183.8

0.0

0.0
1183.8

0.0
1076.7

1183.8

0.0
1076.7

0.0
979.2

2.5
1076.7

0.0
979.2

0.0

1076.7

4.8
979.2

0.0
890.6

10.9
979.2

9.4
890.6

0.0
810.0

979.2

18.7
890.6

18.1
810.0

0.0

27.5
890.6

31.7
810.0

34.9
736.7

890.6

42.9
810.0

52.4
736.7

67.3
670.0

52.7
810.0

65.5
736.7
Â
84.5
670.0

130.0
810

76.3
736.7

97.1
670.0

130.7
609.4

85.6
736.7

107.4
670.0

138.7
609.4

189.8
554.2

116.2
670.0

146.5
609.4

189.3
554.2

245.8

153.6
609.4

192.6
554.2

244.2
504.1

196.8
554.2

243.1
504.1

295.1
458.4

244.0
504.1

293.5
458.4

341.6

292.2
458.4

340.0
416.9

338.4
416.9

382.3
379.2

380.7
379.2

420.8

419.2
344.9

454.3
313.7

486.3
American Call Options

2091.7

0.0

1902.4

0.0
1730.2

1730.2

0.0

0.0
1573.6

1573.6

0.0
1431.2

0.0
1431.2

0.0

1431.2

0.0
1301.6

0.0
1301.6

0.0
1183.8

1301.6

0.0
1183.8

0.0

0.0
1183.8

0.0
1076.7

1183.8

0.0
1076.7

0.0
979.2

2.5
1076.7

0.0
979.2

0.0

1076.7

4.8
979.2

0.0
890.6

10.9
979.2

9.4
890.6

0.0
810.0

979.2

18.8
890.6

18.1
810.0

0.0

27.6
890.6

31.7
810.0

34.9
736.7

890.6

43.1
810.0

52.5
736.7

67.3
670.0

53.0
810.0

65.8
736.7

84.7
670.0

130.0
810.0

76.7
736.7

97.5
670.0

131.1
609.4

86.2
736.7

108.1
670.0

139.5
609.4

190.6
554.2

117.2
670.0

147.6
609.4

190.6
554.2

245.8

155.0
609.4

194.4
554.2

245.8
504.1

198.8
554.2

245.8
504.1

295.9
458.4

246.8
504.1

295.9
458.4

341.6

295.9
458.4

341.6
416.9

341.6
416.9

383.1
379.2

383.1
379.2

420.8

420.8
344.9

455.1
313.7

486.3

American Put Option

2091.7

0.0

1902.4

0.0
1730.2

1730.2

0.0

0.0
1573.6

1573.6

0.0
1431.2

0.0
1431.2

0.0

1431.2

0.0
1301.6

0.0
1301.6

0.0
1183.8

1301.6

0.0
1183.8

0.0

0.0
1183.8

0.0
1076.7

1183.8

0.0
1076.7

0.0
979.2

2.5
1076.7

0.0
979.2

0.0

1076.7

4.8
979.2

0.0
890.6

10.9
979.2

9.4
890.6

0.0
810.0

979.2

18.8
890.6

18.1
810.0

0.0

27.6
890.6

31.7
810.0

34.9
736.7

890.6

43.1
810.0

52.5
736.7

67.3
670.0

53.0
810.0

65.8
736.7

84.7
670.0

130.0
810.0

76.7
736.7

97.5
670.0

131.1
609.4

86.2
736.7

108.1
670.0

139.5
609.4

190.6
554.2

117.2
670.0

147.6
609.4

190.6
554.2

245.8

155.0
609.4

194.4
554.2

245.8
504.1

198.8
554.2

245.8
504.1

295.9
458.4

246.8
504.1

295.9
458.4

341.6

295.9
458.4

341.6
416.9

341.6
416.9

383.1
379.2

383.1
379.2

420.8

420.8
344.9

455.1
313.7

486.3

Stock Price
810
Strike Price
800
Interest Rate
1%
Volatility
30%
Dividends
0%
Time to Maturity in Years
1
Â
Â

d1
0.2247
d2
-0.0753
N (d1)
0.5889
N (d2)
0.4700
N (-d1)
0.4111
N (-d2)
0.5300

Call Option
104.754629
Put Option
86.7944964
Type of Option
Method Used
Option Price
European Call Option
Binomial Pricing Model
\$ 103.52
European Put Option
Binomial Pricing Model
\$ 85.56
American Call Option
Binomial Pricing Model
\$ 103.52
American Put Option
Binomial Pricing Model
\$ 86.17
Â

Â
Call Option
Black Scholes
\$ 104.75
Put Option
Black Scholes
\$ 86.79
According to binomial pricing model the price of European Call Option and American Call option the price is \$ 103.52. However using Black Scholes, we get the price of call option as \$ 104.75. In case of put option the price of European Put option is \$ 85.56 and American Option is \$ 86.17 in Binomial Pricing Model. In case of Black Scholes Model, the put option price is \$ 86.79. Therefore, the price of call and put option is more when computed through Black Scholes. In case of Binomial Pricing Model we can improve our calculations by increasing the number of steps. There can be 25 steps to get precise answer. Since the difference between 25 steps and 1000 steps is very less. Thus we need to have 25 steps to get the precise results.
Question 3
Value at Risk (VaR)
Stock Returns

Â
Bank of...
SOLUTION.PDF