Qus: Amount A is invested at 10% per annum compounded m times per year. What is effective interest rate (Re)?

Nominal interest rate is 10% per annum

Consider r = 10% = 0.10
The value of A at the end of one year with r per annum compounded m times per year = A(1+r/m)^m
The value of A at the end of one year with effective interest rate (Re) per annum = A(1+Re)
A(1+r/m)^m = A(1+Re) --> Re = (1+r/m)^m - 1
Re = (1+0.10/m)^m - 1 for Nominal Rate r = 10% = 0.10
r = m((1+Re)^(1/m) - 1)

Effectiv
Number of e Nominal
Compoundin Effective Interest From Interest
g Interest Rate in Effective Rate in
Per Year m rate Re % to Nominal %
Annual Compounding 1 0.1 10% 0.1 10%
Semiannual
Compounding 2 0.1025 10.25% 0.1 10%
Quaterly Compounding 4 0.103812 10.38% 0.1 10%
Monthly Compounding 12 0.104713 10.47% 0.1 10%
Weekly Compounding 52 0.105064 10.51% 0.1 10%
Daily Compounding 365 0.105155 10.52% 0.1 10%
Qus: Amount A is invested at 10% per annum compounded m times per year. What is effective interest rate (Re)?

Nominal interest rate is 10% per annum

r = 10% = 0.10

Effectiv
Number of e Nominal
Compoundin Effective Interest From Interest
g Interest Rate in Effective Rate in
Per Year m rate Re % to Nominal %
Annual Compounding 1 0.1 10% 0.1 10%
Semiannual
Compounding 2 0.1025 10.25% 0.1 10%
Quaterly Compounding 4 0.103812 10.38% 0.1 10%
Monthly Compounding 12 0.104713 10.47% 0.1 10%
Weekly Compounding 52 0.105064 10.51% 0.1 10%
Daily Compounding 365 0.105155 10.52% 0.1 10%
 Practical -1: CONTINUOUS COMPOUNDING
Initial Deposite 1000
Interest Rate 5% Practical-2: Effect of increasing the compounded frequency at 10 % per annum
Time of Maturity 2
Value of investment at the time of maturit 1105.170918 <--=B2*EXP(B3*B4) Invested Amount (A) 100
Year 1

Compunding Frequency
Value of 100 at the end of 1 yea
1 110
2 110.25
4 110.3812891
12 110.4713067
52 110.5064793
365 110.5155782

100
1
to Calculate Nominal Rate (Two argument are needed ROI, Compounded per 1 110
2 110.25
Effective Rate of Interest Rate 7% 4 110.3812891
Compounded per year 2 12 110.4713067
0.06881608656 52 110.5064793
365 110.5155782
 How to calculate the Duration of bond How to Calculate bond price
Date of settlement 1/30/2025 Date of settlement 1/30/2025
Date of Maturity 1/30/2028 Date of the Maturity 1/30/2028
Face value 100 Coupon Rate 6%
Annual coupon rate 10% Yield 5.80%
Yield on a bond (Annual) 12% Redemption of the Bond 100
Frequency 2 Frequency 2

Duration of bond D is: 2.654845897 Bond's Price is : 100.5435192

How to calculate the Yield of a Bond

Date of settlement 1/15/2025
Maturity Date 1/15/2027
Percent Coupon 6%
Price 98.39
Redemption Value 100
Frequency 2

yield y is: 0.06875352011

 Practical-2
Computing Price and yield of a bond
Q.1 Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are, repectively, 4%, 4.2%, 4.4%, 4.6% and 4.8% per annum with continuous compounding. Estimate the bond price with face value of 100 that will mature in
30-months and pay a coupon of 4% pa semiannually.

Solution
Face Value(F)= 100
Coupon payment pa(c 4%
Time period(T)= 2.5
Frequency= 2

Time period Zero Rate Cash Flows Present Valu PV based on y

0.5 4% 2 1.960397347 1.952165001
1 4.20% 2 1.917739561 1.905474096
1.5 4.40% 2 1.872261729 1.85989992
2 4.60% 2 1.824210299 1.815415765
2.5 4.80% 102 90.46588455 90.37177351

Bond Price(B)=98.0404934897.904728287
Yield= 4.8416334%

Q.2. A 3-year bond provides a coupon of 8% semiannually and has a cash price of 104. What is the bond's yield?

Face Value(F)= 100

Coupon payment pa(c 8%
Time period(T)= 3
Frequency= 2
Bond Price(B)= 104

Yield= 6.5106974449%
Time period Cash Flows PV based on y
0.5 4 3.871882698
1 4 3.747868906
1.5 4 3.627827192
2 4 3.511630334
2.5 4 3.399155182
3 104 85.54734589

103.7057102
Q.3. A 5-year bond with a yield of 7%(cc) pays an 8% coupon at the end of each year. What is the bond's price?

Face Value(F)= 100

Coupon payment pa(c 8%
Time period(T)= 5
Frequency= 1
Yield= 7%

Time period Cash Flows PV based on y

1 8 7.459150559
2 8 6.954865883
3 8 6.484673968
4 8 6.046269932
5 108 76.10631369
Bond Price(B 103.051274
Yield= 7%
 Practical-3
Year Spot rate (%) Spot rate Forward Rate
1 5% 0.05
2 5.50% 0.055 0.0600238095
3 6% 0.06 0.0650236966
4 6.50% 0.065 0.0700235849
5 7% 0.07 0.0750234741
6 7.50% 0.075 0.0800233644
7 8% 0.08 0.0850232558
8 8.50% 0.085 0.0900231481
9 9% 0.09 0.0950230414
10 9.50% 0.095 0.1000229358
11 10% 0.1 0.1050228311
12 10.50% 0.105 0.1100227273
 Practical-4
Q.1. A 5-year bond with a yield of 7% pays an 8% coupon at the end of each year. What is the bond's duration.
Also find its convexity. Using duration calculate the effect on bond's price of a 0.2% decrease in its yield. Do the
i it
Face Value= 100
Coupon= 8%
Yield= 7%
Frequency= 1
Time period= 5 New Yield= 6.800%
Verification
Time interval Coupon Paymen PV of coupon t*PV t^2*PV PV
1 8 7.459150559 7.4591507.459150 7.474083789
2 8 6.954865883 13.9097327.81946 6.98274106
3 8 6.484673968 19.4540258.36206 6.523698969
4 8 6.046269932 24.1850796.74031 6.094834089
5 108 76.10631369 380.53151902.657 76.87119486

Bond Price= 103.051274

Duration= 4.323474
Convexity= 20.31065
Yield change= -0.20%
New Bond Price= 103.9423531 103.9465528 103.9465392
(based on duration) (based on based on
Q.1 Let stock be purchased at Rs 120, call option is excercised at a strike price of Rs 150 and premium is Rs 20. Find the profit when price of share ranges between 60 to 250
when covered call strategy is used.

Current Stock Price(S 120

Strike Price(K)= 150
Put Price(p)= 20

Stock Price(S_T) LongStoc ShortCall ProTotal Profit

60 -60 20 -40
70 -50 20 -30
80 -40 20 -20
Covered Call Strategy

Total Profit
90 -30 20 -10
100 -20 20 0 50
110 -10 20 10
120 0 20 20
130 10 20 30 25
140 20 20 40
150 30 20 50
160 40 10 50
170 50 0 50 0
Stock Price
180 60 -10 50
190 70 -20 50
200 80 -30 50 -25
210 90 -40 50
220 100 -50 50
230 110 -60 50
240 120 -70 50 -50
250 130 -80 50
Q.1 Let a share of Reliance be purchased at Rs 170, put option is excercised at a strike price of Rs 150 and premium is Rs 45. Find the profit when price of share ranges between
30 to 250 when protective put strategy is used.

Current Stock Price(Sₒ 170

Protective Put Strategy

Strike Price(K)= 150

Total Profit
Put Price(p)= 45 15
0
Stock Price(S_T) LongStoc LongPut Prof Total Profit
30 -140 75 -65 10
40 -130 65 -65
50 -120 55 -65 0
60 -110 45 -65
70 -100 45 -55
80 -90 45 -45
50
90 -80 45 -35
100 -70 45 -25
110 -60 45 -15 0 Stock Price
120 -50 45 -5
130 -40 45 5
140
150
-30
-20
45
45
15
25
-50
160 -10 45 35
170 0 45 45 -10
180 10 45 55
190 20 45 65 0
200 30 45 75
210 40 45 85
220 50 45 95
230 60 45 105
240 70 45 115
250 80 45 125
 Q.1 (using calls) An investor buys for $3 a call with strike price of $30 sells for $1 a call with a
strike price of $35, Find profit from bull spread if stock prices are 20,30,31,32,33,34,35,40,45,50.
For Call Option Bull Spread
40
Long call price (c 3 With strike price (K1 30
Short call price(c 1 With strike price(K2) 35 30

Stock price Payoff using long ca Payoff using short c total payoff profit Stock price 20

Profit
20 0 0 0 -2 20
30 0 0 0 -2 30 10
31 1 0 1 -1 31
32 2 0 2 0 32 0
33 3 0 3 1 33
34 4 0 4 2 34 -10
35 5 0 5 3 35 20 30 31 32 33 34 35 40 45 50
40 10 5 15 13 40
45 15 10 25 23 45 Stoc k Pric e
50 20 15 35 33 50

For Put Option Bear Spread

0
Long Put price(p1 28 With Strike Price (K1 100
Short Put price(p 12 With Strike Price (K2 70
-10

Stock Price Payoff using long puPayoff using short p Total payoff profit
10 60 -90 -30 -46 -20
20 50 -80 -30 -46
30 40 -70 -30 -46 -30
40 30 -60 -30 -46
50 20 -50 -30 -46
60 10 -40 -30 -46 -40
70 0 -30 -30 -46
80 0 -20 -20 -36 -50
90 0 -10 -10 -26 10 20 30 40 50 60 70 80 90 100
100 0 0 0 -16
Q.1(using put) An investor buys a put option for $3 with strike price $35 and sells a put option
for $1 with strike price $30. Create a bull spread if stock price ranges between 0 to 120. Find
profit from bear spread if stock prices ranges from 10 to 60.

For Put Option Bear Spread

Long Put price(p1 28 With Strike Price (K1 100 0

Short Put price(p 12 With Strike Price (K2 70

-10
Stock Price Payoff using long puPayoff using short p Total payoff profit
10 60 -90 -30 -46
20 50 -80 -30 -46 -20
30 40 -70 -30 -46
40 30 -60 -30 -46
-30
50 20 -50 -30 -46
60 10 -40 -30 -46
70 0 -30 -30 -46 -40
80 0 -20 -20 -36
90 0 -10 -10 -26
-50
100 0 0 0 -16
10 20 30 40 50 60 70 80 90 100
 PRACTICAL 6
Option pricing [European call option]
strike price(k) 110 Stock price binomial tree
stock price(So) 100
stock price(St) 120 144
80 120
time 2 100 96
risk free rate 5.00% 80
volatility 20.00% 64
number of steps 2,3
option type call

up factor(u) 1.2 Option price binomial tree

down factor(d) 0.8
34
riskneutral probability 0.762927295 23.47105557
16.20266029 0
0
0
 Q A possible sequence stock prices during 8 consecutive trading days is
100,102, 99, 103, 101, 105, 107. Estimate volatility and standard error of the
estimate.

Solution
Day Stock PriceSi/Si-1 Ui=Ln(Si/Si-1)
0 100
1 102 1.02 0.019802627
2 99 0.970588-0.029852963
3 100 1.0101010.0100503358
4 103 1.03 0.0295588022
5 101 0.980582-0.019608471
6 105 1.0396030.0388398333
7 107 1.0190470.018868484

The usual estimate of the standard deviation of Ui(s)= 0.025341

The standard estimate of the daily return= 0.025341
Number of trading days= 252
Time(one trading day)= 0.003968
Number of trading days(n)= 7
The estimate of volatilty= 0.402284
The standard error of estimate= 0.107515
PRACTICAL-8

DAY(i) closing stocSi/Si-1 ui Ui^2

0 20
1 20.1 1.005 0.004987540.00002487 0
2 19.9 0.990049751 -0.0100000 0.00010000 0.0001 n 20
3 20 1.005025126 0.005012540.00002512 0 T 252
4 20.5 1.025 0.0246926 0.00060972 0.0006 15.8745078
5 20.25 0.987804878 -0.0122700 0.00015055 0.0002
6 20.9 1.032098765 0.031594360.00099820 0.001
7 20.9 1 0 0 0
8 20.9 1 0 0 0
9 20.75 0.992822966 -0.0072029 0.00005188 0.0001
10 20.75 1 0 0 0
11 21 1.012048193 0.011976190.00014342 0.0001
12 21.1 1.004761905 0.004750600.00002256 0
13 20.9 0.990521327 -0.0095238 0.00009070 0.0001
14 20.9 1 0 0 0
15 20.9 1 0 0 0
16 21.4 1.023923445 0.023641760.00055893 0.0006
17 21.4 1 0 0 0
18 21.25 0.992990654 -0.0070340 0.00004947 0
19 21.75 1.023529412 0.023256860.00054088 0.0005
20 22 1.011494253 0.011428690.0001306 0.0001
0.09531017 0.0034
0.00908403

S 0.01245158

volatility 0.1976628

standard erro 3.13%

 Q: Consider the stock price is Rs 1225, the strike price is Rs 1250, the risk-free interest rate is 5% per annum, the stock price volatility is 20% per
annum and the time to maturity is 20 weeks. Find the value for European call option and European put option by BSM
Solution:
Stock Price Current S0= 1225
Strike Price K= 1250
Risk-free interest rate r= 0.05
Stock price volatility= 0.2
Time to maturity T= 0.3846153

d1= 0.0541813
d2= -0.0698533
N(d1)= #N/A
N(d2)= #N/A
N(.d1)=1.N(d1) #N/A
N(.d2)=1.N(d2) #N/A

European Call Option Price #N/A

European Put Option Price p #N/A

 Q Consider the stock price is Rs 1225, the strike price is Rs 1250, the risk free interest rate is 5% per annum,
the stock price volatility is 20% per annum and the time to maturity is 20 weeks. Find the Monte Carlo Simulation
for stock price at time to maturity.
Solution: Monte Carlo Simulation
Stock Price Current S0= 1225 1297.61914 1196.706 1100.471 1139.427 1009.066
Strike Price K= 1250 1332.59671 1362.491994.03541421.5461204.722
Risk-free interest rate r= 0.05 1137.62047 1132.656 1138.883 1256.6621146.463
Stock price volatility= 0.2 1096.01502 1279.3971455.7501115.200 1489.216
Time to maturity T= 0.384615 1142.37636 1427.8291491.113 1419.3061237.718
1082.30699 1309.2221388.6651304.011 1110.922
1108.83294 1467.4431560.3871336.1231450.596
1244.17854 1409.4981184.877 1570.4841201.665
Mean = 1223.452 1220.81946 1057.4241121.159 1208.1761450.035
1541.92057 1200.4291341.1311221.0001443.539
1138.98664 1074.8031319.9661121.797 1169.361
1203.03621 1281.4061233.226918.99941177.525
1164.00835 1088.3721022.9081018.4001043.445
1236.66433 1100.915 1299.1221255.6991320.681
1171.58970 1071.324863.93911143.833 1365.255
1426.57517 1239.4681073.3691113.534 1111.895
1194.2042641439.0371130.383 989.58031214.079
1096.44783 1449.2451408.6761364.0851087.601
1075.17411 1178.672 1099.8661004.5151166.590
Qus: Consider the stock price is Rs 1225, the strike price is Rs 1250, the risk-free interest rate is 5% per annum, the
stock price
volatility is 20% per annum and the time to maturity is 20 weeks Find the value for European call option using the
Solution: onte Carlo Simulation for stock price at time Present value of Payoff from call option
Stock Price Current So = 1225 1180.64 1210.35 1594.02 1464.91 1430.18 0.00 0.00 337.47 210.81 176.75
Strike Price K = 1250 1191.62 1164.80 1429.35 1167.78 1429.11 0.00 0.00 175.93 0.00 175.70
Risk-free interest rate r = 5% 1349.32 1343.26 1251.24 806.68 1253.20 97.43 91.48 1.21 0.00 3.14
Stock price volatility = 20% 1444.78 1364.39 1064.44 1192.55 1099.92 191.07 112.21 0.00 0.00 0.00
Time to maturity T = 0.3846 1398.62 1125.03 1223.14 1082.16 1130.50 145.79 0.00 0.00 0.00 0.00
1296.23 1464.02 1104.43 1249.43 1492.98 45.35 209.95 0.00 0.00 238.35
d1= 0.0542 1395.94 1129.01 1137.76 1543.22 1228.93 143.16 0.00 0.00 287.63 0.00
d2= -0.06985 1242.63 1198.56 1255.65 1173.79 1371.52 0.00 0.00 5.54 0.00 119.21
N(d1)= #N/A 1184.38 1250.96 1250.42 1450.84 1288.16 0.00 0.95 0.41 197.02 37.43
N(d2)= #N/A 1218.33 958.11 1236.88 1157.96 1333.71 0.00 0.00 0.00 0.00 82.12
N(-d1)= 1 - N(d1) #N/A 1057.80 1451.16 1184.15 1310.45 1399.80 0.00 197.33 0.00 59.30 146.94
N(-d2)= 1 - N(d2) #N/A 1148.56 1056.71 1287.99 1053.34 1253.32 0.00 0.00 37.27 0.00 3.25
1075.38 908.29 1229.62 972.62 1130.60 0.00 0.00 0.00 0.00 0.00
European Call Option Price by BS #N/A 1407.13 1272.90 1186.29 1218.99 1171.95 154.14 22.46 0.00 0.00 0.00
1186.62 1299.39 1385.58 1491.65 1118.76 0.00 48.45 133.00 237.05 0.00
Mean call option payoff = 45.16 1247.41 1223.65 1375.55 1248.40 1247.72 0.00 0.00 123.16 0.00 0.00
SD call option payoff = 79.34759 958.24 1250.17 1147.01 1223.21 1221.50 0.00 0.16 0.00 0.00 0.00
1145.00 1100.00 1203.15 1111.22 1014.68 0.00 0.00 0.00 0.00 0.00
Value of Euro Call Option by
Simulation c= 45.16 1022.01 1344.29 1158.36 1130.41 1428.33 0.00 92.49 0.00 0.00 174.94
1153.58 991.23 1142.71 1118.24 1189.15 0.00 0.00 0.00 0.00 0.00
Qus: Consider the stock price is Rs 1225, the strike price is Rs 1250, the risk-free interest rate is 5% per annum, the stock
price
volatility is 20% per annum and the time to maturity is 20 weeks Find the value for European put option using the Monte
Solution: onte Carlo Simulation for stock price at time Present value of Payoff from Put option
Stock Price Current So = 1225 1170.36 1385.74 1096.29 1174.46 1066.76 78.12 0.00 150.79 74.10 179.75
Strike Price K = 1250 1316.30 1165.70 1307.97 1283.60 1229.67 0.00 82.70 0.00 0.00 19.94
Risk-free interest rate r = 5% 1323.15 1245.82 1115.35 1111.83 1147.92 0.00 4.11 132.08 135.54 100.14
Stock price volatility = 20% 1031.58 1249.06 1238.01 1223.89 1736.65 214.26 0.92 11.76 25.61 0.00
Time to maturity T = 0.3846 1119.29 1463.81 1468.87 1392.93 1115.61 128.22 0.00 0.00 0.00 131.83
1161.41 1241.82 1059.81 1161.26 1247.56 86.90 8.03 186.57 87.05 2.40
d1= 0.0542 1242.09 1242.23 1507.04 1180.72 1165.63 7.76 7.62 0.00 67.96 82.76
d2= -0.06985 1095.04 1103.45 1382.30 1283.36 1327.76 152.01 143.75 0.00 0.00 0.00
N(d1)= #N/A 1158.71 1168.82 1327.83 1034.71 1377.33 89.55 79.63 0.00 211.19 0.00
N(d2)= #N/A 1399.96 1224.54 1408.90 1448.61 1021.31 0.00 24.97 0.00 0.00 224.34
N(-d1)= 1 - N(d1) #N/A 1270.72 1285.21 1477.86 1212.28 1461.05 0.00 0.00 0.00 37.00 0.00
N(-d2)= 1 - N(d2) #N/A 1493.89 1309.76 1454.47 1137.83 1017.30 0.00 0.00 0.00 110.03 228.27
1469.58 1522.52 1320.13 1226.45 1101.51 0.00 0.00 0.00 23.10 145.66
European Put Option Price by BSM #N/A 1231.26 1062.55 1504.20 1161.22 1163.49 18.38 183.88 0.00 87.09 84.87
1230.68 1347.76 1316.52 1215.66 1184.10 18.95 0.00 0.00 33.69 64.64
Mean Put option payoff = 51.89 1486.48 1567.52 1271.62 1335.28 1126.38 0.00 0.00 0.00 0.00 121.27
SD Put option payoff = 67.76427509 1323.95 1348.96 1226.10 1147.25 1139.77 0.00 0.00 23.45 100.79 108.13
1145.74 1175.09 1187.66 1245.71 1093.91 102.27 73.48 61.16 4.21 153.12
Value of Euro Put Option by
Simulation p= 51.89 1353.79 1368.77 1208.12 1209.93 1291.79 0.00 0.00 41.09 39.31 0.00
1220.95 1155.17 1456.08 1293.70 973.63 28.49 93.02 0.00 0.00 271.11