Derivatives Coursework: Regression, Black-Scholes-Merton Formula, and ANOVA

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Derivatives Coursework
Student Name: Student ID:
Unit Name: Unit ID:

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Table of tables
Table 1: Regression analysis including R................................................................................................3
Table2: Strike vs Cobs-Pobs linear line fit plot.......................................................................................5
Table3: K residual plot for Cobs-Pobs values.........................................................................................5
Table 4: Solution table for σimpl for given K.........................................................................................7
Table 5:σimpl for given K.......................................................................................................................8
Table6: σimplvs strike rate K plot...........................................................................................................9
Table 7:ANOVA for quadratic fit for table 3 data...................................................................................9
Table 8: Regression analysis values excluding three outlier values......................................................11
Table 9: Bull Spread Strategy...............................................................................................................14
Table 10: Bull Spread Payoff matrix.....................................................................................................14
Table 11: Seagull strategy values..........................................................................................................16
Table 12: Seagull payoff values............................................................................................................17
Table 13: Short put butterflypayoff values............................................................................................18
Table 14: Two-period binomial modelwith European vanilla payoff values.........................................19
Table 15: Two-period binomial modelwith Digital payoff values.........................................................21
Table 16: Regression Analysis for Question 1 with residual output......................................................26
Table 17: Question 2 Solution for implied volatility.............................................................................28
Table 18: Regression Analysis for Question 2 with outliers..................................................................30
Table 19: Residual values with outliers for Question 2.........................................................................31
Table 20: Regression Analysis for Question 2 without outliers.............................................................31
Table 21: Black Scholes calculation for Question 3..............................................................................32
Table of figures
Figure 1: Scatter plot excluding outliers 10
Figure 2: Bull spread graph...................................................................................................................16
Figure 3: Seagull payoff graph..............................................................................................................18
Figure 4: Residual Plot for Question 1..................................................................................................27

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ANS:1 Put-call parity requires that the following equation to hold
Cobs−Pobs=S0 e−δT −Ke−rT . .. . .. .. . .. .. . .. .. . .. .. . .(i )
 Where T=11 months
 Cobs , Pobs are observed prices of the call and put options
 δ is continuously compounded dividend per year
 r is continuously compounded risk free interest per annum
 S0 is current spot price of the stock
 T is option maturity
Now linear regression general form of the equation is y=α+ β x . .. . .. .. . .. .. . .. ..(ii )
(i) Comparing equations (i) and (ii) it is obtained that α=S0 e−δT , β =−e−rT
where
y=Cobs −Pobs , x=K
(ii) To fit the linear regression model, help of regression tool in excel has been used
Following results have been obtained:
Table 1: Regression analysis including R

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The marked values are the required values for α , β (
α=int ercept =163 . 01, β=slope=−0 . 995).So the linear regression line is
y=−0 .995 x +163 . 091
Regression Statistics
Multiple R 0.997754
R Square 0.995512
Adjusted R Square 0.995368
Standard Error 3.5137
Observations 33
df SS MS F
Significance
F
Regression 1.000 84904.865 84904.865 6877.068 0.000
Residual 31.000 382.729 12.346
Total 32.000 85287.594
Coefficient
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept 163.091 1.960 83.194 0.000 159.092 167.089 159.092 167.089
X Variable 1 -0.995 0.012 -82.928 0.000 -1.020 -0.971 -1.020 -0.971

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(iii) Required scatter graph plot is given below:
50 100 150 200 250 300
-150
-100
-50
0
50
100
150
f(x) = − 0.995499743567071 x + 163.090566274649
R² = 0.995512493292654
Cobs-Pobs plot
Cobs-Pobs
Linear (Cobs-Pobs)
STRIKE
Table2: Strike vs Cobs-Pobs linear line fit plot
The line of best fit is y=−0 .995 x +163 . 0. The data is almost perfectly negatively correlated,
i.e. for increase in the value of K the value of Cobs−Pobs decreases with almost a slope of
1(which means the angle of the best fit line is 450 .

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50 100 150 200 250 300
-10
-5
0
5
10
15
K Residual Plot
K values
Residuals
Table3: K residual plot for Cobs-Pobs values
From the residual plot it is evident that residual values cluster around the horizontal axis. This
indicates the fact the regression model is fit for linear in nature with almost perfect correlation.
Now for S0=165 . 40 and T=11/12, α =163 . 01 , β=−0. 995 following calculations can be
performed:
α=S0 e−δT
=> α
S0
=e−δT
¿−¿−δT =ln ( α
S0 )
=> δ=−1
T *ln ( α
S0 )
=> δ=−12
11 *ln ( 163 . 01
165 . 40 )
=> δ=0 . 0158
And

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β=−e−rT
=>−rT =ln (−β )
=>−r=1
T *ln (−β )
=> r=−12
11 *ln(0 . 995 )
=> r=0 . 00547
ANS:2 (a) Given values are δ = 1.53% per annum, r = 0.49% per annum, T = 11/12 year and
S0 = 165.40.
Black–Scholes–Merton formula gives the option price as:
CBSM ( S0 , K , r , δ , T , σimp ) =S0 e−δT N ( d1 ) −Ke−rT N ( d2 )
Where
d1= 1
σimpl √ T [ ln S0
K − ( r−δ+ σimpl
2
2 )∗T ]and d2=d1−σimpl∗√ T and N ( . ) is the
standard normal cumulative distribution function.
The governing equation provided as Cobs−CBSM ( S0 , K , r , δ , T , σimp )=0. . .. .. . .. .. .. . .. .. . .. .. . .. .. . .(i)
Using Excel’s add-in solver equation (i) is solved and the solution is as follows:

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Table 4: Solution table for σimpl for given K
K σimpl Cobs
115 27.200000 51.46
120 23.743192 46
125 39.400000 41.78
130 0.268155 37.4
135 0.252982 33
140 0.237571 28.68
145 0.245481 25.64
150 0.237140 22.05
155 0.242571 19.48
160 0.224743 15.8
165 0.220736 13.2
170 0.215498 10.8
175 0.207808 8.53
180 0.210950 7.18
185 0.205065 5.55
190 0.205596 4.5
195 0.198816 3.3
200 0.198853 2.6
205 0.199595 2.06
K σimpl Cobs
210 0.203694 1.73
215 0.206262 1.42
220 0.203330 1.04
230 0.202240 0.6
240 0.202373 0.35
255 0.210170 0.2
Note: Detailed calculations attached in the Appendix
b) (i)The K versus σ impl values table is as follows:
Table 5:σimpl for given K
K σimpl
115 27.200000
120 23.743192
125 39.400000
130 0.268155
135 0.252982