Putcall parity requires that the following equation holds
Cobs − Pobs = S0e
−dT − Ke−rT, (1)
where Cobs, Pobs are respectively the observed prices of the call and put options, d is the continuously compounded dividend yield per annum, r the continuously compounded riskfree rate of interest per annum, S0 the current spot price of the stock, K the strike price and T the option maturity.
a) Putcall parity (1) may be viewed as a simple linear regression with general form y = a + bx,
where y corresponds to the (Cobs − Pobs) price differences and x to the strike prices K.
i) Express the intercept a and the slope b in terms of S0, d,r, T.
ii) Fit the linear regression model to the observed option price differences and provide estimates for the intercept and the slope.
 Putcall parity requires that the following equationto hold
 Where T=11 months
 are observed prices of the call and put options
 is continuously compounded dividend per year
 r is continuously compounded risk free interest per annum
 is current spot price of the stock
 T is option maturity
Now linear regression general form of the equation is
 Comparing equations (i) and (ii) it is obtained thatwhere
 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
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 
Pvalue 
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 
The marked values are the required values for ( ).So the linear regression line is
Table2: Strike vs CobsPobs linear line fit plot
The line of best fit is. The data is almost perfectly negatively correlated, i.e. for increase in the value of K the value of decreases with almost a slope of 1(which means the angle of the best fit line is.
Table3: K residual plot for CobsPobs 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 forand T=11/12, following calculations can be performed:
And
 (a) Given values are = 1.53% per annum, r = 0.49% per annum, T = 11/12 year and S_{0} = 165.40.
Black–Scholes–Merton formula gives the option price as:
Where and andis the standard normal cumulative distribution function.
The governing equation provided as
Using Excel’s addin solver equation (i) is solved and the solution is as follows:
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
 (i)The K versus 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 
140 
0.237571 
145 
0.245481 
150 
0.237140 
155 
0.242571 
160 
0.224743 
165 
0.220736 
170 
0.215498 
175 
0.207808 
180 
0.210950 
185 
0.205065 
190 
0.205596 
195 
0.198816 
200 
0.198853 
205 
0.199595 
210 
0.203694 
215 
0.206262 
220 
0.203330 
230 
0.202240 
240 
0.202373 
255 
0.210170 
The graphical plot between K and is as follows:
Table6: σimplvs strike rate K plot
(ii) The quadratic fit for the data in table (3) in the form is as follows:
Table 7:ANOVA for quadratic fit for table 3 data
Regression Statistics 






Multiple R 
0.720815475 





R Square 
0.519574949 





Adjusted R Square 
0.475899945 





Standard Error 
7.381231069 





Observations 
25 





ANOVA 







df 
SS 
MS 
F 
Significance F 

Regression 
2 
1296.292 
648.146 
11.89639 
0.000314689 

Residual 
22 
1198.617 
54.48257 



Total 
24 
2494.909 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
130.305705 
31.25254 
4.169443 
0.000399 
65.49189564 
195.1195144 
X Variable 1 
1.328255552 
0.356351 
3.72738 
0.00117 
2.067281355 
0.58922975 
X Variable 2 
0.003307286 
0.000982 
3.366778 
0.002783 
0.001270059 
0.005344513 
From the ANOVA calculations it is evident that the intercept values are and the second order polynomial fit is .
(iii) From table 2 it can be identified that three outlier values. Excluding them the trend of the data is almost quadratic in nature and can be identified from the scatter plot.
Figure 1: Scatter plot excluding outliers
Excluding the outliers the regression analysis provides a well behaved intercept values.
Table 8: Regression analysis values excluding three outlier values
Regression Statistics 






Multiple R 
0.972541562 





R Square 
0.94583709 





Adjusted R Square 
0.940135731 





Standard Error 
0.0049415 





Observations 
22 





ANOVA 







df 
SS 
MS 
F 
Significance F 

Regression 
2 
0.008101874 
0.004050937 
165.897 
9.33617E13 

Residual 
19 
0.00046395 
2.44184E05 



Total 
21 
0.008565824 





Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
0.597016263 
0.030053865 
19.8648748 
3.6E14 
0.534112801 
0.659919725 
X Variable 1 
0.003684844 
0.000325616 
11.31653952 
6.9E10 
0.00436637 
0.003003322 
X Variable 2 
8.5387E06 
8.60305E07 
9.925210419 
5.9E09 
6.73807E06 
1.03393E05 
(a) Given data values
Black–Scholes–Merton formula gives the option price as:
Where and andis the standard normal cumulative distribution function.
Now for,
Hence
So,
Now for,
Hence
Now,
Note: The C(BSM) value got evaluated in Excel using the above formulae.
Definition: Bull call spread is for moderate rise in asset price. It is an option strategy which guides to purchase call options at particular strike rate and sell equal number of calls at a higher strike rate for same expiration period(Brown 2012).
Explanation of the strategy: In this strategy put call option has higher strike rate than long call options. Therefore the policy requires an initial cash flow. The maximum gain will be difference of strike price of long call and short call minus the net cost. The maximum loss though is limited, which equals to the net premium paid for the options.
The profit for this option increases up to the strike of short call option. Hence gain remains stationary for security price going above short call strike price. Losses will be occurred for fall in security prices but becomes stationary if security price goes below long call strike price.
Call Option Value 

12.28 
Intrinsic Value 

0.00 
Speculative Prem. 

12.28 
Put Option Value 

6.90 
Intrinsic Value 

0.00 
Speculative Prem. 

6.90 
 Using the problem of 3(a), it can be calculated using excel sheet (calculation attached) that for strike value of 70,
And for strike rate 80,
Call Option Value 

8.18 
Intrinsic Value 

0.00 
Speculative Prem. 

8.18 
Put Option Value 

12.03 
Intrinsic Value 

10.00 
Speculative Prem. 

2.03 
Bull spread payoff for all three possible cases calculations:
Table 9: Bull Spread Strategy
Europe market 
current value 
70 
Buy ITM 
strike price 
70 

premium 
12.28 
SELL OTM 
strike price 
80 

premium 
8.18 

Net premium paid 
4.1 

Breakeven point 
74.1 
Table 10: Bull Spread Payoff matrix
On expiry 
Net Payoff from Call buy 
Net Payoff from Call Sold 
Net Payoff 
68.60 
12.28 
8.18 
4.1 
69.10 
12.28 
8.18 
4.1 
69.60 
12.28 
8.18 
4.1 
70.00 
12.28 
8.18 
4.1 
70.10 
12.18 
8.18 
4 
70.60 
11.68 
8.18 
3.5 
71.10 
11.18 
8.18 
3 
71.60 
10.68 
8.18 
2.5 
72.10 
10.18 
8.18 
2 
72.60 
9.68 
8.18 
1.5 
On expiry 
Net Payoff from Call buy 
Net Payoff from Call Sold 
Net Payoff 




73.10 
9.18 
8.18 
1 
73.60 
8.68 
8.18 
0.5 
74.10 
8.18 
8.18 
0 
74.60 
7.68 
8.18 
0.5 
75.10 
7.18 
8.18 
1 
75.60 
6.68 
8.18 
1.5 
76.10 
6.18 
8.18 
2 
76.60 
5.68 
8.18 
2.5 
77.10 
5.18 
8.18 
3 
77.60 
4.68 
8.18 
3.5 
78.10 
4.18 
8.18 
4 
78.60 
3.68 
8.18 
4.5 
79.10 
3.18 
8.18 
5 
79.60 
2.68 
8.18 
5.5 
80.00 
2.28 
8.18 
5.9 
80.10 
2.18 
8.08 
5.9 
80.60 
1.68 
7.58 
5.9 
81.10 
1.18 
7.08 
5.9 
81.60 
0.68 
6.58 
5.9 
82.10 
0.18 
6.08 
5.9 
82.60 
0.32 
5.58 
5.9 
83.10 
0.82 
5.08 
5.9 
8.18
0 70 74.1 80
12.8 Break Even Point
 The cost for implementing the strategy will be =(12.28+8.18)=(4.1)
Hence outlay cost will be 4.1 X (the number of shares each contract has)
(c) (i) For the seagull strategy the following holds:
Table 11: Seagull strategy values
Europe Market 
Current Market Price 
70 
Buy 2 ATM Call Option 
Strike Price 
70 
pays 
Premium (2*12.28) 
24.56 
Sells 1 ITM Call Option 
Strike Price 
60 
receives 
Premium 
17.87 
Sells 1 OTM Call Option 
Strike Price 
80 
receives 
Premium 
8.18 

Break Even Point upper 
78.51 

Break Even Point lower 
61.49 
The payoff table for the policy is:
Table 12: Seagull payoff values
On expiry market Closes at 
Net Payoff from ATM Calls purchased 
Net Payoff from ITM Call sold 
Net Payoff from OTM Call sold 
Net Payoff 
52.5 
24.56 
17.87 
8.18 
1.49 
55 
24.56 
17.87 
8.18 
1.49 
57.5 
24.56 
17.87 
8.18 
1.49 
60 
24.56 
17.87 
8.18 
1.49 
61.49 
24.56 
16.38 
8.18 
0 
62.5 
24.56 
15.37 
8.18 
1.01 
65 
24.56 
12.87 
8.18 
3.51 
67.5 
24.56 
10.37 
8.18 
6.01 
70 
24.56 
7.87 
8.18 
8.51 
72.5 
19.56 
5.37 
8.18 
6.01 
75 
14.56 
2.87 
8.18 
3.51 
77.5 
9.56 
0.37 
8.18 
1.01 
78.51 
7.54 
0.64 
8.18 
0 
80 
4.56 
2.13 
8.18 
1.49 
82.5 
0.44 
4.63 
5.68 
1.49 
85 
5.44 
7.13 
3.18 
1.49 
87.5 
10.44 
9.63 
0.68 
1.49 
90 
15.44 
12.13 
1.82 
1.49 
92.5 
20.44 
14.63 
4.32 
1.49 
95 
25.44 
17.13 
6.82 
1.49 
97.5 
30.44 
19.63 
9.32 
1.49 
100 
35.44 
22.13 
11.82 
1.49 
Cost of seagull strategy (short call) is = (1 sell ITM + 2 buy ATM + 1 sell OTM) X (the number of shares each contract has). Hence the cost of seagull strategy will be less than bull strategy because of the bear effect in seagull.
 The alternative model is short put butterfly with following strategy:
Table 13: Short put butterfly payoff values
Europe Market 
Current Market Price 
70 
Sells 1 ITM put Option 
Strike Price K_{p} 
60 
receives 
Premium 
3.252707516 
Buy 2 ATM put Option 
Strike Price K_{1} 
70 
pays 
Premium (2*6.90024289527396) 
13.80048579 
Sells 1 OTM put Option 
Strike Price K_{2} 
80 
receives 
Premium 
12.03187329 

Break Even Point upper 
78.51590498 

Break Even Point lower 
61.48409502 

premium profit 
1.484095018 



Here the investor would get the extra cash as the premium received for initiating the position.
ANS: 4. (a). Here u is stock price moveup factor per period and d= stock price movedown factor per period, q is risk neutral probability of an upward movement
 For binomial stock pricing with oneyear time period split into two sixmonth intervals and assuming a twoperiod binomial model, the tree is obtained as below:
Table 14: Twoperiod binomial model with European vanilla payoff values



solution 



price 
100 

u 
1.236311 
=>magnitude 
Of up jump 
strike (assume) 
100 

d 
0.808858 
=>magnitude 
Of down jump 
time(years) 
0.5 

a 
1.002503 


volatility 
30% 

q 
0.453021 
=>probability 
Of up jump 
risk free rate 
0.50% 

1q 
0.546979 
=>probability 
Of down jump 
dividend 
0% 













time point 
0 

0.5 

1 

stock 




152.8465 

option 




52.84652 

stock 


123.6311 



option 


23.8808 



stock 
100 



100 

option 
10.79149 



0 

stock 


80.88579 



option 


0 



stock 




65.42511 

option 




0 
The value of the parameters, which are u, d, q were obtained from answer 4a (i).
There are two options in binomial model of pricing. Either the price will go up with probability 0.453021 or will go down with probability 0.546979. Jump magnitudes are 1.236311 and 0.808858 for the up and down jump for each unit. The tree is calculated for two periods that is for three time points that are 0, 0.5 and 1 year. Hence going up prices were calculated by multiplying previous step price by q and subsequently down prices were calculated by multiplying previous step price by 1q.
(b) The option prices at the end of one year (t=1) were calculated by taking the maximum value between zero and the difference between current stock price (S_{T}) and strike price (K). The calculation of the option prices at time T=1 was obtained from the payoff profile. But for the in the money values option prices were all zero. Consequently previous option prices were evaluated by the formula [q × Option up + (1−q) × Option down] × exp ( r × Δt). Hence for example at time t=0.5, the option price for the first leg of the tree for time 0.5 was calculated as [0.453021*52.84652+0.546979*0]*exp (0.50%*0.5) = 23.8808. Option price at time t=0 was 10.79149.
(c) For binomial stock pricing with oneyear time period split into two sixmonth intervals and assuming a twoperiod binomial model, the tree is obtained as below. The value of the parameters u, d, q were obtained from answer 4a (i).
Table 15: Twoperiod binomial model with Digital payoff values



solution 



price 
100 

u 
1.236311 
=>magnitude 
Of up jump 
strike 
100 

d 
0.808858 
=>magnitude 
Of down jump 
time(years) 
0.5 

a 
1.002503 


volatility 
30% 

q 
0.453021 
=>probability 
Of up jump 
risk free rate 
0.50% 

1q 
0.546979 
=>probability 
Of down jump 
dividend 
0% 













time point 
0 

0.5 

1 






152.8465 






1 




123.6311 






1 



stock 
100 



100 

option 
0.45189 



0 




80.88579 






0 

65.42511 






0 
The option prices at the end of one year (t=1) were calculated by the payoff profile of digital option. Hence option price will be 1 where stock price is greater than strike price (100) and 0 where it is less than strike price(Bali 2011).
Consequently previous option prices were evaluated by the formula [q × Option up + (1−q) × Option down] × exp ( r×Δt). Hence for example at time t=0.5, the option price for the first leg of the tree for time 0.5 was again 1 for stock price 123.6311, but for 80.88579 the option price calculated as [0*0.453021+0*0.546979]*=0. For time t=0 the option price for 100 is [1*0.453021+0*0.546979]* =0.45189]. It is to be noted that instead of stock price being 100, option price is calculated by discounting rate.
(d) The payoff profile of the paylater strategy is as follows:
where V is the price of pay later option.
Given payoff is
(i) Now from part (b) payoff of vanilla call option was and from part (c) payoff profile of digital call option was. Strike price here is K=100.
Combining the facts it is obtained that for paylater strategy the payoff profile can be rewritten as: which clearly expresses payoff paylater strategy as linear combination of vanilla call and digital call payoff values.
 At initiation of contract at time t=0 pay later Option is zero.Which implies that at t=0
 The buyer pays price ‘c’ at time if vanilla option has any value. The pay later option is priced at t=0. To get the premium seller will wait till time. Hence it can be said that where is probability of finishing in the money by Black Scholes formula. So which implies.
(e)
(i) At time t, the standard put call parity equation is
, where
C= call premium
K= strike rate of call and put
r=annual interest rate
T=time in years
S_{0}=initial price of underlying
The putcall parity relationship comes nicely from some simple steps. The true expression considering the payoff of pay later calls and put options:
(1) At expiration time, we get:
(2) Now multiplying each side by the discount factor e−r (T−t):
Taking the conditional expectations for risk neutral measure regarding the stock price:
From risk neutral pricing theory the discounted value of a risky asset is a Martingale. Hence the first term is the price of a Call option for Pay later at time t, the first term in RHS of the equality is the price of a Pay later Put option at time t (Mencia 2013).The second term on the LHS is the price of Call option for Digital call and the second term on the RHS is price of Put option for Digital call stock at time t. The second expectations on both the sides are simply a deterministic function and therefore expectation goes out of calculation.
Hence the putcall parity relationship is:
Where ‘c>0’ and ‘p>0’ are the option premiums for call and put options for Paylater strategy.
(ii) The European pay later put option has the following payoff:
, where ‘p>0’.
At time t=0 the value of the put option is zero for pay later, i.e. .
The previous part of the question gives:
Again from Table 15, since
And from Table 14
Hence
Bali, T.G., Brown, S.J. and Caglayan, M.O., 2011. Do hedge funds' exposures to risk factors predict their future returns?. Journal of financial economics, 101(1), pp.3668.
Brown, R., 2012. Analysis of investments & management of portfolios.
Mencia, J. and Sentana, E., 2013. Valuation of VIX derivatives. Journal of Financial Economics, 108(2), pp.367391.
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