# IF2209 Derivatives Coursework For Regression Analysis And Black Scholes Calculation

• Put-call 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 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

The marked values are the required values for ( ).So the linear regression line is

Table2: Strike vs Cobs-Pobs 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 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 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 S0 = 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 add-in 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

1. (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 P-value 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 9.33617E-13 Residual 19 0.00046395 2.44184E-05 Total 21 0.008565824 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 0.597016263 0.030053865 19.8648748 3.6E-14 0.534112801 0.659919725 X Variable 1 -0.003684844 0.000325616 -11.31653952 6.9E-10 -0.00436637 -0.003003322 X Variable 2 8.5387E-06 8.60305E-07 9.925210419 5.9E-09 6.73807E-06 1.03393E-05

(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 Speculative Prem. 12.28 Put Option Value 6.9 Intrinsic Value 0 Speculative Prem. 6.9
• 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 Speculative Prem. 8.18 Put Option Value 12.03 Intrinsic Value 10 Speculative Prem. 2.03

Bull spread payoff for all three possible cases calculations:

 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 Kp 60 receives Premium 3.252707516 Buy 2 ATM put Option Strike Price K1 70 pays Premium (2*6.90024289527396) 13.80048579 Sells 1 OTM put Option Strike Price K2 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 move-up factor per period and d= stock price move-down factor per period, q is risk neutral probability of an upward movement

• For binomial stock pricing with one-year time period split into two six-month intervals and assuming a two-period binomial model, the tree is obtained as below:

Table 14: Two-period 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% 1-q 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 1-q.

(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 (ST) 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 one-year time period split into two six-month intervals and assuming a two-period binomial model, the tree is obtained as below. The value of the parameters u, d, q were obtained from answer 4a (i).

Table 15: Two-period 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% 1-q 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 pay-later 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 pay-later strategy the payoff profile can be rewritten as: which clearly expresses payoff pay-later 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

K= strike rate of call and put

r=annual interest rate

T=time in years

S0=initial price of underlying

The put-call 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 put-call parity relationship is:

Where ‘c>0’ and ‘p>0’ are the option premiums for call and put options for Pay-later 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

Reference Lists

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.36-68.

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.367-391.

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