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Question:
Consider the data given in the sheet named ‘Question 1’ of workbook ‘IF2209cwk_data.xlsx’. These represent the prices of European plain vanilla call and put options on stock S of the Goldman Sachs Group, Inc.(GS) traded on NYSE. The market option prices for maturity T = 11 months (i.e., 11/12 year) are available for the indicated range of strikes in sheet ‘Question 1’.
Put-call 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 risk-free rate of interest per annum, S0 the current spot price of the stock, K the strike price and T the option maturity.
a) Put-call 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.
Answer:
  • 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

 

0.00

   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 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

C= call premium

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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