Options Pricing And Strategies

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.

 

	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 (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 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 

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.