Description
Explains topics like derivatives and risk management,theoretical future price for index and stocks
T.A.Pai Management Institute
Comparison of theoretical and market future prices in India
DERIVATIVES AND RISK MANAGEMENT- THEME 5
GROUP 20 Manish M Shreyas Srinivasan Sudhanva S Shetty Vijay Pandey 08129 08151 08156 08164
The objective of the study is to test the theoretical models of future prices in the Indian context. The study was done for index futures, stock futures and commodities. The underlying considered were Nifty 50 from National Stock exchange. The underlying stocks considered were ICICI bank and HUL listed in NSE. The commodities considered were gold and crude oil traded in the MCX. The methodology followed is as follows: 1. 2. 3. 4. 5. Literature study of different theoretical models for future prices. Data collection for spot and future prices from data sources (NSE, MCX.) Calculation of theoretical futures price. Calculate the difference between the actual future prices and theoretical futures price. Analyses of the differences and conclusions.
Theoretical Future Price for Index: A Futures contract is specified for a period of time, at the end of which it is settled. Theoretical Future price should factor the current price and holding costs. In order to compensate the seller for waiting till expiry for realizing the sale proceeds the buyer has to pay some interest which is reflected in the form of cost of carry. Futures Price = Spot Price + Cost of Carry The Cost of carry is the sum of all costs incurred if a similar position is taken in cash market and carried to maturity of the futures contract less any revenue which may result in this period The most popular models used for finding the theoretical future prices is i. ii. Where F0 is the theoretical future price S0 is the spot price of the underlying r is the risk free interest rate T is the time to expiry q is the yield on Index F0= S0*e^(rT) F0= S0*e^(r-q)T (for dividend yield q)
If the yield on the index is to be considered, then it has to be discounted from the theoretical future prices as in equation (ii). Theoretical Future Price for Stocks: This is similar to the index theoretical future prices. But for dividend consideration, the data was available in absolute terms and hence the formula changes as in equation (iv.) iii. iv. F0= S0*e^(rT) F0=(S0 –d)*e^(rT) (for absolute dividend ‘d’)
Equity Index The equity index considered for studying the difference between theoretical and futures price is the broad based 50 – stock Nifty index. For the equity index Nifty, the theoretical models considered were the following: ? ? ? F0= S0*e^(rT) F0= S0*e^(r-q)T (for dividend yield q) Log(Ft/St)=a + b*r + c*(sigma)+error
(Hemler and Longstaff Model)
Stocks The stock equities considered for studying the difference between theoretical and futures price are the following: ? ? ICICI Bank Hindustan Unilever Limited
For the stock equity, the theoretical models considered were the following: ? ? ? F0= S0*e^(rT) F0=(S0 –d)*e^(rT) (for absolute dividend ‘d’) Log(Ft/St)=a+ b* r+ c*(sigma)+error (Hemler and Longstaff Model)
Time series data for the 3 – month contracts were considered for the index & stock equity. Data was collected for the period from 1st November 2007 to 30th October 2009. The contracts were rolled-over for every near month contract. For the period from 1st November to 28th November 2007, the near term contract expiring on 29th November, 2007 is considered. On its expiry, the next near term contract expiring on 27th December, 2007 is considered for the period from 29th November to 26th December 2007. The contracts were rolled-over for every near term contract because the futures market is most liquid for a near term contract. This liquidity results in efficient price determination in
the market. The contract was rolled over on the previous day of the expiry date because of the volatility of the contract on the expiry day. Roll-over of the contract for every subsequent contract 1st Nov – 28th Nov ‘07 1– month 29 Nov ’07 29th Nov – 26th Dec ‘07 27 Dec ‘07 27th Dec – 30th Jan ‘08 31 Jan ‘08 28th Feb – 26th Mar ‘08 28 Feb ‘08 … 30th Oct ‘09
<= Data collected for near month contract
2– month 3– month
27 Dec ‘07 31 Jan ‘08
31 Jan ‘08 28 Feb ‘08
28 Feb ‘08 27 Mar ‘08
27 Mar ‘08 24 Apr ‘08
Results for time series from Nov-2007 to Oct 2009 Using formula S0*e^(rT) Using dividend (S0 –d)*e^(rT)
Security name Hindustan Unilever
MAPE
Standard Deviation
MAPE
Standard deviation
0.585659
0.683131 0.527161 0.3592638
0.584418 0.394727 0.3274482
0.681685 0.526219 0.3594415
ICICI Bank 0.3955753 NIFTY 0.3276939
Results for a single contract of 3 months from March 2009 to June 2009 Using formula S0*e^(rT) Using dividend d)*e^(rT) MAPE (S0 –
Security name
MAPE
Standard
Standard
Deviation Hindustan Unilever 1.000141 ICICI Bank NIFTY 0.82589 0.153422 0.61979 0.512115 0.147486 0.999366 0.824014 0.153402
deviation 0.618436 0.510252 0.147507
2 – Month & 3 – Month roll-over In order to study the effect of time to maturity on the futures prices, 2-month and 3-month rollover data was considered for ICICI stock and the MAPE as well as standard deviation of the error was studied to see whether the error increased with increase in time to maturity. The methodology adopted was as follows: Time series data for the 3 – month contracts were considered for the ICICI equity. Data was collected for the period from 1st November 2007 to 30th October 2009. In addition to the rollover for every near month contract, roll-over for 2 – month & 3 – month contract was also done and the difference between theoretical & market price was studied. In the 2 – month roll-over, every second month contract is considered. For the period from 1st November to 28th November 2007, the contract expiring on 27th December, 2007 is considered. On its expiry, the next second month contract expiring on 31st January, 2008 is considered for the period from 29th November to 26th December 2007, and so on. 1st Nov – 28th Nov ‘07 1– month 2– month 29 Nov ’07 27 Dec ‘07 29th Nov – 26th Dec ‘07 27 Dec ‘07 31 Jan ‘08 27th Dec – 30th Jan ‘08 31 Jan ‘08 28 Feb ‘08 28th Feb – 26th Mar ‘08 28 Feb ‘08 27 Mar ‘08 <= Data collected for second month contract … 30th Oct ‘09
3– month
31 Jan ‘08
28 Feb ‘08
27 Mar ‘08
24 Apr ‘08
10
20
30
0
The chart showing the difference between the theoretical price and settlement price for ICICI stock for 1 month rollover contract is shown below. The error plot ie Theoretical price – Settlement price is shown below
-40 200 400 600 800 0
-30
-20
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
-10
1000
1200
1400
1600
ICICI 1 month chart
Error 1 month
S*e(rt)
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
difference w/o dividend
difference with dividend
Settle Price
Difference between theoretical price and market price for a 2 – month roll-over of the contract for ICICI Bank Using formula S0*e^(rT) Security name MAPE Standard Deviation 0.5446 Using dividend (S0 –d)*e^(rT) MAPE Standard deviation 0.5431
ICICI Bank
0.51126
0.5091
1600 1400 1200 1000 800 600 400 200
ICICI 2 month contract
Theoretical price Settle Price
50 40 30 20 10 0 -10
01/… 01/… 01/J… 01/… 01/… 01/… 01/… 01/J… 01/J… 01/… 01/… 01/… 01/… 01/… 01/J… 01/… 01/… 01/… 01/… 01/J… 01/J… 01/… 01/… 01/…
0
Error 2 month rollover
difference without dividend difference with dividend
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
In the 3 – month roll-over, every third month contract is considered. For the period from 1st November to 28th November 2007, the contract expiring on 31st January, 2008 is considered. On its expiry, the next third month contract expiring on 28th February, 2008 is considered for the period from 29th November to 26th December 2007, and so on. 1st Nov – 28th Nov ‘07 1– month 2– month 3– month 29 Nov ’07 27 Dec ‘07 31 Jan ‘08 29th Nov – 26th Dec ‘07 27 Dec ‘07 31 Jan ‘08 28 Feb ‘08 27th Dec – 30th Jan ‘08 31 Jan ‘08 28 Feb ‘08 27 Mar ‘08 28th Feb – 26th Mar ‘08 28 Feb ‘08 27 Mar ‘08 24 Apr ‘08 <= Data collected for second month contract … 30th Oct ‘09
Difference between theoretical price and market price for a 3 – month roll-over of the contract for ICICI Bank Using formula S0*e^(rT) Security name MAPE Standard Deviation 5.2782 Using dividend (S0 –d)*e^(rT) MAPE Standard deviation 5.2778
ICICI Bank
5.13487
5.1345
1600 1400 1200 1000 800 600 400 200 0
ICICI 3 month contract
Settle Price Theoretical price
150 100 50 0
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
Difference 3 month contract
difference w/o dividend -50 -100 -150 -200 -250 difference with dividend
From the values of MAPE and Standard deviation for 1-month, 2-month and 3-month contracts, we see that the MAPE and Standard deviation keep increasing as the contract duration increases from 1-month to 2-month to a 3-month contract. Hence we can conclude that time to maturity of a contract also plays a significant role in determining the futures prices. The difference between the theoretical price and the settlement price is also visible in the charts where we see that both the price curves overlap in case of a 1-month contract, A slight deviation is observed in case of a 2-month rolling contract and in a 3- month contract we can distinctly see two different coloured graphs on several occasions showing that the difference is quite high in a far month contract. The primary reason for this difference could
be the liquidity of each contract. We find that the liquidity is high for the near month contract as compared to a far month contract. Commodities The calculation of theoretical futures prices for commodities is different from that of equities as we have to account for the storage cost as well as the convenience for holding for the commodity. Hence we use a model which is termed as the cost of carry model for commodities. The model is given below F = S0*e(r+u-y)*t Where F is the Futures price for the commodity S0 is the Spot price of the commodity r is the risk free rate of interest t is the time to expiry u is the storage cost for the commodity y is the convenience yield resulting from holding the commodity For the empirical analysis of the model examples of Gold and Oil commodities are taken which are traded in the MCX exchange. The time duration of the contract is for 1 year from 16th October 2007 to 4th October 2008. Initially a simple model (S0ert) is considered without considering the storage cost and the convenience yield to see if the Mean Absolute percentage error (MAPE) differs by a significant amount as compared to that with equity stocks as underlying. The following table shows the results of comparison of the theoretical futures prices of gold with the actual prices traded in the same period. MAPE Standard deviation 2.38% 1.355%
As compared to equity underlying this value of Mean absolute percentage error (MAPE) and standard deviation are quite high. Hence it is evident that storage cost and convenience yield plays a significant role in determining the futures prices. But for gold the storage cost is minimal and hence can be neglected. Hence only the convenience yield plays a major role in determining the futures prices. By trial and error method we try to determine the value of convenience yield ‘y’, which results in the minimum MAPE. The following table shows the result of the trial and error analysis.
Y (in%) 1 2 3 4 5 6
MAPE 1.912 1.453 1.074 0.895 0.912 1.123
Std dev(%) 1.165 1.030 0.912 0.8 0.832 0.980
Y (in%) 4.25 4.5 4.75
MAPE 0.878 0.874 0.886
Std dev(%) 0.796 0.801 0.813
We see from trial and error method that the minimum value of MAPE is attained when the convenience yield is 4.5%. Doing the same process for Oil commodity we find the below values of MAPE and Standard deviation when storage cost and convenience yield are not considered to check the relevance of these factors in determining the future price. The values are as given in the table below MAPE Standard deviation 2.71% 1.744%
In case of oil we see that the storage cost also plays a major role along with the convenience yield. Determining individual values can give many combinations of ‘u’ and ‘y’ which
ultimately gives the same value of ‘u-y’ for a minimum MAPE. By using the trial and error approach we get the value as -20%. This negative value suggests that Oil is costlier in the Spot market as compared to the futures market. The same result can also seen by observing the spot prices and future prices of oil in that period. Hemler Longstaff model Hemler and Longstaff derived a continuous time model wherein they tried to determine the futures price of any underlying based on the following equation Lt =?+?rt+??St+?t Where, Lt = log (Ft a/St) Ft a is the dividend-adjusted futures price St is the stock price ?St is the asset price volatility rt is the risk free interest rate ?t is the residual value If the regression coefficients ?, ? and ?t become zero then the equation reduces to the normal equation Ft=St*ert. Thus the Hemler Longstaff equation says that the asset volatility also plays a major role in determining the futures prices. This model has been tested to be successful in U.S. market. We try to test the significance of this model in Indian markets by running a regression equation with Lt is the dependant variable and all other parameters are the independent variables. The model is tested for NIFTY index and ICICI stock. The results are as follows:
For ICICI (Significance of model 0.693) Model Summary
Model R 1
R Square Adjusted R Square Std. Error of the Estimate -.002 .2576071
.044a .002
a. Predictors: (Constant), std_dev, t_bill
Coefficientsa Unstandardized Coefficients Standardized Coefficients Model B Std. Error .042 6.616 .829 -.002 -.044 Beta t Sig.
1 (Constant) .048 t_bill std_dev -.284 -.784
1.145 .253 -.043 .966 -.947 .344
a. Dependent Variable: log_ft_st
For NIFTY (Significance of model 0.561) Model Summary
Model R 1 .049a
R Square .002
Adjusted R Square -.002
Std. Error of the Estimate .004171
a. Predictors: (Constant), std_dev, t_bill
Coefficientsa Unstandardized Coefficients Model 1 B Std. Error Standardized Coefficients Beta t .237 Sig. .813
(Constant) 9.742E.000 5
t_bill std_dev
0.016 -0.014
.106 .013
.007 -.044
.154 -1.070
.877 .285
a. Dependent Variable: log_ft_st We see that for both ICICI and NIFTY the models are not significant at all .Moreover the R 2 value is very low which shows that in Indian Markets Asset volatility does not play any role in determining the futures prices.
Conclusions: The difference between the theoretical and settlement prices reduces overall when dividend is adjusted in the model for Index and stock futures. From the ICICI futures charts, it is evident that the theoretical future price is close to the actual price in case of the near month contract as compared to far month contracts. This is because of inadequate liquidity in the far month contract. Moreover the variance in this error also increases as the contract duration increases. Mean and variance of the difference between actual and theoretical futures prices is higher in case of commodities because of the lack of liquidity in the commodities market. In case of commodities contract, storage costs and convenience to carry has a significant effect on commodity pricing in India. Helmer and Longstaff model is not proved to be valid to estimate the theoretical prices in the Indian Market as the model is not significant with the given data. References: 1. www.nseindia.com 2. www.mcxindia.com 3. COMPARISON OF FUTURES PRICING MODELS IN A NEW MARKET: THE CASE OF
INDIVIDUAL SHARE FUTURES By, T. J. BRAILSFORD A. J. CUSACK The Journal of Futures Markets, Vol. 17, No. 5, 515-541 (1997)
doc_615419910.docx
Explains topics like derivatives and risk management,theoretical future price for index and stocks
T.A.Pai Management Institute
Comparison of theoretical and market future prices in India
DERIVATIVES AND RISK MANAGEMENT- THEME 5
GROUP 20 Manish M Shreyas Srinivasan Sudhanva S Shetty Vijay Pandey 08129 08151 08156 08164
The objective of the study is to test the theoretical models of future prices in the Indian context. The study was done for index futures, stock futures and commodities. The underlying considered were Nifty 50 from National Stock exchange. The underlying stocks considered were ICICI bank and HUL listed in NSE. The commodities considered were gold and crude oil traded in the MCX. The methodology followed is as follows: 1. 2. 3. 4. 5. Literature study of different theoretical models for future prices. Data collection for spot and future prices from data sources (NSE, MCX.) Calculation of theoretical futures price. Calculate the difference between the actual future prices and theoretical futures price. Analyses of the differences and conclusions.
Theoretical Future Price for Index: A Futures contract is specified for a period of time, at the end of which it is settled. Theoretical Future price should factor the current price and holding costs. In order to compensate the seller for waiting till expiry for realizing the sale proceeds the buyer has to pay some interest which is reflected in the form of cost of carry. Futures Price = Spot Price + Cost of Carry The Cost of carry is the sum of all costs incurred if a similar position is taken in cash market and carried to maturity of the futures contract less any revenue which may result in this period The most popular models used for finding the theoretical future prices is i. ii. Where F0 is the theoretical future price S0 is the spot price of the underlying r is the risk free interest rate T is the time to expiry q is the yield on Index F0= S0*e^(rT) F0= S0*e^(r-q)T (for dividend yield q)
If the yield on the index is to be considered, then it has to be discounted from the theoretical future prices as in equation (ii). Theoretical Future Price for Stocks: This is similar to the index theoretical future prices. But for dividend consideration, the data was available in absolute terms and hence the formula changes as in equation (iv.) iii. iv. F0= S0*e^(rT) F0=(S0 –d)*e^(rT) (for absolute dividend ‘d’)
Equity Index The equity index considered for studying the difference between theoretical and futures price is the broad based 50 – stock Nifty index. For the equity index Nifty, the theoretical models considered were the following: ? ? ? F0= S0*e^(rT) F0= S0*e^(r-q)T (for dividend yield q) Log(Ft/St)=a + b*r + c*(sigma)+error
(Hemler and Longstaff Model)
Stocks The stock equities considered for studying the difference between theoretical and futures price are the following: ? ? ICICI Bank Hindustan Unilever Limited
For the stock equity, the theoretical models considered were the following: ? ? ? F0= S0*e^(rT) F0=(S0 –d)*e^(rT) (for absolute dividend ‘d’) Log(Ft/St)=a+ b* r+ c*(sigma)+error (Hemler and Longstaff Model)
Time series data for the 3 – month contracts were considered for the index & stock equity. Data was collected for the period from 1st November 2007 to 30th October 2009. The contracts were rolled-over for every near month contract. For the period from 1st November to 28th November 2007, the near term contract expiring on 29th November, 2007 is considered. On its expiry, the next near term contract expiring on 27th December, 2007 is considered for the period from 29th November to 26th December 2007. The contracts were rolled-over for every near term contract because the futures market is most liquid for a near term contract. This liquidity results in efficient price determination in
the market. The contract was rolled over on the previous day of the expiry date because of the volatility of the contract on the expiry day. Roll-over of the contract for every subsequent contract 1st Nov – 28th Nov ‘07 1– month 29 Nov ’07 29th Nov – 26th Dec ‘07 27 Dec ‘07 27th Dec – 30th Jan ‘08 31 Jan ‘08 28th Feb – 26th Mar ‘08 28 Feb ‘08 … 30th Oct ‘09
<= Data collected for near month contract
2– month 3– month
27 Dec ‘07 31 Jan ‘08
31 Jan ‘08 28 Feb ‘08
28 Feb ‘08 27 Mar ‘08
27 Mar ‘08 24 Apr ‘08
Results for time series from Nov-2007 to Oct 2009 Using formula S0*e^(rT) Using dividend (S0 –d)*e^(rT)
Security name Hindustan Unilever
MAPE
Standard Deviation
MAPE
Standard deviation
0.585659
0.683131 0.527161 0.3592638
0.584418 0.394727 0.3274482
0.681685 0.526219 0.3594415
ICICI Bank 0.3955753 NIFTY 0.3276939
Results for a single contract of 3 months from March 2009 to June 2009 Using formula S0*e^(rT) Using dividend d)*e^(rT) MAPE (S0 –
Security name
MAPE
Standard
Standard
Deviation Hindustan Unilever 1.000141 ICICI Bank NIFTY 0.82589 0.153422 0.61979 0.512115 0.147486 0.999366 0.824014 0.153402
deviation 0.618436 0.510252 0.147507
2 – Month & 3 – Month roll-over In order to study the effect of time to maturity on the futures prices, 2-month and 3-month rollover data was considered for ICICI stock and the MAPE as well as standard deviation of the error was studied to see whether the error increased with increase in time to maturity. The methodology adopted was as follows: Time series data for the 3 – month contracts were considered for the ICICI equity. Data was collected for the period from 1st November 2007 to 30th October 2009. In addition to the rollover for every near month contract, roll-over for 2 – month & 3 – month contract was also done and the difference between theoretical & market price was studied. In the 2 – month roll-over, every second month contract is considered. For the period from 1st November to 28th November 2007, the contract expiring on 27th December, 2007 is considered. On its expiry, the next second month contract expiring on 31st January, 2008 is considered for the period from 29th November to 26th December 2007, and so on. 1st Nov – 28th Nov ‘07 1– month 2– month 29 Nov ’07 27 Dec ‘07 29th Nov – 26th Dec ‘07 27 Dec ‘07 31 Jan ‘08 27th Dec – 30th Jan ‘08 31 Jan ‘08 28 Feb ‘08 28th Feb – 26th Mar ‘08 28 Feb ‘08 27 Mar ‘08 <= Data collected for second month contract … 30th Oct ‘09
3– month
31 Jan ‘08
28 Feb ‘08
27 Mar ‘08
24 Apr ‘08
10
20
30
0
The chart showing the difference between the theoretical price and settlement price for ICICI stock for 1 month rollover contract is shown below. The error plot ie Theoretical price – Settlement price is shown below
-40 200 400 600 800 0
-30
-20
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
-10
1000
1200
1400
1600
ICICI 1 month chart
Error 1 month
S*e(rt)
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
difference w/o dividend
difference with dividend
Settle Price
Difference between theoretical price and market price for a 2 – month roll-over of the contract for ICICI Bank Using formula S0*e^(rT) Security name MAPE Standard Deviation 0.5446 Using dividend (S0 –d)*e^(rT) MAPE Standard deviation 0.5431
ICICI Bank
0.51126
0.5091
1600 1400 1200 1000 800 600 400 200
ICICI 2 month contract
Theoretical price Settle Price
50 40 30 20 10 0 -10
01/… 01/… 01/J… 01/… 01/… 01/… 01/… 01/J… 01/J… 01/… 01/… 01/… 01/… 01/… 01/J… 01/… 01/… 01/… 01/… 01/J… 01/J… 01/… 01/… 01/…
0
Error 2 month rollover
difference without dividend difference with dividend
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
In the 3 – month roll-over, every third month contract is considered. For the period from 1st November to 28th November 2007, the contract expiring on 31st January, 2008 is considered. On its expiry, the next third month contract expiring on 28th February, 2008 is considered for the period from 29th November to 26th December 2007, and so on. 1st Nov – 28th Nov ‘07 1– month 2– month 3– month 29 Nov ’07 27 Dec ‘07 31 Jan ‘08 29th Nov – 26th Dec ‘07 27 Dec ‘07 31 Jan ‘08 28 Feb ‘08 27th Dec – 30th Jan ‘08 31 Jan ‘08 28 Feb ‘08 27 Mar ‘08 28th Feb – 26th Mar ‘08 28 Feb ‘08 27 Mar ‘08 24 Apr ‘08 <= Data collected for second month contract … 30th Oct ‘09
Difference between theoretical price and market price for a 3 – month roll-over of the contract for ICICI Bank Using formula S0*e^(rT) Security name MAPE Standard Deviation 5.2782 Using dividend (S0 –d)*e^(rT) MAPE Standard deviation 5.2778
ICICI Bank
5.13487
5.1345
1600 1400 1200 1000 800 600 400 200 0
ICICI 3 month contract
Settle Price Theoretical price
150 100 50 0
01/Nov/07 01/Dec/07 01/Jan/08 01/Feb/08 01/Mar/08 01/Apr/08 01/May/08 01/Jun/08 01/Jul/08 01/Aug/08 01/Sep/08 01/Oct/08 01/Nov/08 01/Dec/08 01/Jan/09 01/Feb/09 01/Mar/09 01/Apr/09 01/May/09 01/Jun/09 01/Jul/09 01/Aug/09 01/Sep/09 01/Oct/09
Difference 3 month contract
difference w/o dividend -50 -100 -150 -200 -250 difference with dividend
From the values of MAPE and Standard deviation for 1-month, 2-month and 3-month contracts, we see that the MAPE and Standard deviation keep increasing as the contract duration increases from 1-month to 2-month to a 3-month contract. Hence we can conclude that time to maturity of a contract also plays a significant role in determining the futures prices. The difference between the theoretical price and the settlement price is also visible in the charts where we see that both the price curves overlap in case of a 1-month contract, A slight deviation is observed in case of a 2-month rolling contract and in a 3- month contract we can distinctly see two different coloured graphs on several occasions showing that the difference is quite high in a far month contract. The primary reason for this difference could
be the liquidity of each contract. We find that the liquidity is high for the near month contract as compared to a far month contract. Commodities The calculation of theoretical futures prices for commodities is different from that of equities as we have to account for the storage cost as well as the convenience for holding for the commodity. Hence we use a model which is termed as the cost of carry model for commodities. The model is given below F = S0*e(r+u-y)*t Where F is the Futures price for the commodity S0 is the Spot price of the commodity r is the risk free rate of interest t is the time to expiry u is the storage cost for the commodity y is the convenience yield resulting from holding the commodity For the empirical analysis of the model examples of Gold and Oil commodities are taken which are traded in the MCX exchange. The time duration of the contract is for 1 year from 16th October 2007 to 4th October 2008. Initially a simple model (S0ert) is considered without considering the storage cost and the convenience yield to see if the Mean Absolute percentage error (MAPE) differs by a significant amount as compared to that with equity stocks as underlying. The following table shows the results of comparison of the theoretical futures prices of gold with the actual prices traded in the same period. MAPE Standard deviation 2.38% 1.355%
As compared to equity underlying this value of Mean absolute percentage error (MAPE) and standard deviation are quite high. Hence it is evident that storage cost and convenience yield plays a significant role in determining the futures prices. But for gold the storage cost is minimal and hence can be neglected. Hence only the convenience yield plays a major role in determining the futures prices. By trial and error method we try to determine the value of convenience yield ‘y’, which results in the minimum MAPE. The following table shows the result of the trial and error analysis.
Y (in%) 1 2 3 4 5 6
MAPE 1.912 1.453 1.074 0.895 0.912 1.123
Std dev(%) 1.165 1.030 0.912 0.8 0.832 0.980
Y (in%) 4.25 4.5 4.75
MAPE 0.878 0.874 0.886
Std dev(%) 0.796 0.801 0.813
We see from trial and error method that the minimum value of MAPE is attained when the convenience yield is 4.5%. Doing the same process for Oil commodity we find the below values of MAPE and Standard deviation when storage cost and convenience yield are not considered to check the relevance of these factors in determining the future price. The values are as given in the table below MAPE Standard deviation 2.71% 1.744%
In case of oil we see that the storage cost also plays a major role along with the convenience yield. Determining individual values can give many combinations of ‘u’ and ‘y’ which
ultimately gives the same value of ‘u-y’ for a minimum MAPE. By using the trial and error approach we get the value as -20%. This negative value suggests that Oil is costlier in the Spot market as compared to the futures market. The same result can also seen by observing the spot prices and future prices of oil in that period. Hemler Longstaff model Hemler and Longstaff derived a continuous time model wherein they tried to determine the futures price of any underlying based on the following equation Lt =?+?rt+??St+?t Where, Lt = log (Ft a/St) Ft a is the dividend-adjusted futures price St is the stock price ?St is the asset price volatility rt is the risk free interest rate ?t is the residual value If the regression coefficients ?, ? and ?t become zero then the equation reduces to the normal equation Ft=St*ert. Thus the Hemler Longstaff equation says that the asset volatility also plays a major role in determining the futures prices. This model has been tested to be successful in U.S. market. We try to test the significance of this model in Indian markets by running a regression equation with Lt is the dependant variable and all other parameters are the independent variables. The model is tested for NIFTY index and ICICI stock. The results are as follows:
For ICICI (Significance of model 0.693) Model Summary
Model R 1
R Square Adjusted R Square Std. Error of the Estimate -.002 .2576071
.044a .002
a. Predictors: (Constant), std_dev, t_bill
Coefficientsa Unstandardized Coefficients Standardized Coefficients Model B Std. Error .042 6.616 .829 -.002 -.044 Beta t Sig.
1 (Constant) .048 t_bill std_dev -.284 -.784
1.145 .253 -.043 .966 -.947 .344
a. Dependent Variable: log_ft_st
For NIFTY (Significance of model 0.561) Model Summary
Model R 1 .049a
R Square .002
Adjusted R Square -.002
Std. Error of the Estimate .004171
a. Predictors: (Constant), std_dev, t_bill
Coefficientsa Unstandardized Coefficients Model 1 B Std. Error Standardized Coefficients Beta t .237 Sig. .813
(Constant) 9.742E.000 5
t_bill std_dev
0.016 -0.014
.106 .013
.007 -.044
.154 -1.070
.877 .285
a. Dependent Variable: log_ft_st We see that for both ICICI and NIFTY the models are not significant at all .Moreover the R 2 value is very low which shows that in Indian Markets Asset volatility does not play any role in determining the futures prices.
Conclusions: The difference between the theoretical and settlement prices reduces overall when dividend is adjusted in the model for Index and stock futures. From the ICICI futures charts, it is evident that the theoretical future price is close to the actual price in case of the near month contract as compared to far month contracts. This is because of inadequate liquidity in the far month contract. Moreover the variance in this error also increases as the contract duration increases. Mean and variance of the difference between actual and theoretical futures prices is higher in case of commodities because of the lack of liquidity in the commodities market. In case of commodities contract, storage costs and convenience to carry has a significant effect on commodity pricing in India. Helmer and Longstaff model is not proved to be valid to estimate the theoretical prices in the Indian Market as the model is not significant with the given data. References: 1. www.nseindia.com 2. www.mcxindia.com 3. COMPARISON OF FUTURES PRICING MODELS IN A NEW MARKET: THE CASE OF
INDIVIDUAL SHARE FUTURES By, T. J. BRAILSFORD A. J. CUSACK The Journal of Futures Markets, Vol. 17, No. 5, 515-541 (1997)
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