1. Arbitrage Page 1 of 5
Arbitrage - Safety with Smartness
Arbitrage by definition is a financial transaction that makes an immediate profit without involving
any risk. Technically it consists of purchasing a commodity or security in one market for immediate
sale in another market (deterministic arbitrage). However popular usage has expanded the
meaning of the term to include any activity which attempts to buy a relatively underpriced item
and sell a similar, relatively overpriced item, expecting to profit when the prices resume a more
appropriate theoretical or historical relationship (statistical arbitrage).
The recently introduced futures index by the BSE and NSE is an instrument which can be used for
arbitrage. Arbitrage opportunities are said to exist whenever the futures price moves away from
the fair value. Fair value is the summation of spot price and the holding cost . Holding costs could
be Cost of financing plus Storage costs plus Insurance purchased, etc.(In case of commodities) and
holding cost = Cost of financing minus Dividend returns, which could be in the form of dividends in
case of equities futures.
For example, suppose a futures contract is traded on two different exchanges. If, Futures price =
Spot price + Holding
F=S+C
However if
If F> (S + C) or F< (S + C), then arbitrage opportunity exists.
The futures price is calculated as follows:-
Example
Futures price of 100 gms of silver one-month down the line i.e. a contract expiring 30th November
is computed as follows:
What is the spot price of silver? The spot price of silver, S= Rs. 7000/kg
What is the cost of financing for a month? rT, cost of financing for a month, 15% annualized = ln
(1.15)*30/365
What are the holding costs? Assume storage cost, C = 0
The fair value of futures price, F=S*exp(rT) + C = 700 * exp(ln(1.15)*30/365) = Rs. 708
If the contract was for a three-month period i.e. expiring on 30th January, the cost of financing
would increase the futures price. Therefore, the futures price would be
F = 700 * exp(ln (1.15)*90/365) = Rs. 725
In case of calculation of the price of future contracts on equities there is no cost of storage
considered in holding paper, however equity paper comes with a dividend stream, which is a
negative cost if you are long on the stock, and a positive cost if you are short the stock.
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C = financing cost - dividends
Thus, a crucial aspect of dealing with equity futures as opposed to commodity futures is an
accurate forecasting of dividends. The better the forecast of dividend offered by a security, the
better is the estimate of the futures price.
For example
What is the fair value of a two-month S&P CNX Nifty futures contract expiring on April 25?
What is the annual dividend yield on S&P CNX Nifty index? The dividend yield on S&P CNX Nifty,
2% annualized = ln(1.02)*60/365
What is the spot value of S&P CNX Nifty? Current value of S&P CNX Nifty is 910
What is the cost of financing for two months? RT, cost of financing for a month, 15% annualized =
ln(1.15)*60/365
What are the holdings costs? Assume storage cost, C=0
The fair value of futures price, F=S*exp(ln(1+r-q))*T + C = 910 * exp(ln(1.13)*60/365) = Rs.
928.47
Arbitrage helps investors to lend funds into the stock market, without suffering the slightest risk. In
the traditional methods of loaning money into the stock market there is a price risk or credit risk
involved. But through the index futures market an investor can hedge both the price and credit
risk.
The basic idea is simple. The lender buys all 50 stocks of S&P CNX Nifty on the cash market, and
simultaneously sells them at a future date on the futures market. There is no price risk since the
position is perfectly hedged.
There is no credit risk since the counter party on both legs is the National Securities Clearing
Corporation (NSCC) which supplies clearing services on NSE. It is an ideal lending vehicle for
entities which are shy of price risk and credit risk, such as traditional banks and the most
conservative corporate treasuries.
Hedging the Price Risk
One buys a portfolio in which all the 50 stocks in S&P CNX Nifty are in correct proportion, (i.e.
where the money invested in each stock is proportional to its market capitalization.) on the cash
market. Simultaneously sell S&P CNX Nifty futures of equal value. Now you are completely hedged,
so fluctuations in S&P CNX Nifty do not affect you.A few days later, you will have to take delivery of
the 50 stocks and pay for them. This is the point at which you are "loaning money to the
market".Some sell your portfolio and reverse your future position. A few days later, you will have
to make delivery of the 50 stocks and receive money for them. This is the point at which "your
money is repaid to you".The interest rate that you will receive is the difference between the futures
price and the cash S&P CNX Nifty plus any dividends earned minus the transactions costs (impact
cost, brokerage) in doing these trades.
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Suppose the S&P CNX Nifty spot is at 1000 and the two-month futures are at 1040. Suppose the
transactions costs involved are 0.4% per month and dividends over the two months are nil. Then
the rate of return in loaning money to the market is 1.5% (1040/1000 over two months is near
1.9% per month. Subtract out 0.4% as transaction costs to get 1.5% per month.
On 1 August, S&P CNX Nifty is at 1200. A futures contract is trading with 27 August expiration for
1230. Ashish wants to earn this return (30/1200 for 27 days).
He buys Rs. 3 million of S&P CNX Nifty on the spot market. In doing this, he places 50 market
orders and ends up paying slightly more. His average cost of purchase is 0.3% higher, i.e. he has
obtained the S&P CNX Nifty spot for 1204.
He sells Rs. 3 million of the futures at 1230. The futures market is extremely liquid so the market
order for Rs. 3 million goes through at near-zero impact cost. He takes delivery of the shares and
waits.
While waiting; a few dividends come into his hands. The dividends work out to Rs. 7,000.
On 27 August, at 3:15, Ashish puts in market orders to sell off his S&P CNX Nifty portfolio, putting
50 market orders to sell off all the shares. S&P CNX Nifty happens to have closed at 1210 and his
sell orders (which suffer impact cost) goes through at 1207.
The futures position spontaneously expires on 27 August at 1210 (the value of the futures on the
last day is always equal to the S&P CNX Nifty spot).
Ashish has gained Rs. 3 (0.255) on the spot S&P CNX Nifty and Rs. 20 (1.63%) on the futures for a
return of near 1.88%. In addition, he has gained Rs. 7,000 or 0.23% owing to the dividends for a
total return of 2.11% for 27 days, risk free.
Arbitrage also offers an investor the opportunity to lend securities to the market and earn
revenues. The mechanism is simple -you sell off your certificates and contract to buy them back in
the future at a fixed price. The basic idea is quite simple. You would sell all 50 stocks in S&P CNX
Nifty and buy them back at a future date using the index futures. You would soon receive money
for the shares you have sold. You can deploy this money as you like until futures expiration. On
this date, you would buy back your shares, and pay for them.
Suppose you have Rs. 5 million of the S&P CNX Nifty portfolio (in their correct proportion, with
each share being present in the portfolio with a weight that is proportional to its market
capitalization).
Sell off all 50 shares on the cash market. This can be done using a single keystroke (offline order
entry) using the NEAT software.
Buy index futures of an equal value.
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A few days later, you will receive money and have to make delivery of the 50 shares.
Deploy this money at the riskless interest rate.
On the date that the futures expire, at 3:15 PM, put in 50 orders (using NEAT again) to buy the
entire S&P CNX Nifty portfolio.
A few days later, you will need to pay in the money and get back your shares.
This is possible when the spot-futures basis (the difference between spot S&P CNX Nifty and the
futures S&P CNX Nifty) is smaller than the riskless interest rate that you can find in the economy.
If the spot-futures basis is 2.5% per month and you are loaning out the money at 1.5% per month,
it is not profitable. Conversely, if the spot-futures basis is 1% per month and you are loaning out
money at 1.2% per month, this stocklending could be profitable.The stock lending rate is calculated
as follows:- we assume that transactions cost account for 0.4%. Suppose the spot-futures basis is
x% and suppose the rate at which funds can be invested is y%. Then the total return is y - x -
0.4%, over the time that the position is held.
Example
Suppose Akash has Rs. 4 million of the S&P CNX Nifty portfolio which he would like to lend to the
market.
Akash puts in sell orders for Rs. 4 million of S&P CNX Nifty using the feature in NEAT to rapidly
place 50 market orders, in quick succession. The seller always suffers impact cost; suppose he
contains an actual execution at 1098.
A moment later, Akash puts in a market order to buy Rs. 4 million of the S&P CNX Nifty futures.
The order executes at 1110. At this point, he is completely hedged.
A few days later, Akash makes delivery of shares and receives Rs. 3.99 million (assuming an
impact cost of 2/1100)
Suppose Akash lends this out at 1% per month for two months.
At the end of two months, the money comes back to him as Rs. 4,072,981. Translated in terms of
S&P CNX Nifty, this is1098 * 1.012 or 1120.
On the expiration date of the futures, he puts in 50 orders, using NEAT, placing market orders to
buy back his S&P CNX Nifty portfolio. Suppose S&P CNX Nifty has moved up to 1150 by this time.
This makes shares costlier in buying back, but the difference is exactly offset by profits on the
futures contract.
When the market order is placed, suppose he ends up paying 1153 and not 1150, owing to impact
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cost. He has funds in hands of 1120, and the futures contract pays 40 (1150-1110) so he ends up
with a clean profit, on the entire transaction, of 1120+40-1153 = 7. On a base of Rs. 4 million, this
is Rs. 25,400.
Aru Srivastava
http://www.karvy.com/articles/arbitrage.htm 1/12/2009