# Delta neutral hedge with futures

#### Benamed

##### Newbie
4 0
Dear all,

I'm wondering if my calculations of the following problem is correct:

10 000 call options on S&P500
Characteristics of the option:
Start date: March 31st 2011
Type: European
Maturity: May 31st 2011
Strike Price: 1400
1 option contract gives the right to buy 1 S&P500 index

1) So, what is the amount of CME S&P 500 Futures traded on the Chicago Mercantile Exchange (www.cme.com ) that I need, in order to make this portfolio delta neutral?

This is how far I got:

Delta of a call option is positive.
∆ (call)= N(d1)
d1 = (ln(S0/K)+(r+σ^2/2)T)/(σ√T)

First we need to find the respective parameters

S0 = 1325,83 on 31/03/2011
K = 1400
r = 0,0015 is the annual treasury bill rate at 31 March
σ= 0,149 (Annualizing gives 0,0094*SQRT252 = 0,149)
T = 2/12

Now we can calculate delta:
d1 = (ln((1325,83 )/1400)+(0.0015+〖0,149〗^2/2)*(2/12))/(0,149√((2/12)))

D1 = -0,86034

N(d1)= 0,194801

Delta of a futures contract is e^rT

e^(0,0015)*(2/12) = 0,166916854

Number of future contracts =
= (number of options * delta call)/delta future
= (10000*0,194801472167245)/0,166916854
= 11670,56934

So in order to create a delta neutral portfolio the investor should go short in 11670,56934.

Is this correct or should I somehow take into account the contract size of the future (\$250 x S&P 500 futures price) ?

Your help is much appreciated.

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#### tripleogstar

##### Established member
659 29
Dear all,

I'm wondering if my calculations of the following problem is correct:

10 000 call options on S&P500
Characteristics of the option:
Start date: March 31st 2011
Type: European
Maturity: May 31st 2011
Strike Price: 1400
1 option contract gives the right to buy 1 S&P500 index

1) So, what is the amount of CME S&P 500 Futures traded on the Chicago Mercantile Exchange (www.cme.com ) that I need, in order to make this portfolio delta neutral?

This is how far I got:

Delta of a call option is positive.
∆ (call)= N(d1)
d1 = (ln(S0/K)+(r+σ^2/2)T)/(σ√T)

First we need to find the respective parameters

S0 = 1325,83 on 31/03/2011
K = 1400
r = 0,0015 is the annual treasury bill rate at 31 March
σ= 0,149 (Annualizing gives 0,0094*SQRT252 = 0,149)
T = 2/12

Now we can calculate delta:
d1 = (ln((1325,83 )/1400)+(0.0015+〖0,149〗^2/2)*(2/12))/(0,149√((2/12)))

D1 = -0,86034

N(d1)= 0,194801

Delta of a futures contract is e^rT

e^(0,0015)*(2/12) = 0,166916854

Number of future contracts =
= (number of options * delta call)/delta future
= (10000*0,194801472167245)/0,166916854
= 11670,56934

So in order to create a delta neutral portfolio the investor should go short in 11670,56934.

Is this correct or should I somehow take into account the contract size of the future (\$250 x S&P 500 futures price) ?

Your help is much appreciated.

the Emini S&P future is twice the size of the options so if you are buying 10k call options which have a delta of 0.19, you need to sell 3,800 futures to be delta hedged

#### tripleogstar

##### Established member
659 29
you can also hedge by doing combos, which is basically a synthetic futures position. So if you are selling futures you would sell an ATM Call and buy the ATM put. In your example you would do 1,900 combos. So sell 1,900 ATM calls and buy 1900 ATM puts, this would make your position delta hedged. The maturity month you would do this in would be the same as the futures expiry. So at the moment the front month Emini S&P future is Jun11, so you would execute your combos in the Jun11 SPX options.

#### Benamed

##### Newbie
4 0
Thank you for your advice. I need to find the number of future contracts though, so then I have to divide 11670,56934 by the contract size of the S&P 500 future (\$250) = 46,6822774

So I need to short 46 contracts, right?

#### tripleogstar

##### Established member
659 29
I have already told you what your hedge would be in terms of futures. Another thing,your calculations are all wrong, I don't know why on earth you are calculating the delta of a futures contract???!!!! The delta of a futures contract is 1! (100%) The same for any underlying of an option, the delta is 1. When you adjust for how many futures you need to hedge with, ALL you need to take into account of is the size of the futures in relation to the size of the options. If the size is the same, then your hedge would simply be :

Number of options * delta.

But in this case, if you are hedging with Emini s&p futures, which are half the size of SPX Options, the calculation is:

(Number of options*delta)/0.5

In the example you gave, it would be 3,800.

#### tripleogstar

##### Established member
659 29
Just so you know, you do know you are exposed to other greeks, not just delta. You have vega, gamma and theta. All are very important, vega I would say is most important but gamma and theta become much more important as expiry becomes closer

#### maxima

##### Established member
606 31
there are not a lot of SPX options traded on CME. are you sure you can get a proper quote on it? also if you will need to rid of them (for example you suddenly realised that all you calculus are just ******** after you opened the position).. you might not be able to do so as (see above) there not a lot of SPX options traded on CME..

also CME options are american as long as I remember.. so what options are we talking here exactly ?

#### etem_tezcan

##### Junior member
42 2
Please note that whenever underlying price changes, you have a different delta and number of SP futures contract you must hedge your option position changes. Thus you should adjust your position in futures to remain delta neutral. I cannot follow your delta calculation, it is so boring. There are online tools which calculates delta from streaming underlying prices.ThinkorSwim platform does that.

Why did you use 31 march value of SP futures on a post made 26 april. This gives me a slight discomfort implying that you may not be fully aware of mathematical logic running behind pricing of options. Also at least in the beginning focus on e-mini futures and their options.

#### Benamed

##### Newbie
4 0
@tripleogstar: Thank you for your reply. I wasn't aware of the fact that the delta of future contracts was always one in this case. In class I always had to calculate the amount of shares to go long or short in not the amount of futures that's why I was confused. I'm aware of the other Greeks which influence the portfolio.

@maxima: I forgot to mention in my opening post that the question is pure hypothetical, I didn't come up with it so I guess they just made up the option. I guess they chose European options since they are easier to work with. The future which I need to short should be a real one from the CME though.

@etem_tezcan: Since the question is pure hypothetical I need to assume that it is the 31st of March when I want to make the portfolio delta neutral. That's why I didn't use the real-time S&P500 value but the historical value on the 31st of March. In terms of the characteristics of the option: it is European, 1 option contract gives the right to buy 1 S&P500 index, it expires on May 31st 2011 and its strike price is 1400. Again, sorry for not mentioning that the question is hypothetical.

#### etem_tezcan

##### Junior member
42 2
I plugged your numbers into a option workbook. Your delta is 0.19. I used implied volatility of 15%.

You have a long call option position of 10000. This 10000 number is tricky. What does it correspond to. If it corresponds to call option of 10000 SP500 futures (which is very big position, 1 sp500 future (250 usd/point)is exposure to 350k stocks at 1400. Your 10,000 position is right to buy some stocks at 3.5 billion USD !!

Adjusting by delta you need to have a -1900 futures position to be delta neutral.
Just multiply/divide by your "hypothetical" contract values, to determine number of contracts to go short.

#### Benamed

##### Newbie
4 0
Thanks for all the input. In the end, I came up with the following:

The 10000 long call option position gives the right to buy 10000 S&P500 index.
So the underlying of the option is not the future contract on the S&P500 but just the real index.
So the option is a non-existing one which is made up however the S&P500 index future, which I need to use to make the position delta-neutral, should be an existing one from the CME site.

To delta hedge my portfolio I need to use S&P500 index futures.
Since I have a (positive) 1900 delta, I can use E-mini S&P 500 Futures with a contract size of \$50 to arrive at a total number of 1900/50 = 38 futures. These 38 futures will provide me with a negative delta of 1900.

Hence, I take a SHORT position in 38 E-mini S&P 500 Futures to make my portfolio delta-neutral on the 31st of March. On the 1st of April I have to rebalance this portfolio by shorting more/less futures in order to keep it delta-neutral.

#### maxima

##### Established member
606 31
i missed the whole point. are you trying to pass some kind of exam to be an option desk trader or what?

#### devvvy

##### Newbie
8 0
option position value * delta = future position value

option position value = c x number of contracts
(where c is warrant/option price, you can use market price, or compute with say BlackShoals, in which case you're using market neutral value of option/warrant position as suppose to market price. Just caution - market neutral valuation means valuation don't make any directional assumption on future/anticipated movements - as such it's quite useless - for risk/compliance! To setup real hedge, use "Anticipated Price" instead)

future contract value = (current index level * multiplier * discount factor - quoted future closing)

# futures contract = future position value / future contract value

Get a copy of Hull's text of Derivative valuation

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#### Uclmak

##### Newbie
3 0
Can some one help me in finding the solution.

Q1. Consider a one year put swaption, which has an underlying swap as a four-year swap. A put swaption gives its buyer the right to enter into a swap as a floating rate payer and if he exercises the option, receives fixed rate from the swap. The strike price of the put swaption is 9% and the notional principal is \$10 million. At expiration of the swaption, the spot rates on zero coupon bonds of various maturities turned out as below:

Year Yield on Zero Coupon Bond
1 7.5%
2 8.0%
3 8.4%
4 8.7%

You are required to calculate the payoff from the put swaption (Assume 360 days in a year)

Q2. An Indian Bank has sold three-month European call option on \$2 million with a strike price of Rs 45.10. The current rupee dollar exchange rate is Rs 45.30/\$. The annual volatility of rupee-dollar exchange rates is 6%. The 91-day T-Bill rate in India is 4% p.a and 91-day US T-Bill rate is 1% p.a.

You are required to find out the position the bank should take (using options) to make the position delta neutral.

Q3. The Current market price of ABB's stock if Rs 290. The Following European call an put options are available in the market.

Option Strike Price Premium Expiration(Months)
Call 280 24 6
Put 280 3 6
The risk-free interest rate is 6% p.a

You are required to find out whether there is any arbitrage opportunity available in the put and call prices. If no, justify why not. If yes, show how you can make arbitrage profit.