Hello, I’m new to future and option. Today I came across Delta Hedged portfolio and I was very confused. If I have 1 call option, is it I need one future contract in order to build a delta hedged portfolio. Say for example: Simulation date: Mars 31st 2011 Using 10 000 call options on S&P500 whose characteristics are Exercise type: European Maturity: May 31st 2011 Strike: 1400 What should I do if I want to know how many future contract CME S&P 500 Futures I need to get a delta hedged portfolio. And say at after a month from the simulation date, what is the impact of the delta hedged portfolio if the strike of S&P500 go up or down (±1%,±10%). And do you think this portfolio is good? I am really appreciate for any help. Thanks
Options traders primarily trade options based on volatility, not so much about strike price (which they can more or less choose when they buy or sell the option. So if the implied volatility is higher than you think the realized volatility is (through some model or simplistically one might take the historical average), then you would sell the option because the premium is a little bit juicier than normal.
However, the option could still expire either in or out of the money, so just because you sold one with a juicy premium or bought one with a cheap premium doesnt’ mean you are going to make any moeny based on your good volatility decision (let’s assume that was a good decision, because volatility can change too.)
To avoid this, you would delta-hedge the portfolio, which means that you would hold an offsetting portfolio that is a combination of cash and the underlying asset (you can also use a future because a future+cash is similar to being long the asset, but it only works if the account sizes are large enough for a futures contract to work).
In the short term, holding call option is like holding a portfolio that is allocated as (delta) to the stock + (1-delta) to cash (a put is the same, but short the stock instead of long), so what you would do if you wanted to sell a call is sell the call and collect (optionContractSize*premium), then buy (delta)*(optionContractSize*strikePrice) woth of stock + (1-delta)*(optionContractSize*strikePrice) in cash. The idea is that this portfolio will go up and down in value (over the very short term) approximately as much as the option’s value, so if you are making money by collecting a juicy premium, you won’t then lose it by the underlying’s price moving against you, because the cash+stock portfolio will go up and down by the same amount. However, once the price has changed, delta will be different, so you have to readjust the portfolio over time. This results in transaction costs and is one of the reasons that implied volatility is almost always higher than historical volatility (the other reason is the possibility of a volatility risk premium).
If you are using a future instead of the underlying, then you would be substituting futures for the underlying stock to create a synthetic stock position. A synthetic stock position with a future consists of (futuresMargin%) allocated to the future and (1-futuresMargin%) allocated to cash. One futures contractis going to be equivalent to holding contractSize of stock, but it’s divided into (futuresContractSize*stockPrice)*(margin%) in futures margin and (futuresContractSize*stockPrice)*(1-margin%) in cash.
Putting these together, a delta hedged options contract is going to have (delta)*(optionsContractSize*strikePrice) allocated to the underlying stock. But you are going to use futures, so that is going to mean (delta)*(optionsContractSize*strikePrice)/(futuresContractSize*futuresPrice*margin%) placed in futures, and everything else in cash (I wont add up the remaining terms).
Mathematically, it’s a pain to crunch through the equations, but it is a good question to test if you understand how delta hedging works and understand the use of futures to create synthetic exposures.
In practice, the option contract size and the futures contract sizes are often equal at 100, but that is not always the case.
OP:
School project? Non current dates, your first post ever, and the use of 1 and 10 seems to be very academic
BChad:
How do you know this? Is this deep quant derivative information a part of the curriculum or is this because of your quant background? All very interesting but after the second line you’ve lost me.
I have to make a comment here too. Why is it that almost every person i talk to (non Charterholders and certainly not bchad) that thinks they are so quantitatively advanced has blown up their accounts with options strategies gone wrong? They are almost all of a non-american decent and probably brilliant in their professions, but when the market is returning 18% they are -30%…why???
Thank for reply. I understand up to 3rd paragraph. But one thing I understand is that we need to find delta of the option and future contract in order to find the contract size. As I understand delta for future contract is always equal 1. So here is how i calculate delta for option positon:
∆ (call)= N(d1) d1 = (ln(S0/K)+(r+σ^2/2)T)/(σ√T) I can find these informations 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
And i get stuck here. I dont have any idea of what future contract we get from CME?
Just come across a problem in my study, I post the question else where and somebody recommend for me this forum so I figure I just try it out.
PDub… the greeks stuff is in the L2 curriculum. When I took the exams, the dynamic hedging stuff was mostly in the L3 curriculum, but the idea of the option replicating portfolio was in L2. And then some of the strategy stuff (i.e. buying options when implied volatility is low and selling them when IV is high) just seeped in through various things I’ve come across.
I am still a little shy of using options, because I don’t have a really good intuitive feel for how they work, although I probably shouldn’t be. Ohai and LPoulin and (I think) jmh are the AFers that really know their options stuff. Lockheed talks big, but I can’t really tell if that’s based on deep knowledge or not.
Delta pops out of the Black Scholes Merton model. You need a fairly advanced calculator or excel sheet to get it. Sounds like you got that, lovefinance, and the delta of yoru call is about 0.2, which means the call is OTM (because delta is <0.5 when the call is OTM).
So you’d have a portfolio that is 1400*multiplier*optionsContract size, of which 0.2 is supposed to be the S&P 500, and 0.8 is supposed to be Tbills. I think the multiplier is 100 and the contract size is 1 for index options, so that means the portfolio is 140,000, of which 28000 is S&P (owning the constituents, or maybe allocated to an ETF like SPY), and $112,000 is in T-bills.
But now, instead of holding the S&P in an ETF, you want to use futures to get your exposure. So you are going to try to get $28000 of exposure via a futures contract. The multiplier for an S&P500 future is 250, so that means you want $28000 / 250 or 448 contracts. To secure that, you are going to put down 5% margin, or $5600, and keep $22400 in cash/TBills.
When all is said and done, you will have $5600 in a futures contract, and $140,000 - $5600 or $134400 in cash in order to keep your options contract hedged at this level. That’s a lot of money earning almost no premium, so hopefully you are making money off of the implied volatility of the option.
Just remember that tomorrow, the S&P index will move, and so delta will change. This means you may have to recalculate the portfolio delta and change the number of futures you hold if the index has moved enough.
The danger of having all that cash lying around is that you will probably want to put it to use, when in fact it’s already in use hedging your portfolio. That is a slippery slope where someone says, “what’s the harm in having a little more ‘put to use’ to generate extra return.” On the other hand, every 1 point drop in the S&P 500 costs you $250, and you are likely to get frequent margin calls with the future, so you definitely need to have some cash around for that.
What happens next is that people say “well, we have all this cash undeployed, and we know it’s supposed to be there to hedge our portfolios, but what’s the likelihood of all our hedges going bad?” Then people try to maximize the use of that cash by having it hedge several options position simultaneously, which is fine if positions don’t move against you at the same time. But in a true crisis, all correlations go 1, and so you find you don’t ahve enough cash around to hedge your options, and then, all of a sudden, you find that you don’t have enough money around to cover all the options being exercised.
Anyway, I don’t claim to be an options guru, so maybe there’s a little detail I got wrong here or there, but that’s the basic idea. In particular, I don’t remember if you use the strike price or the current price to calculate the total size of the replicating portfolio. I think it’s the strike price because that stays the same over the life of the option, but it might be the current price because as the option goes way into the money, it needs to behave like having pure stock exposure.
I am really appriciated your effort. And hope you can bear with me a lillte bit more.
I dont really understand what you mean after the Black Scholes Merton mode paragraph.
From what I understanding, please correct me if I am wrong, you said we only need $5600 to have a delta neutral hedged, if so how many future contract CME S&P 500 Futures I need?
From what I think, we have the call option delta =.19 so with 10000 option, so we have a adjust delta of 1900. And as I mentioned in my last post, delta for future contract is 1.
So the number of future of contract CME S&P 500 Futures I need is 1900.
I think the challenge is that you don’t really understand what it means to have a replicating portfolio. This is something that comes up on the L2 and L3 exams, so maybe you just aren’t there yet. But if you are trying to answer the question you posed, then someone - at least - is expecting you to know this.
Basically an option’s replicating portfolio is a portfolio that consists of (delta%) in stock and (1-delta)% in cash. If you hold that portfolio for a short time, its change in price will be the same as holding the option for the same time. The only thing to remember is that as the stock price changes, the delta changes too, so you have to continually readjust the portfolio for that replicating effect to remain calibrated to the option’s current price. To some extent, the premium charged by an options seller might be thought of as the price they extract for managing that replicating portfolio for you - changing the amount in cash vs stock as delta changes - so you don’t have to.
Futures are actually simpler: a portfolio that is 5% allocated to the futures contract and 95% allocated to cash is equivalent to 100% allocated to the stock (except for the dividends due before expiration, which you only get if you actually hold the stock). The 5% number comes from the margin requirement. If your broker requires 10% margin for a futures contract, then you will need 10% in futures margin and 90% in cash to get the same replicating effect.
The answer I gave you is a combination of these two ideas. You create the replicating portfolio for the option, and then use a future to replicate the % of the option’s portfolio that needs to be in the underlying stock.
Thanks for the love bchad, but my skills are more in stats and portfolio management than options. At best, I can say that I took a class at Courant in them and have read some books. I recommend the OP picks up Wilmott Introduces Quantitative Finance. It is by far the clearest explanation of options pricing I have ever seen.
I don’t think I had any questions like this when I took the CFA exam, but a few things to note, without getting into too much detail. The call is reasonably out of the money, based on the value of the S&P 500 on March 31st, 2011. Increasing the strike by 1% or 10% is not going to make the call go from out of the money to in the money. Based on the relationship between the stock price and delta, this suggests that the delta will be some amount below 0.5 in the base case and a larger number, but still less than 0.5 in the 1% or 10% cases. This would mean that an option delta hedged by shorting the futures would require a greater dollar value of futures as the strike increases.
As to whether it is a “good portfolio”. Risk neutral pricing means that under the risk neutral measure you would not expect to make any money from a delta-hedged portfolio. However, that may not be the case with real world probabilities. Whether the portfolio looks good depends on whether you think volatility was expensive or cheap at that point in time.
Thank for the help and I know that I arent there yet. Because I need to understand problem before the due time, that’s why I am looking for help. I will be sure to come back and study it. And because the question that I have is how many future contract CME S&P500 Futures traded on Chicago Mercantile Exchange do I need to build a delta hedged portfolio. That’s why I want to stick close to how many future contract we actually need.
Thank for your help and I just wonder do you think my calculation is correct? I need 1900 future contract to build a delta hedged portfolio?
And for this one, do you think I calculate its right?
σ= 0,149 (Annualizing gives 0,0094*SQRT252 = 0,149).
Thanks
I’m not sure where that number is from. I didn’t read bchad’s posts that carefully, but he seemed to go through it in a lot of detail. I would re-read what he said a few times until you understand it better. Or pick up books.
You’re looking at the actual SP500 index value on 03/31/11, the actual futures settled at 1321, not the 1325.83 you’re using. This means “10 000 call options on S&P500” is an arbitrary amount, you need to figure out the contract specifications of whatever you want to trade.
Yes, to get delta neutral you’d have to buy/sell ~19% of the notional exposure. You can find all the contract and sizing information you need here: http://www.cmegroup.com/trading/equity-index/us-index/sandp-500_contractSpecs_options.html
There are different contracts you can use of various sizing, but yes if you held 10K long SPX call options you’d need to sell 1900 SPX futures contracts to get delta neutral - you would also be a baller, and hoping for a rise in volatility (or the inverse if you sold the options/long futures).
I went through bchad’s post but I didnt mention about that before so he didnt talk about it in his post. And sure I will come re-read and pick up the something this weekend to learn more about this topics. Thank for you help.
Thank for your respond. So if I use this future contract, how many contract do you think I should get
http://www.cmegroup.com/trading/equity-index/us-index/euro-emini-sp500_contract_specifications.html
I think of 1900/50 = 38 contracts.
And also related to delta option, do you think that I computed it right? do you know how can I find σ the volatility of returns of the underlying assset.
And as for the change in the strike price after a month how should I adjust it if it move up/down 1 to 10%
Your help is much appriciated