Hedging using options...

Corey said: "If I am long, or short, an equity, is there a way to calculate the optimal number of options to buy (puts or calls), based on time, to perform the optimal hedge? Obviously it is a loaded question...if there were a way, it would be a more common practice. But assume you could represent the movement of a stock as the function F. If we don't need the hedge, we allow our contracts to expire worthless -- and write-off the loss as part of the hedging strategy. But, when the option is IN THE MONEY, it moves exponentially with respect to F (where the exponent is > 1). So if we could somehow figure out the probability of the option being executed (presumably by the implicit volatility of the equity) -- you could figure out how much money to partition for the hedge. I am probably talking out of my ass ... as well as about the basic issues that option players deal with ... but I figured I might as well try to educate myself..."

AnonBroker said: "I think Hull's book covers this. If not, there are MANY papers published on how to find an optimal hedge."

lil dickie said: "For sure this is a decent strategy. Of course the technical way to do it eludes me at this hour."

Corey said: "The majority of models don't deal with 'optimum' but rather simply with methods of hedging, such as covered calls and puts, and combinations to create collars. These are all basic methods, and are sub-average if you are talking about absolute optimal hedging strategy based on risk and return. Of course, the question is loaded. Risk is impossible to measure exactly, and return is just as elusive -- both can only be guessed -- and unless they both come true, the hedging strategy will be sub-optimal. Considering most options seemed to be priced by the Black-Scholes model, you could figure out a stocks 'implied' volatility based on the current market price for an option at a given strike date. Of course, this 'implied' volatility would change daily -- so unless you feel like adjusting your hedge daily (which only increases costs), you will always be sub-optimal. I would expect that the option's price would move exponentially with respect to momentum (the second derivative of a smoothed price curve) once it is in-the-money. However, trying to figure out how many contracts to purchase, when to re-balance, et cetera, seems to be more of an art than a science, given that there would be so many variables involved in trying to determine optimal value, risk, and potential return. All in all, it seems rather futile..."

SnowPro said: "Huh? Corey, you seem to make it more complicated than it has to be. What is your objective? Hedging completely to collar and capture dividend, appreciating collars to capture mispricings? Delta neutral with appropriate adjustments, or Married put/Dynamic hedging strategies. What type of risk do you want to hedge? Lots of ways to skin this monkey. For example I put together a number of perfect calendar hedges this summer. But yes, one can get wishboned when some hedges don't work out. LTCM. j"

Corey said: "The optimal hedge would be the amount, based on implied volatility, that maximizes the return of an investment while minimizing potential risk. Theoretically you can hedge anywhere between 0-100%. However, if a stock has low volatility and a strong uptrend, it seems sub-optimal to hedge 100% -- you would only be wasting money on contracts. That is the sort of thing I was trying to get at."

SnowPro said: "Ok, that works for me. The best book I have read on the subject is: Stock, Options, and Collars by J.L. Lord I have put together a spread sheet for Dynamic Hedging. It takes into account: Stock Price, Historical Vol, Days trading until Expiration, and Put Strike to Purchase. If you want the spread sheet, drop me PM with your email and I will send it to you. It is not perfect, but it is pretty darn good. The above book will be best teach you how to do this type of hedging strategy. I also built a straddle adjuster to find the best strikes to hedge with and get your best $/delta. trade well, J"

AJJ said: "Snowpro, I am very interested in your spreadsheet. IT will give me the opportunity to test my straddleplanner option strategy calculator. Especially in calculating 5 year duration conversions and reversals with about 10 discrete dividends. Regards, AJJ"

TheOptionClub said: "[quote=Corey;57152]If I am long, or short, an equity, is there a way to calculate the optimal number of options to buy (puts or calls), based on time, to perform the optimal hedge? Obviously it is a loaded question...if there were a way, it would be a more common practice. But assume you could represent the movement of a stock as the function F. If we don't need the hedge, we allow our contracts to expire worthless -- and write-off the loss as part of the hedging strategy. But, when the option is IN THE MONEY, it moves exponentially with respect to F (where the exponent is > 1). So if we could somehow figure out the probability of the option being executed (presumably by the implicit volatility of the equity) -- you could figure out how much money to partition for the hedge. I am probably talking out of my ass ... as well as about the basic issues that option players deal with ... but I figured I might as well try to educate myself...[/quote] Wow. This is getting complicated.... I don't know what an "optimal" hedge is, but why not just cut to the heart of things and make this simple. [B]How Many Options To Buy?[/B] You're long, or short, an equity. Fine. Your delta is +/- the number of shares you're long or short. Let's say you're long 50 shares and you want to neutralize your directional (i.e., downside) risk. Buy an at-the-money put option. The put will have a delta of .45 to .50 and allows you to sell 100 shares of stock at the current market price. Your risk is limited to current market price, less the premium paid for the put option. If your stock trades lower, your put will be in-the-money. Sell the put to offset the loss on your stock. You can then replace the put option with another at-the-money put. If the stock trades higher, your loss on the put option will be offset by the price increase in the 50 shares of stock. Oh, but we only need protection for 50 shares and we don't want to pay to insure 100 shares. Fine. You can't buy half a put, though. What we can do is sell the 50 shares, then buy the call option with a .50 delta. If the market tanks, your risk is still limited to the premium and if it rallies you can exercise the call, sell off 50 shares, and you're back in your long position. Alternatively, you sell the call option for a profit and buy the 50 shares. This later approach is better if you still have time value left in the contract. At-the-money too expensive? You can move further out-of-the-money, buy a cheaper option, but you'll lose some of that protection. That's true whether you're buying a put to protect the stock or replacing the stock with a call option. The reverse applies for a short position. You can buy a call option to negate the risk if you're short stock. If the stock trades higher you can 1.) sell the call to offset your loss on the 50 short shares; or 2.) exercise the call to close your short position and be left long 50 shares. Alternatively, you can replace the short stock with a long put option. Your risk is limited to the premium paid for the put option. If you want to approximate 50 short shares, buy the put with a -.50 delta. [B]Probability Of Option Expiring Worthless / In-The-Money[/B] Now, you also talk about establishing a probability for your option expiring in-the-money. A real easy way to approximate that is to simply look at delta. The delta of the option can be used to approximate the probability an option will expire in-the-money. For example, an option with a .45 delta has a 45% chance of expiring in-the-money. Conversely, if the delta is .45, you then know that there is approximately a 65% chance the option will expire out-of-the-money. [B]What's Optimal?[/B] So, what's optimal? I say that depends upon your gut. You have an equity position and you need to feel good about it so that you can get some sleep at night. If you're feeling comfortable, you've probably got the balance between protection and cost optimized for your particular need. Or, you just don't understand what you did and you're enjoying the fool's sense of security. On the other hand, if you're queezy, maybe you need to re-think things and move that put to a higher strike or, if you're concerned about the negative theta, you need to reduce the premium outlay."

TheOptionClub said: "[quote=Corey;59156]The optimal hedge would be the amount, based on implied volatility, that maximizes the return of an investment while minimizing potential risk. Theoretically you can hedge anywhere between 0-100%. However, if a stock has low volatility and a strong uptrend, it seems sub-optimal to hedge 100% -- you would only be wasting money on contracts. That is the sort of thing I was trying to get at.[/quote] Ironically, it's when IV is low that investors are most complacent and the market is most likely to experience significant deterioration."

AJJ said: "Hi, Optimal delta hedging with gamma trading Reading all reactions with regard to my previous posting, I think it is best to explain what I mean by working out an example. To better understand my explanation, you can set up, manage and work out the example yourself at straddleplanner. com. (my site). Gamma trading example: Position to set up: -Trading date: 14-Jun-08 -By 1000 stock XYZ @ USD 8.58 -By 25 puts “strike 8.50 exp:19-Dec-08 trading at Impl.vol of 41%. -Interest rate 5% How to enter straddleplannerdotcom: 1. open straddleplanner .com (unfortunately it is not yet “googleable”, thus type in the url) 2. Chose fundname: “XYZ” (click on NEW… and enter XYZ) 3. By 1000 stock XYZ @ USD 8.58 ( click field below XYZ and enter 1000) 4. Set portfolio date to 14 JUN 08 (Click on agenda icon. Portfolio date is the date on which you want to analyse your position. You can change this date the measure the time decay on your portfolio) 5. Enter interest rate 5% (click in box marked interest on add. Click on the date and change it to 19-Dec-08, click on 0.00 and enter 5). 6. Enter strike (click in box marked STRIKES on add. Click on the date and change it to 19-Dec-08, click on the 0.00 under Volatility to enter 41 and click on the 0.00 under Strike to enter 8.50) 7. Enter number of puts (click in column marked long next to 8.50 and enter 25) You have now entered the entire strategy and can start analysing: Press on calculate and all values are shown on screen. -On the left of volatility you will see the call values (Price: 1.14, Delta: 60.54, Gamma: 15.21, Theta:0.31, Rho: 2.09, Vega: 2.37) -On the right of Strike you will see the put values (Long: 25, Price: 0.86, Delta: -40.72, Gamma: 16.04, Theta: 0.21, Rho:-1.70, Vega: 2.37) -On the bottom you will see the totals of the position: (Delta: 18, Gamma: 401, Theta: 5, Rho: -42, Vega: 59.) -Premium: 2161 (Option price*contract size* number of option) -Value: 10741 (Stock * stock price + Premium) (this number is the amount of USD or GBP or EU you need to invest to set up this particular position. If you have the same values on your screen than “Congratulations” . (you might want to save your position now. ) Now what can we do with this? As you can see, the delta (18 long) of your portfolio is almost completely hedged. But what will happen when the stock moves down from 8.58 to 8.02? (click on 8.58 and enter 8.02 and press calculate) Now you will see that all values will have changed. I will not mention all values again, but just concern you with the important ones (for this strategy). First the delta of the position (total delta) is now short 257 and the position value has increased to 10818. You can now re-hedge your postion again by buying 257 stock XYZ @ 8.02. (click on 1000 and enter 1257 and press calculate) You will see that the the total delta is now 0 and your portfolio value is now 12879. This process now continues as long as the stock price keeps on changing. If the stock XYZ for example decreases further to 7.60 you can buy an other 196 stock to re-hedge your position and thereby locking in profits. Important note; in this example I have let the stock price decrease, but the same will happen when the stock price will increase. (in that case you can sell stock to re-hedge). Also important is to decide upfront when to re-hedge your position. For example whenever your total delta position is long/short 250, 500…. Hope this helps you in getting a better understanding of what gamma trading is about and how a tool like straddleplannerdotcom can help you to manage positions. Thanks. If you have any other strategies, you want to analyse or discuss then please let me know. AJJ alexander@"

TheOptionClub said: "AJJ, Nice post on gamma trading. I didn't get that Corey's original post was focused on gamma trading, but more so on the idea of having portfolio of stocks and wanting to hedge against a downturn. I could be wrong, though."