Delta Neutral Dynamic Hedging

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Delta Neutral Dynamic Hedging

October 23, 2009This article discusses Delta (δ) neutral hedging, and Delta (δ) neutral dynamic hedging. The

article disusses the theory, the mathematics and a theoretical example. Future revisions to thisarticle will include some concrete examples. If you would like to contact the author, send amessage to . If there are any errors, typos, or defects with this articleplease make the effort to contact the author.

1 Delta Neutral Hedging

Having a portfolio that contains an option on an underlying stock contains some risk.Specifically, the option will have risk that it’s value will change as a function of time, and as afunction of the underlying stock price. Hedging is to reduce or ’hedge’ away risk. When onetalks of Delta Hedging, what one is saying is that they own an option on an underlying (e.g.

options on a stock), and would like to reduce the risk to the portfolio due to changes in theunderlying stock’s price. For small changes in the stock’s price this can be accomplished bygoing long or short on the underlying. The choice of going long or short on the underlying willdepend if you are long or short on a call or put option. Delta hedging is more complex for largechanges in the underlying stocks price. Finally, a discussion of risks associated with assemblingthese various portfolios will be presented. Future revisions to this document will contain adiscussion of Value at Risk, Catostrophic Failure, Mark to Market and Mark to Model aspects of risk.

1.1 Definition

Delta neutral hedging is defined as keeping a portfolio’s value neutral to small changes in theunderlying stock’s price. Delta is the sensitivity of an option’s value to the stock price while allother variables remain unchanged. Because the option pricing equations are partial differentialequations, Delta is mathematically represented as:

this can be read as the partial derivative of the options value with respect to changes in the

underlying stock’s price. Delta ( ) must be between 0 and 1 for call options and -1 to 0 forput options. This previous statement can be though of quite intuitively as: a call option gives you

the right to buy a stock at a predetermined price (called the strike price) therefore, as the price of the stock moves up the value of the option moves up and as the value of the stock moves downthe value of the option moves down. In other words, for call options the value of the option willmove in the same direction as the value of the stock. Now, intuitively thinking about a put optionwhich gives one the right to sell a stock at a predetermined price, as the price of the stock goesup the value of the option goes down (e.g. if the put option’s strike price is $100, and the stocksprice goes from $90 to $95 on the day of expirey then the put option’s value dropped from $10 to$5). In other words, the put options price will move in the opposite direction of the stock price.



NOTE: all the options discussed in this section were long options (i.e. long put options or longcall options). A long options is a contract that gives the holder the right to buy or sell anunderlying stock at a predetermined price on or before a predetermined day. Shorting a calloption means to sell a contract giving another party the right to buy a stock at a predeterminedprice on or before a predetermined day. There are caveats related to the diferences between

American options, European options and other more exotic options.

Example 1

An options trader is looking to make some money, and decides to look at FSLR (First SolarCorpration).

Selecting a stock to add to a portfolio is a bit like selecting your next move on the chess board.You have to look at both tactical as well as stretegic details. This article is covering only thetactical aspect. However, just pick up any good book about portfolio management and you willquickly see that the strategic decisions are every bit as important as the tactical decisions.

Q. 1.1 How would the trader calculate the value of δ (Delta)?

Q. 1.2 Let’s say the stock’s price is volitile through out the day, moving more than 10% aboveand below the open. What could you say about the value of δ (Delta) calculated above? Let’s saythat very few options contracts traded hands for the strike price and expirey of interested, whatwould that say about the error associated with the traders determination of δ.

Q. 1.3 What are all the other variables that could affect the option’s price, and how could theyaffect the accuracy and correctness of the δ calculation above?

A. 1.1 There are many techniques to calculate this. However, a simple technique would be togenerate a table containing the value of the option, and the value of the underlying stock. Thenuse this table to generate a scatter chart plotting the option’s value on one axis and theunderlying stock’s price on the other. The slope of the line would represent Delta (δ). Theinteresting thing about doing this activity is that you could start to see how Delta (δ) changes asthe price changes. Delta (δ) is not constant, but is a function of the underlying’s stock price.Therefore as the stock’s price changes the sensitivity of the option’s value to the stock’s pricewill change. This concept is complex and is really the second derrivative of option’s value with

respect to stock price or mathematically it is .

A. 1.2 δ may not be acurately determined because if few or no options actually trade hands then

there is effectively no market for the options themselves. Therefore, the error in the value of δcould be quite large.

A. 1.3 Options are less liquid than stocks. Option’s value is based on more than just theunderlying stock price, it is based on the stock’s volitility time to expirey, price of theunderlying, and possibly other factors such as liquidity and open interest.

1.2 A Simple Delta Neutral Strategy



Assembling a strategy and portfolio is where the theory and application of hedge or derivativeinvestment strategies come together. Here we will look at a very simple strategy in which theinvestor will buy a call option, and short the actual underlying stock. This is just about thesimplest strategy available.

The investor will hold a call option on an underlying stock; that means that the investor will holdthe rights to purchace shares of stock on or before a predetermined day at a predetermined price.The investor will also sell shares of the underlying stock that he does not own. This is calledshorting. Before going any further with this example, lets take a moment to think about thisposition. By purchasing the call option the investor is betting on the stock to go up in price. Calloptions increase in value as the underlying stock increases in price. On the contrary by sellingshort stocks of the underlying the investor is betting on the stock to go down in price. Now, whatdoes the δ neutral aspect of a Delta Neutral strategy mean? This is the somewhat subtle point of aδ neutral strategy. At this point the astute student of quantitative finance would also pose thequestion how many shared of the underlying should one short, and how many options contractsshould be bought? The very astute student of quantitative finance would further pose the

question of how far out should I purchace an options contract and what stike price should Ichoose? These are actually quite complex questions to answer, and in some cases requireexperince and vision not just mathematical analysis and simplification. Since these questions arenot being answered presently in this section, they will be presented in a list for future reference.Questions to be answered for a sucssessful delta neutral hedging strategy:

1. What does the δ neutral aspect of delta neutral hedging mean?2. How many shares of the underlying should one short?3. How many options should be bought?4. How far out should I purchase the options expirey for?5. What strike price should I choose for the options?6. How often should I re- balance the portfolio to make it δ neutral?

Taking a completely contrived example let’s say that an option and it’s underlying exhibit thefollowing behavior:

1. , or the options value changes 50 cents for every dollar change in theunderlying stock’s price.

2. , or δ will change by 2 cents for every dollar change in the price of theunderlying.

3. S, the stock price at T0 = $1004. V, the option price at T0 = $10

Assumptions:

The option’s expirey is a long way’s out. The underlying stock has a constant and predictable volatility. γ the sensitivity of δ to the stock price is a constant and will not change due to either the

passage of time, or the fluctuation of the underlying stock price.



The options value will only change do to δ based changes. In other words the passage of time, change in volatility, etc will not affect the value of the option.

Therefore, it is time to assemble the portfolio. Answering question 1 from above, what does the δneutral aspect of delta neutral hedging mean? An observation that should be clear by now is that

the value of the options purchased will move in the oposite direction to the profit or loss due tothe short sell bet that is made. Being δ neutral means that for every dollar increase in the optionsvalue due to change in the underlying stock price, there will be an exactly equal and offsettingdecrease in the value of the short sell bet. If the stock is moving down in price, then for everydollar decrease in the options value, the short sell bet will increase by an equal amount.

There is an important caveat to note at this point. The value of the short sale is very real, thestock is absolutely liquid in nature and therefore the value of the short sale is quantitativelydetermined based on the movements of the underlying stock. However, the value of the option isbased on a large number of fuzzy variables which make it take on a model value. The marketvalue of the option (i.e. the value of the option if it were executed) would be 0, due to the fact

that δ neutral hedging tends to start with out-of-the-money options.

Therefore, the initial portfolio will start with options contracts and short sells in a ratio of 2 : 1 or2 options for every stock which is shorted. For the sake of mathematics, decimal shares can beused. In a real world example these normalized values would have to translated into values thatmake much more sense once balancing frequency and trading transaction costs are factored in.

Prior to starting the cash flow analysis, we will derive some simple equations that will allows usto do the math without thinking too much. One thing to keep in mind as we are doing thisdifferential calculus exersize, is that the real world is a more discrete environment. Firstly, wewill need to know what the Mark-To-Model value of the options are worth.

Remembering that δ(P) and knowing that γ represents the sensitivity of δ to price, We can makethe very academic sounding statememt that we can assume that γ is constant.

which leads to:

therefore:



Assuming that γ is a constant, allows for γ to be pulled out of the integral and reduces theequation to:

This equation could be simplified a bit by defining Pinit - P = ΔP, which would give:

In addition to being able to determine the value of the option at each price movement, we need todetermined the value of the short position and we need to determine the number of shares to buyor sell. Finally, we will need to come up with a nice tabular view to encapsulate each transaction.As a famous mutual fund manager once said, the definition of a good investor is to sell high andbuy low. So, keeping that in mind, we will do just exactly what that fund manager indicated weshould. Every time the underlying stock goes down a buck, we will buy, and every time the

underlying stock goes up we will sell. The amount that we buy or sell will be enough to keep theportfolio neutral to movements in the stock price. Let’s assume that the stock has the followingmovements:

P1 100P2 99P3 98P4 97P5 96P6 97P7 98

P8 99P9 100P10 101

ΔP1 = N∕A ACTION: Initial Entry buy 1 Option Sell 0.5 shares short.

cost value Value - Cost

options 10.00 10.00 0.00

short -50.00 -50.00 0.00

cash in 0.00 0.00 0.00

Total -40.00 -40.00 0.00

ΔP2 = -1 ACTION: Buy .02 shares.



Shares Sort Will be 0.48


options 10.00 9.49 -short -50.00 -47.52 -

Cash IN 1.98 0.0 -

Total -38.02 -38.03 -0.01




options 10.00 8.96 -

short -50.00 -45.08 -

Cash IN 3.94 0.0 -

Total -36.06 -36.12 -0.06




options 10.00 8.41 -

short -50.00 -42.68 -

Cash IN 5.88 0.0 -

Total -34.12 -34.27 -0.15






options 10.00 7.84 -

short -50.00 -40.32 -

Cash IN 7.80 0.0 -

Total -32.20 -32.48 -0.28

ΔP6 = 1 ACTION: Sell .02 shares.



options 10.00 8.41 -

short -50.00 -42.68 -

Cash IN 5.86 0.0 -

Total -34.14 -34.27 -0.13




options 10.00 8.96 -

short -50.00 -45.08 -

Cash IN 3.90 0.0 -

Total -36.10 -36.12 -0.02




options 10.00 9.49 -

short -50.00 -47.52 -

Cash IN 1.92 0.0 -



Total -38.08 -38.03 0.05

ACTION: Sell .02 shares.



options 10.00 10.0 -

short -50.00 -50.00 -

Cash IN -.08 0.0 -

Total -40.08 -40.00 0.08




options 10.00 10.51 -

short -50.00 -52.52 -

Cash IN -2.10 0.0 -

Total -42.10 -42.01 0.09

1.3 Risks

Asembling a portfolio as the one above has some risks. These risks will be discussed briefly.

1. The underlying stock move strongly in one direction. In otherwords the stock does notfluctuate around a given price, but instead trends up or down. Delta δ neutral strategiestend to work best if the stock moves equaly up and down in a Markov like process.

2. The volatility suddenly changes (or really the ’Implied Volatility’ changes) δ neutral

strategies are not typically protecting against this type of change. However, the volitiliycan be a much more significant term in the value of an option than δ.3. Counter party risk. You will be buying an option from a counter party. If, at expirey, the

option is ’in the money’ you will need to exercise the option. This will require thecounter party which sold you the option to be solvent and able to service theirobligations.

4. There are other risks, however they will be discussed in a later revision to this article.



References

Delta Neutral Dynamic Hedging

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