Delta Neutral Dynamic Hedging

Delta Neutral Dynamic Hedging


October 23, 2009
This 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 this article will include some concrete examples. If you would like to contact the author, send a message to q.boiler@QuantP rinciple.com  . If there are any errors, typos, or defects with this article please 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 a function of the underlying stock price. Hedging is to reduce or ’hedge’ away risk. When one talks 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 the underlying stock’s price. For small changes in the stock’s price this can be accomplished by going long or short on the underlying. The choice of going long or short on the underlying will depend if you are long or short on a call or put option. Delta hedging is more complex for large changes in the underlying stocks price. Finally, a discussion of risks associated with assembling these various portfolios will be presented. Future revisions to this document will contain a discussion 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 the underlying stock’s price. Delta is the sensitivity of an option’s value to the stock price while all other variables remain unchanged. Because the option pricing equations are partial differential equations, Delta is mathematically represented as:

δ = ∂V-,
    ∂S

this can be read as the partial derivative of the options value with respect to changes in the underlying stock’s price. Delta (    ∂V-
δ = ∂S  ) must be between 0 and 1 for call options and -1 to 0 for put 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 down the value of the option moves down. In other words, for call options the value of the option will move in the same direction as the value of the stock. Now, intuitively thinking about a put option which gives one the right to sell a stock at a predetermined price, as the price of the stock goes up the value of the option goes down (e.g. if the put option’s strike price is $100, and the stocks price 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 long call options). A long options is a contract that gives the holder the right to buy or sell an underlying stock at a predetermined price on or before a predetermined day. Shorting a call option means to sell a contract giving another party the right to buy a stock at a predetermined price 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 Solar Corpration).

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 the tactical aspect. However, just pick up any good book about portfolio management and you will quickly 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% above and below the open. What could you say about the value of δ (Delta) calculated above? Let’s say that very few options contracts traded hands for the strike price and expirey of interested, what would 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 they affect 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 to generate a table containing the value of the option, and the value of the underlying stock. Then use this table to generate a scatter chart plotting the option’s value on one axis and the underlying stock’s price on the other. The slope of the line would represent Delta (δ). The interesting thing about doing this activity is that you could start to see how Delta (δ) changes as the 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 price will change. This concept is complex and is really the second derrivative of option’s value with respect to stock price or mathematically it is     ∂δ   ∂2V-
γ = ∂S = ∂S2   .

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 δ could be quite large.

A. 1.3 Options are less liquid than stocks. Option’s value is based on more than just the underlying stock price, it is based on the stock’s volitility time to expirey, price of the underlying, 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 derivative investment strategies come together. Here we will look at a very simple strategy in which the investor will buy a call option, and short the actual underlying stock. This is just about the simplest strategy available.

The investor will hold a call option on an underlying stock; that means that the investor will hold the 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 called shorting. Before going any further with this example, lets take a moment to think about this position. By purchasing the call option the investor is betting on the stock to go up in price. Call options increase in value as the underlying stock increases in price. On the contrary by selling short stocks of the underlying the investor is betting on the stock to go down in price. Now, what does the δ neutral aspect of a Delta Neutral strategy mean? This is the somewhat subtle point of a δ neutral strategy. At this point the astute student of quantitative finance would also pose the question how many shared of the underlying should one short, and how many options contracts should 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 I choose? These are actually quite complex questions to answer, and in some cases require experince and vision not just mathematical analysis and simplification. Since these questions are not 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 the following behavior:

  1. δ = ∂∂VS-= 0.5  , or the options value changes 50 cents for every dollar change in the underlying stock’s price.
  2.     ∂δ   ∂2V-
γ = ∂S = ∂S2 = .02  , or δ will change by 2 cents for every dollar change in the price of the underlying.
  3. S, the stock price at T0 = $100
  4. 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 δ neutral 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 to the short sell bet that is made. Being δ neutral means that for every dollar increase in the options value due to change in the underlying stock price, there will be an exactly equal and offsetting decrease in the value of the short sell bet. If the stock is moving down in price, then for every dollar 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, the stock is absolutely liquid in nature and therefore the value of the short sale is quantitatively determined based on the movements of the underlying stock. However, the value of the option is based on a large number of fuzzy variables which make it take on a model value. The market value 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 or 2 options for every stock which is shorted. For the sake of mathematics, decimal shares can be used. In a real world example these normalized values would have to translated into values that make 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 us to do the math without thinking too much. One thing to keep in mind as we are doing this differential calculus exersize, is that the real world is a more discrete environment. Firstly, we will need to know what the Mark-To-Model value of the options are worth.

                 ∫ P
Vmodel(P) = Vinit + Pnitδ∂P
                   i

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

∂δ-= γ
∂P

which leads to:

            ∫ P
δ(P) = δinit +     γ∂P
              Pinit

therefore:

          ∫ P          ∫ P  ∫ P    2
V = Vinit + P   δinitP +  P    P   γ∂ P
            init          init  init

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

V = Vinit + δinit(Pinit - P )+ γ1(Pinit - P )2
                          2

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

                    1    2
V = Vinit + δinitΔP + γ2ΔP

In addition to being able to determine the value of the option at each price movement, we need to determined the value of the short position and we need to determine the number of shares to buy or 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 and buy low. So, keeping that in mind, we will do just exactly what that fund manager indicated we should. 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 the portfolio neutral to movements in the stock price. Let’s assume that the stock has the following movements:

P1100
P2 99
P3 98
P4 97
P5 96
P6 97
P7 98
P8 99
P9 100
P10101

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





cost valueValue - 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.
                     1   2                      1     2
V2 = Vinit + δinitΔP + γ2ΔP   = 10.00+ .5(- 1)+ - .02(2)(- 1) = 9.49
Shares Sort Will be 0.48





cost valueValue - Cost




options 10.00 9.49 -




short -50.00-47.52 -




Cash IN 1.98 0.0 -




Total -38.02-38.03 -0.01




ΔP3 = -1 ACTION: Buy .02 shares.
                     1   2                      1     2
V3 = Vinit + δinitΔP + γ2ΔP   = 10.00+ .5(- 2)+ - .02(2)(- 2) = 8.96
Shares Sort Will be 0.46





cost valueValue - Cost




options 10.00 8.96 -




short -50.00-45.08 -




Cash IN 3.94 0.0 -




Total -36.06-36.12 -0.06




ΔP4 = -1 ACTION: Buy .02 shares.
V4 = Vinit + δinitΔP + γ12ΔP 2 = 10.00+ .5(- 3)+ - .02(12)(- 3)2 = 8.41
Shares Sort Will be 0.44





cost valueValue - Cost




options 10.00 8.41 -




short -50.00-42.68 -




Cash IN 5.88 0.0 -




Total -34.12-34.27 -0.15




ΔP5 = -1 ACTION: Buy .02 shares.
V5 = Vinit + δinitΔP + γ12ΔP 2 = 10.00+ .5(- 4)+ - .02(12)(- 4)2 = 7.84
Shares Sort Will be 0.42





cost valueValue - Cost




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.
                     1   2                      1     2
V6 = Vinit + δinitΔP + γ2ΔP   = 10.00+ .5(- 3)+ - .02(2)(- 3) = 8.41
Shares Sort Will be 0.44





cost valueValue - Cost




options 10.00 8.41 -




short -50.00-42.68 -




Cash IN 5.86 0.0 -




Total -34.14-34.27 -0.13




ΔP7 = 1 ACTION: Sell .02 shares.
                     1   2                      1     2
V7 = Vinit + δinitΔP + γ2ΔP   = 10.00+ .5(- 2)+ - .02(2)(- 2) = 8.41
Shares Sort Will be 0.46





cost valueValue - Cost




options 10.00 8.96 -




short -50.00-45.08 -




Cash IN 3.90 0.0 -




Total -36.10-36.12 -0.02




ΔP8 = 1 ACTION: Sell .02 shares.
V8 = Vinit + δinitΔP + γ12ΔP 2 = 10.00+ .5(- 1)+ - .02(12)(- 1)2 = 9.49
Shares Sort Will be 0.48





cost valueValue - Cost




options 10.00 9.49 -




short -50.00-47.52 -




Cash IN 1.92 0.0 -




Total -38.08-38.03 0.05




ΔP9 = 1  ACTION: Sell .02 shares.
V9 = Vinit + δinitΔP + γ12ΔP 2 = 10.00+ .5(0) + - .02(12)(0)2 = 10.0
Shares Sort Will be 0.50





cost valueValue - Cost




options 10.00 10.0 -




short -50.00-50.00 -




Cash IN -.08 0.0 -




Total -40.08-40.00 0.08




ΔP10 = 1 ACTION: Sell .02 shares.
V10 = Vinit + δinitΔP + γ1ΔP 2 = 10.00+ .5(1)+ .02(1)(1)2 = 10.51
                      2                       2
Shares Sort Will be 0.52





cost valueValue - Cost




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 not fluctuate around a given price, but instead trends up or down. Delta δ neutral strategies tend 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 volitiliy can 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 the counter party which sold you the option to be solvent and able to service their obligations.
  4. There are other risks, however they will be discussed in a later revision to this article.

References

[1]   De Weert F. An Introduction to Options Trading. John Wiley & Son’s Chichester, UK, 2006.

[2]   Wilmott P. Paul Wilmott Introduces Quantitative Finance. John Wiley & Son’s Chichester, UK, 2007.

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