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©David Dubofsky and 19-1 Thomas W. Miller, Jr. Chapter 19 Using Options for Risk Management I. The Greeks II. Riskless Hedging III. Delta Hedging and Delta-Gamma Hedging IV. Position Deltas and Position Gammas V. Time Spreads VI. Caps, Floors, and Collars

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©David Dubofsky and 19-2 Thomas W. Miller, Jr. The Greeks Delta Gamma Theta 1 Vega Rho 1 Theta is often expressed as a negative number

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©David Dubofsky and 19-3 Thomas W. Miller, Jr. We Know that the Value of a Call Option Depends on the Values of: –Stock Price –Time to Expiration –Volatility –Riskless interest rate (generally assumed to be constant) –Dividend yield (generally assumed to be constant) –Strike Price (never changes)

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©David Dubofsky and 19-4 Thomas W. Miller, Jr. Accordingly, as These Factors Change over time: The value of the call option will change over time. This is important for risk management because the composition of a “riskless” hedge must, therefore, change over time.

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©David Dubofsky and 19-5 Thomas W. Miller, Jr. Some Intuition From Using a Taylor-Series Expansion Given value at x 0, we can use derivatives of function to approximate value of function at x: f(x) = f(x 0 ) + f ' (x 0 ) (x-x 0 ) + (1/2) f '' (x 0 )(x-x 0 ) Incorporate curvature effects via Second-order Taylor series expansion of option price around the stock price (quadratic approximation): C(S + dS) = C(S) + ( C/ S) dS + (0.5) ∂ 2 C/∂S 2 ( dS) 2 = C(S) + dS (dS) 2 So, the change in the call price, given a change in S is: C(S + dS) - C(S) = dS (dS) 2 + … dC = dS (dS) 2 + …

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©David Dubofsky and 19-6 Thomas W. Miller, Jr. Delta AKA: Hedge Ratio Delta ( ) is the rate of change of the option price with respect to the underlying. (I.e., How does the option price change as the underlying price changes?) Option price A B Slope = Stock price

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©David Dubofsky and 19-7 Thomas W. Miller, Jr. Gamma Addresses Delta Hedging Errors Caused By Curvature S C Stock price S’S’ Call price C’ C’’

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©David Dubofsky and 19-8 Thomas W. Miller, Jr.

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©David Dubofsky and 19-9 Thomas W. Miller, Jr.

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©David Dubofsky and Thomas W. Miller, Jr. Example: An investment bank has sold (for $300,000) a European call option on 100,000 shares of a non-dividend paying stock: i.e, S 0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks The Black-Scholes value of the option is $240,000. How does the investment bank hedge its risk?

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©David Dubofsky and Thomas W. Miller, Jr. Two Alternatives: “Naked” position: Take no action (other than deep breath and hope for the best). Risk: Stock price is greater than $53 at expiration. Covered position: Buy 100,000 shares today. Risk: Stock price is less than $46 at expiration.

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©David Dubofsky and Thomas W. Miller, Jr. Stop-Loss Strategy Which Involves: Buying 100,000 shares as soon as price reaches $50. Selling 100,000 shares as soon as price falls below $50. NB: This deceptively simple hedging strategy does not work well. Main failure: Bid-Ask Spreads. (Think about “bid-ask bounce”)

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©David Dubofsky and Thomas W. Miller, Jr. Delta Hedging This involves maintaining a “delta neutral” portfolio. The delta of a European call on a non-dividend paying stock is N (d 1 ). The delta of a European call on a stock paying dividends at rate q is N (d 1 )e – qT. The delta of a European put is on a non-dividend paying stock is: [N (d 1 ) – 1]. The delta of a European put on a stock paying dividends at rate q is e – qT [N (d 1 ) – 1].

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©David Dubofsky and Thomas W. Miller, Jr. Delta Hedging, Cont. N.B. The hedge position must be rebalanced “frequently.” (Delta Hedging is a concept born from Black-Scholes, I.e., in continuous time.) Delta hedging a written option involves a “buy high, sell low” trading rule. (I.e., When S increases, so does Delta, and vice versa.) Data from above: S 0 = 49, K = 50, r = 0.05, = 0.20, T = 20 weeks (0.3846) Option written on 100,000 shares, selling price: $300,000 Black-Scholes value of this option: $240,000

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©David Dubofsky and Thomas W. Miller, Jr. A Closer Look at Delta Hedging Important: –Option on 100,000 shares –Volatility is assumed constant. –“Rebalancing” assumed weekly.

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©David Dubofsky and Thomas W. Miller, Jr. Aim of Delta Hedging: Keep Total Wealth Unchanged. Let’s look at the end of Week 9. Value of written option at start: $240,000 (per B-S.) Value of option at week 9 ( =20%): $414,500. Option Position Loss: $(174,500). Cash Position Change, (Measured by Cum. Cost: $2,557,800 – 4,000,500 = $(1,442,700). Value of Shares held: –At start: 49 * 52,200 = $2,557,800 –At end of week 9: 53 * (67, ,300) = $4,171,100. –Increase of: $1,613,300. Net: $1,613,300 – 1,442,700 – 174,500 = $(3,900).

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©David Dubofsky and Thomas W. Miller, Jr. More on Delta Hedging Delta is the (fractional) number of shares required to hedge one call. Positions in the fractional shares and call have opposite signs. For calls, delta lies between 0 and 1. For puts, delta lies between –1 and 0. Strictly speaking, the riskless hedge exists only for small changes in the stock price and over very small time intervals. As time passes and/or the stock price changes, the of the call changes (as measured by gamma). As changes, shares of stock must be bought or sold to maintain the riskless hedge.

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©David Dubofsky and Thomas W. Miller, Jr. Gamma Gamma ( ) is the rate of change of delta ( ) with respect to the price of the underlying asset. This varies with respect to the stock price for a call or put option. ( reaches a maximum at the money). As a rule, we would prefer to have a positive Gamma (more on this later…)

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©David Dubofsky and Thomas W. Miller, Jr. Interpretation of Gamma For a delta neutral portfolio, d dt + ½ dS 2 dd dSdS Negative Gamma dd dSdS Positive Gamma

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©David Dubofsky and Thomas W. Miller, Jr. Position, or Portfolio, Deltas The position delta determines how much a portfolio changes in value if the price of the underlying stock changes by a small amount. The portfolio might consist of several puts and calls on the same stock, with different strikes and expiration dates, and also long and short positions in the stock itself. The delta of one share of stock that is owned equals The delta of a share of stock that is sold short is –1.0. Why?

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©David Dubofsky and Thomas W. Miller, Jr. Calculating a Position Delta Assuming that each option covers one share of stock, the position delta is calculated as a weighted sum of individual deltas. That is, where n i is the number of options of one particular type, or the number of shares of stock. The sign of n i is positive if the options or stock is owned, and negative if the options have been written or the stock sold short. The delta of the i th option or stock is given by i.

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©David Dubofsky and Thomas W. Miller, Jr. An Example of a Position Delta Suppose you have positions in the following assets: Position and number of options or shares n i___ delta/unit Total Deltas=n i i long 300 shares long 40 puts short 150 calls long 62 calls p = This assumes that each option is on one share of stock

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©David Dubofsky and Thomas W. Miller, Jr. In this example, the position delta of is positive. This means that if the stock price were to increase by one dollar, the value of this portfolio would rise by $ If the stock price were to decline by one dollar, then the value of this portfolio would fall by $ (Remember that this example assumes that the underlying asset of an option is one share of stock.) Knowing your portfolio's position delta is as essential to intelligent option trading as knowing the profit diagrams of your portfolio.

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©David Dubofsky and Thomas W. Miller, Jr. Position Gammas Similar to position deltas, a position gamma is the weighted sum of the gammas of the elements of the portfolio: where n i is the number of options of one particular type. The sign of n i is positive if the option is owned, and negative if the option has been written. The gamma of the i th option is given by i. Note that the gamma of a stock is zero, because the delta of a stock is always equal to one.

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©David Dubofsky and Thomas W. Miller, Jr. Ideally, a delta neutral hedger would like to maintain a portfolio with a positive gamma. Recall that the gamma of a call or a put are given by the same formula and that gammas cannot be negative. However, a portfolio can have a positive or negative gamma. It is important to note that a portfolio with a positive gamma increases in value if the underlying stock value changes. Further, a portfolio with a negative gamma decreases in value if the underlying stock value changes.

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©David Dubofsky and Thomas W. Miller, Jr. Time Spreads, I. Time spreads are also sometimes called calendar spreads or horizontal spreads. Time spreads are identified by the ratio of options written and purchased. To create a 1:1 time spread, the investor buys one option and sells one option. –The two options must either be both calls, or both puts, and share the same strike price on the same underlying asset. –However, unlike most of the option portfolios discussed previously, the time spread portfolio of options has options with different maturity dates.

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©David Dubofsky and Thomas W. Miller, Jr. Time Spreads, II. The maximum profit on a time spread occurs when the stock price equals the strike price on the expiration date of the nearby call. The short-term written option expires worthless, and the longer term option can be sold with some time value remaining. –An important risk to be kept in mind is that if the written, short-term option is in-the-money as its expiration date nears, it could be exercised early. –Thus, in-the-money time spreads using American options will often have to be closed out early in order to preclude this possibility.

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©David Dubofsky and Thomas W. Miller, Jr. Neutral Time Spreads A neutral time spread is a strategy designed to capitalize on two options that are somehow “mis- priced” relative to each other. The two options differ only in their times to expiration. Usually the short-term option is written and the longer option is purchased. The number of each option traded is designed to create a delta neutral portfolio.

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©David Dubofsky and Thomas W. Miller, Jr. Neutral Time Spreads, Example Suppose the implied standard deviations of the two options were (for a September call), and (for a December call.) Thus, assuming that the BSOPM is the proper pricing formula, the short-term call is overvalued relative to the December call. Assume that the "true" volatility is 0.40/year. Then, the hedge ratios of the two options are = for the September call, and = for the December call. A delta neutral portfolio of these two options is: p = 0 = n 1 1 + n 2 2

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©David Dubofsky and Thomas W. Miller, Jr. Rearranging: Using the above deltas, and denoting the September and December calls as assets 1 and 2, respectively: Equation (19-6) says that to create a delta neutral position: 1.22 September calls should be sold for each December call purchased. This allows the investor to arbitrage the relative mispricing of the two options, and not speculate on price movements.

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©David Dubofsky and Thomas W. Miller, Jr. Caveats Of course, there are real risks and costs that must be included. For example, we have ignored transactions costs. –In practice, the position will have to be closed out early if the options are in the money as September 21 nears, because the September call could be exercised early. –If they are exercised, the trader will bear additional transactions costs of having to purchase the stock and making delivery. In addition, there are rebalancing costs of trading options to maintain delta neutrality. For this reason, neutral hedgers prefer low gamma positions, if possible. We assumed there are no ex-dividend dates between August 1 and September 21. Dividends would affect the computed deltas and also introduce early exercise risk. We assumed constant interest rates and constant volatility. These also affect the computed deltas and indeed, may be the source of the apparent mispricing.

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©David Dubofsky and Thomas W. Miller, Jr. Caps A cap is a purchased call option in which the underlying asset is an interest rate. Note that a call on an interest rate is equivalent to a put on a debt instrument. The interest rate cap is actually established when the interest rate call is combined with an existing floating rate loan. The call caps the interest that can be paid on the floating rate liability.

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©David Dubofsky and Thomas W. Miller, Jr. Buying a cap provides insurance against an interest rate increase that hurts the firm ΔVΔV Δr Δr long call (at the money) A rise in interest rates increases interest expense.

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©David Dubofsky and Thomas W. Miller, Jr. Buying a cap provides insurance against an interest rate increase that hurts the firm VV Δ rΔ r long call resulting exposure The firm benefits from a decline in r, but has bought insurance to protect itself from a rise in r.

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©David Dubofsky and Thomas W. Miller, Jr. Caps An interest rate call is purchased along with a variable rate loan to create the cap. The borrower then pays: if r < r c r c if r > r c The interest rate call payments themselves are: (r - r c ) (notional principal) (d/360) if r > r c 0 if r < r c where: r c is the cap rate (strike price), and d is the # of days in the settlement period.

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©David Dubofsky and Thomas W. Miller, Jr. Effective Interest Cost under a Cap r rcrc Interest Expense

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©David Dubofsky and Thomas W. Miller, Jr. An Example of a “Caplet” Suppose the cap rate is 7.00%. If: – the reference interest rate (3-month LIBOR) is 7.90% on the settlement date, –there are 90 days in the settlement period, –and, NP = $15,000,000. Then, the payment is: ( ) ($15,000,000) (90/360) = $33,750 A cap is a portfolio of caplets, each having a different expiration (reset) date.

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©David Dubofsky and Thomas W. Miller, Jr. The Cap Premium When purchased along with a floating rate loan, the cap premium is often amortized, and just added to the periodic interest rates. Example. –A firm borrows $15,000,000 at LIBOR, with 8 semiannual payments. –A cap will cost $300,000. PV = -300,000, i =.035 *, n = 8 => caplet PMT = $43,643. This will add 43,643/15,000,000 = 29 basis points per semiannual payment = 58 bp per year to the loan That is, the firm will pay LIBOR + 58bp, capped at 7.58% per annum.

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©David Dubofsky and Thomas W. Miller, Jr. Floors An interest rate floor is a put on an interest rate ( = a call on a debt instrument). Lenders, or institutions that own floating rate debt (an asset), purchase interest rate floors. The floor is actually created when the interest rate put is combined with the floating rate asset. The put places a floor (a minimum) on how low the institution’s interest income can be.

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©David Dubofsky and Thomas W. Miller, Jr. Firm buys an at the money interest rate put VV rr Firm is exposed to the risk that interest rates will decline raw exposure long put

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©David Dubofsky and Thomas W. Miller, Jr. The Interest Rate Floor is Insurance Against Declining Interest Rates VV rr Firm is exposed to the risk that interest rates will decline. Buying the interest rate put = buying insurance. Raw Exposure long put Net Exposure

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©David Dubofsky and Thomas W. Miller, Jr. Floors Combine a floating rate asset with a floor. The lender then receives: if r > r f r f if r < r f The floor payments themselves are: (r f - r) (notional principal) (d/360) if r < r f 0 if r > r f where: r f is the floor rate (strike price), and d is the # of days in the settlement period.

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©David Dubofsky and Thomas W. Miller, Jr. Effective Interest Income with a Floor r rFrF Interest Income

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©David Dubofsky and Thomas W. Miller, Jr. Interest Rate Collars A collar combines the purchase of a cap at a high strike price with the sale of a floor at a low strike price. Frequently, the cost of the cap equals the selling price of the floor, so that a zero cost collar is “purchased”. A collar allows a firm with a floating rate liability to insure itself from the undesired impact of an interest rate rise, but pays for it by giving up the benefits of a rate decline.

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©David Dubofsky and Thomas W. Miller, Jr. Interest Rate Collar r Profit Long Call rcrc rfrf Written put

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©David Dubofsky and Thomas W. Miller, Jr. Interest Rate Collar rfrf rcrc Profit r

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©David Dubofsky and Thomas W. Miller, Jr. Interest Rate Collar rfrf rcrc Profit r The underlying risk exposure is to rising interest rates

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©David Dubofsky and Thomas W. Miller, Jr. Interest Rate Collar: rfrf rcrc Profit r underlying risk exposure

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©David Dubofsky and Thomas W. Miller, Jr. Effective Interest Expense Under a Collar Interest Expense r rfrf rcrc

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©David Dubofsky and Thomas W. Miller, Jr. Interest Rate Corridors Corridors are purchased by a firm with a floating rate liability. The firm buys an interest rate call with a (relatively) low strike price, and sells a cheaper one with a higher strike price. Profit r buy low r c cap (in-the-money) sell cheaper higher r c cap (out-of-the-money)

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©David Dubofsky and Thomas W. Miller, Jr. Interest Rate Corridors underlying risk exposure profit r r The firm is hedged only within a range of interest rates

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©David Dubofsky and Thomas W. Miller, Jr. Cap - Floor Parity Recall that C - P = S - PV(K), and that the zero profit point for buying a call and selling a put is the theoretical forward price. The zero profit point for buying a cap and selling a floor is the theoretical price for a fixed - for - floating interest rate swap.

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©David Dubofsky and Thomas W. Miller, Jr. Extra Material What follows is are examples of a delta neutral strategy and a delta-gamma neutral strategy

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©David Dubofsky and Thomas W. Miller, Jr. Example: Suppose S = K = $100, r = 8%, = 30%, and T = 180 days. Then, by using the BSOPM, one can generate the following information: VariableCall OptionPut OptionStock Price$10.30 $6.44$100 Delta Gamma

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©David Dubofsky and Thomas W. Miller, Jr. Now, suppose a trader is considering using either calls or puts to maintain a delta neutral portfolio. This can be accomplished with the purchase of puts or by writing calls. Using Puts: Delta Neutral Strategy: Long 1 put at $6.44; long shares at $100/share. Positive Position Gamma: (1* ) + (0.3849* 0) = Value of Stock Put0.3849Port. Pct. Chg. Price PriceSharesValue (From S=100)

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©David Dubofsky and Thomas W. Miller, Jr. Using Calls: Delta Neutral Strategy: Short 1 call at $10.30; long shares at $100/share. Note that this results in a Negative Position Gamma: (-1* ) + (0.6151* 0) = Value of Stock Call0.6151Port. Pct. Chg. Price PriceSharesValue (From S=100)

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©David Dubofsky and Thomas W. Miller, Jr. Note that when the delta hedger constructs a portfolio using puts, the value of the portfolio increases, regardless of the direction of the change in the underlying asset price. However, in the case where calls are used, the portfolio value falls whether the underlying asset price increases or decreases. From a cost standpoint, a delta neutral hedger would like to have a portfolio with a low, but positive gamma. Recall that gamma measures changes in delta, and that deltas change as S, T, , and/or r change. Thus, a portfolio that is delta neutral portfolio today may not be delta neutral tomorrow. A low position gamma will mean that the delta neutral investor will conserve on the transactions costs of readjusting the portfolio delta back to zero if S changes. But a positive gamma at least compensates a trader for bearing the risk of fluctuating delta, because the value of the portfolio will increase if the underlying asset’s price changes, all else equal.

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©David Dubofsky and Thomas W. Miller, Jr. Generalized Portfolio Sensitivities Traders can create portfolios that are neutral (insensitive) to any single “Greek” or combination of “Greeks.” From the examples above, you can see that a trader needs one option to remove one effect. In the example above, the trader made the portfolio insensitive to changes in the stock price using a delta neutral strategy (i.e., recall that traders can use either calls or puts in a delta neutral strategy). To neutralize additional effects, traders must add additional securities to their portfolios.

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©David Dubofsky and Thomas W. Miller, Jr. Delta-Gamma Hedge To remove delta effects via a delta-neutral hedging strategy using options, the trader employed one option. To remove another effect, i.e., gamma, the trader will need to use a second option. Suppose S = K 1 = $100, r = 8%, = 30%, T = 180 days, and K 2 =110. Then, by using the BSOPM, one can generate the following information. Call Option Variable(K=100) (K=110)Stock Price$10.30$6.06$100 Delta Gamma

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©David Dubofsky and Thomas W. Miller, Jr. Suppose the trader decides to sell 200 call options with the strike of 100, each on 100 shares of stock. Because the trader wants to create a delta-gamma hedged portfolio, the trader must simultaneously solve for the number of shares to purchase, S, and the number of K=110 calls to purchase, K 110. This is straightforward because the gamma of the stock is zero, and the portfolio gamma is a weighted sum of the constituent gammas. The trader wants a gamma of zero. That is, 0 = (-200*100*0.0181) + (S*0) + (K 110 *100*0.0187) which yields K 110 = for any value of S.

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©David Dubofsky and Thomas W. Miller, Jr. Assuming the trader buys 194 of the calls having a strike price of 110. Solving for the number of shares is accomplished by noting that the delta of a share equals one and that the trader also wants to hold a delta neutral portfolio. Accordingly, 0 = (-200*100*0.6151) + (S*1) + (194*100*0.4365) Therefore S = 3,834 makes the portfolio delta-gamma neutral, given the other positions. To examine the performance of this delta-gamma hedge, suppose the stock price were to change to either $101 or to $99.

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©David Dubofsky and Thomas W. Miller, Jr. StockK=100 CallK=110 Call Portfolio Pct. Chg. Price Price Price Value(S 0 = 100) $294, $294, $294, ( Note how the value of the portfolio remains nearly unchanged, even when the stock price changes up or down by 1%. ) At the same time, note that the portfolio delta of zero remains (virtually) unchanged: Original S = 100 = (3834)(1) + (194)(100)(0.4365) + (-200)(100)(0.6151) = S = 99, = (3834)(1) + (194)(100)(0.4178) + (-200)(100)(0.5967) = S = 101, = (3834)(1) + (194)(100)(0.4552) + (-200)(100)(0.6330) = 4.88

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