1、David Dubofsky and 19-1Thomas 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,David Dubofsky and 19-2Thomas W. Miller, Jr.,T
2、he Greeks,1 Theta is often expressed as a negative number,David Dubofsky and 19-3Thomas 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 assum
3、ed to be constant) Strike Price (never changes),David Dubofsky and 19-4Thomas 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
4、over time.,David Dubofsky and 19-5Thomas W. Miller, Jr.,Some Intuition From Using a Taylor-Series Expansion,Given value at x0, we can use derivatives of function to approximate value of function at x:f(x) = f(x0) + f (x0) (x-x0) + (1/2) f (x0)(x-x0)2 + . Incorporate curvature effects via Second-orde
5、r Taylor series expansion of option price around the stock price (quadratic approximation):C(S + dS) = C(S) + (C/S) dS + (0.5) 2C/S2 (dS)2= C(S) + D dS + 0.5 G (dS)2So, the change in the call price, given a change in S is:C(S + dS) - C(S) = D dS + 0.5 G (dS)2 + dC = D dS + 0.5 G (dS)2 + ,David Dubof
6、sky and 19-6Thomas W. Miller, Jr.,Delta AKA: Hedge Ratio,Delta (D) 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?),David Dubofsky and 19-7Thomas W. Miller, Jr.,Gamma Addresses Delta Hedging Errors Cau
7、sed By Curvature,S,C,Stock price,S,Call price,C,C,David Dubofsky and 19-8Thomas W. Miller, Jr.,David Dubofsky and 19-9Thomas W. Miller, Jr.,David Dubofsky and 19-10Thomas W. Miller, Jr.,Example:,An investment bank has sold (for $300,000) a European call option on 100,000 shares of a non-dividend pay
8、ing stock: i.e,S0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeksThe Black-Scholes value of the option is $240,000.How does the investment bank hedge its risk?,David Dubofsky and 19-11Thomas W. Miller, Jr.,Two Alternatives:,“Naked” position:Take no action (other than deep breath and hope for the best).R
9、isk: Stock price is greater than $53 at expiration.Covered position:Buy 100,000 shares today.Risk: Stock price is less than $46 at expiration.,David Dubofsky and 19-12Thomas W. Miller, Jr.,Stop-Loss Strategy Which Involves:,Buying 100,000 shares as soon as price reaches $50.Selling 100,000 shares as
10、 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”),David Dubofsky and 19-13Thomas W. Miller, Jr.,Delta Hedging,This involves maintaining a “delta neutral” portfolio. The delta of a European cal
11、l 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.,David Dubo
12、fsky and 19-14Thomas 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 do
13、es Delta, and vice versa.)Data from above:S0 = 49, K = 50, r = 0.05, s = 0.20, T = 20 weeks (0.3846)Option written on 100,000 shares, selling price: $300,000Black-Scholes value of this option: $240,000,David Dubofsky and 19-15Thomas W. Miller, Jr.,A Closer Look at Delta Hedging,Important: Option on
14、100,000 shares Volatility is assumed constant. “Rebalancing” assumed weekly.,David Dubofsky and 19-16Thomas W. Miller, Jr.,Aim of Delta Hedging: Keep Total Wealth Unchanged. Lets look at the end of Week 9.,Value of written option at start: $240,000 (per B-S.) Value of option at week 9 (s=20%): $414,
15、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 * (67,400 + 11,300) = $4,171,100. Increase of: $1,613,300. Net: $1,613,300 1,442,700 174,500 = $(3,9
16、00).,David Dubofsky and 19-17Thomas 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 speak
17、ing, 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 D of the call changes (as measured by gamma). As D changes, shares of stock must be bought or sold to maintain the riskless hedge.,David Dub
18、ofsky and 19-18Thomas W. Miller, Jr.,Gamma,Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset. This G varies with respect to the stock price for a call or put option. (G reaches a maximum at the money). As a rule, we would prefer to have a positive Gamma (
19、more on this later),David Dubofsky and 19-19Thomas W. Miller, Jr.,Interpretation of Gamma,For a delta neutral portfolio, dP = Q dt + GdS2,dP,dS,Negative Gamma,dP,dS,Positive Gamma,David Dubofsky and 19-20Thomas W. Miller, Jr.,Position, or Portfolio, Deltas,The position delta determines how much a po
20、rtfolio 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 i
21、s owned equals +1.0. The delta of a share of stock that is sold short is 1.0. Why?,David Dubofsky and 19-21Thomas 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 ni
22、 is the number of options of one particular type, or the number of shares of stock. The sign of ni 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 ith option or stock is given by Di.,David Dubofsky and 19-22Thomas
23、W. Miller, Jr.,An Example of a Position Delta,Suppose you have positions in the following assets:Position and number of options or shares ni_ delta/unit Total Deltas=niDi long 300 shares +300 1.00 300.00 long 40 puts +40 -0.46 -18.40 short 150 calls -150 0.80 -120.00 long 62 calls +62 0.28 17.36Dp =
24、 178.96,This assumes that each option is on one share of stock,David Dubofsky and 19-23Thomas W. Miller, Jr.,In this example, the position delta of 178.96 is positive. This means that if the stock price were to increase by one dollar, the value of this portfolio would rise by $178.96. If the stock p
25、rice were to decline by one dollar, then the value of this portfolio would fall by $178.96. (Remember that this example assumes that the underlying asset of an option is one share of stock.) Knowing your portfolios position delta is as essential to intelligent option trading as knowing the profit di
26、agrams of your portfolio.,David Dubofsky and 19-24Thomas 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 ni is the number of options of one particular type. The sign of ni is positive if the option i
27、s owned, and negative if the option has been written. The gamma of the ith option is given by Gi. Note that the gamma of a stock is zero, because the delta of a stock is always equal to one.,David Dubofsky and 19-25Thomas W. Miller, Jr.,Ideally, a delta neutral hedger would like to maintain a portfo
28、lio 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
29、value changes. Further, a portfolio with a negative gamma decreases in value if the underlying stock value changes.,David Dubofsky and 19-26Thomas W. Miller, Jr.,Time Spreads, I.,Time spreads are also sometimes called calendar spreads or horizontal spreads. Time spreads are identified by the ratio o
30、f 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,
31、the time spread portfolio of options has options with different maturity dates.,David Dubofsky and 19-27Thomas 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 op
32、tion 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 opti
33、ons will often have to be closed out early in order to preclude this possibility.,David Dubofsky and 19-28Thomas 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
34、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.,David Dubofsky and 19-29Thomas W. Miller, Jr.,Neutral Time Spreads, Example,Suppose the implied standard d
35、eviations of the two options were 0.4043 (for a September call), and 0.3262 (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 t
36、he two options are D = 0.4439 for the September call, and D = 0.5415 for the December call. A delta neutral portfolio of these two options is:Dp = 0 = n1D1 + n2D2,David Dubofsky and 19-30Thomas W. Miller, Jr.,Rearranging: Using the above deltas, and denoting the September and December calls as asset
37、s 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.,David Dubofsky and 19-31
38、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 exerci
39、sed 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
40、. 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 app
41、arent mispricing.,David Dubofsky and 19-32Thomas 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 r
42、ate call is combined with an existing floating rate loan. The call caps the interest that can be paid on the floating rate liability.,David Dubofsky and 19-33Thomas W. Miller, Jr.,Buying a cap provides insurance against an interest rate increase that hurts the firm,V,r,long call (at the money),A ris
43、e in interest rates increases interest expense.,David Dubofsky and 19-34Thomas W. Miller, Jr.,Buying a cap provides insurance against an interest rate increase that hurts the firm,DV, r,long call,resulting exposure,The firm benefits from a decline in r, but has bought insurance to protect itself fro
44、m a rise in r.,David Dubofsky and 19-35Thomas 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 rc The interest rate call payments themselves are:(r - rc) (notional principal) (d/360) if r rc0 if r rcwhere: rc is the
45、 cap rate (strike price), and d is the # of days in the settlement period.,David Dubofsky and 19-36Thomas W. Miller, Jr.,Effective Interest Cost under a Cap,r,rc,Interest Expense,David Dubofsky and 19-37Thomas W. Miller, Jr.,An Example of a “Caplet”,Suppose the cap rate is 7.00%. If:the reference in
46、terest 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:(0.079 - 0.070) ($15,000,000) (90/360) = $33,750A cap is a portfolio of caplets, each having a different expiration (reset) date.,David Dubofsky and 19
47、-38Thomas 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
48、= 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.,David Dubofsky and 19-39Thomas W. Miller, Jr.,Floors,An interest rate floor is a put on an interest ra
49、te ( = 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 institutions interest income can be.,David Dubofsky and 19-40Thomas W. Miller, Jr.,
