©david dubofsky and 19-1 thomas w. miller, jr. chapter 19 using options for risk management i. the...

©David Dubofsky and 19-1 Thomas W. Miller, Jr.

Chapter 19Using 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


The Greeks

Delta

Gamma

Theta1

Vega

Rho

1 Theta is often expressed as a negative number

1)N(d Δ0 ΔS

C1CC

01-)N(d Δ1- Δ

S

P1PP

0 Γ S

C

S

ΔC2

2C

0 Γ S

P

S

ΔP2

2P

0 Θ T

CC

0 Θ

T

PAmericanP,

0 Vσ

CC

0 Vσ

PP

0 ρ r

CC

0 ρ r

PP


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)


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.


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-order 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) + dS + 0.5 (dS)2

• So, the change in the call price, given a change in S is:

C(S + dS) - C(S) = dS + 0.5 (dS)2 + …

dC = dS + 0.5 (dS)2 + …


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?)

Optionprice

A

BSlope =

Stock price


Gamma Addresses Delta Hedging Errors Caused By Curvature

S

CStock price

S’

Callprice

C’C’’


Stock Price: 68Strike Price: 65Volatility: 0.50 Black-Scholes Call: 8.52Time to Expiration: 0.25 Black-Scholes Put: 4.96Riskless Rate: 0.035Dividend Rate: 0

Stock Call50 1.1451 1.3452 1.5653 1.8154 2.0755 2.3756 2.6957 3.0358 3.4059 3.8060 4.2261 4.6762 5.1563 5.6564 6.1765 6.7266 7.3067 7.9068 8.5269 9.1770 9.8371 10.5272 11.2273 11.9574 12.6975 13.45

Call Price as a Function of Stock Price, All Else Equal

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

40 45 50 55 60 65 70 75 80

Stock Price

Ca

ll P

rice


Delta as a Function of the Stock Price

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

35 45 55 65 75 85 95

Stock Price

De

lta

Stock Delta36 0.013838 0.023440 0.037442 0.056344 0.080646 0.110748 0.146250 0.186952 0.231954 0.280456 0.331458 0.383760 0.436462 0.488464 0.539066 0.587568 0.633370 0.676072 0.715474 0.751376 0.783978 0.813180 0.8390


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,

• S0 = 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?


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.


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”)


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].


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:S0 = 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


A Closer Look at Delta Hedging

• Important:

– Option on 100,000 shares

– Volatility is assumed constant.

– “Rebalancing” assumed weekly.

Stock Delta * Starting Shares Cost (in Cumula- Interest Week Price Delta 100,000 Shares B/S 000's) tive Cost (in 000's)

0 49.000 0.522 52200 0 52,200 2,557.80 2,557.8 2.51 48.125 0.458 45800 52,200 -6,400 -308.00 2,252.3 2.22 47.375 0.400 40000 45,800 -5,800 -274.78 1,979.7 1.93 50.250 0.596 59600 40,000 19,600 984.90 2,966.5 2.94 51.750 0.693 69300 59,600 9,700 501.98 3,471.3 3.35 53.125 0.774 77400 69,300 8,100 430.31 3,904.9 3.86 53.000 0.771 77100 77,400 -300 -15.90 3,892.8 3.77 51.875 0.706 70600 77,100 -6,500 -337.19 3,559.3 3.48 51.375 0.674 67400 70,600 -3,200 -164.40 3,398.4 3.39 53.000 0.787 78700 67,400 11,300 598.90 4,000.5 3.8


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 * (67,400 + 11,300) = $4,171,100.

– Increase of: $1,613,300.

• Net: $1,613,300 – 1,442,700 – 174,500 = $(3,900).


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.


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…)


Interpretation of Gamma

For a delta neutral portfolio, d dt + ½dS2

d

dS

Negative Gamma

d

dS

Positive Gamma


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 +1.0. The delta of a share of stock that is sold short is –1.0. Why?


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 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 i.

i

N

iip n

1


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=nii

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 17.36

p = 178.96

This assumes that each option is on one share of stock


• 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 price 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 portfolio's position delta is as essential to intelligent option trading as knowing the profit diagrams of your portfolio.


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 is owned, and negative if the option has been written.

• The gamma of the ith 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.

i

N

iip n

1


• 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.


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.


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.


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.


Neutral Time Spreads, Example

• Suppose the implied standard deviations 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 the two options are = 0.4439 for the

September call, and = 0.5415 for the December call. • A delta neutral portfolio of these two options is:

p = 0 = n11 + n22


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.

1

2

2

1

Δ

Δ

n

n

6)-(19 1.220.4439

0.5415

n

n

2

1


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.


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.


Buying a cap provides insurance against an interest rate increase that hurts the firm

ΔV

Δr

long call (at the money)

A rise in interest rates increases interest expense.


Buying a cap provides insurance against an

interest rate increase that hurts the firm

V

Δ 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.


Caps

• An interest rate call is purchased along with a variable rate loan to create the cap. The borrower then pays:

if r < rc

rc if r > rc

• The interest rate call payments themselves are:

(r - rc) (notional principal) (d/360) if r > rc

0 if r < rc

where: rc is the cap rate (strike price), and d is the # of days in the settlement period.

r~


Effective Interest Cost under a Cap

rrc

Interest Expense


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:

(0.079 - 0.070) ($15,000,000) (90/360) = $33,750

• A cap is a portfolio of caplets, each having a different expiration (reset) date.


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.


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.


Firm buys an at the money interest rate put

V

r

Firm is exposed to the risk that interest rates will decline

rawexposure

long put


The Interest Rate Floor is Insurance Against Declining Interest Rates

V

r

Firm is exposed to the risk that interest rates will decline. Buying the interest rate put = buying insurance.

RawExposure

long put

Net ExposureNet Exposure


Floors

• Combine a floating rate asset with a floor. The lender then receives:

if r > rf

rf if r < rf

• The floor payments themselves are:

(rf - r) (notional principal) (d/360) if r < rf

0 if r > rf

where: rf is the floor rate (strike price), and d is the # of days in the settlement period.

~r


Effective Interest Income with a Floor

rrF

Interest Income


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.


Interest Rate Collar

r

ProfitLong Call

rc

rf

Written put



rfrc

Profit

r



rfrc

Profit

r

The underlying risk exposure is to rising interest rates


Interest Rate Collar:

rfrc

Profit

r

underlying risk exposure


Effective Interest Expense Under a Collar

Interest Expense

rrf rc


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 rc cap (in-the-money)

sell cheaper higher rc cap (out-of-the-money)


Interest Rate Corridors

underlying risk exposure

profit

r

profit

r

The firm is hedged only within a range of interest rates


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.


Extra Material

• What follows is are examples of a delta neutral strategy and a delta-gamma neutral strategy


Example:

• Suppose S = K = $100, r = 8%, = 30%, and T = 180 days.

• Then, by using the BSOPM, one can generate the following information:

Variable Call Option Put Option Stock

Price $10.30 $6.44 $100

Delta 0.6151 -0.3849 1

Gamma 0.0181 0.0181 0


• 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 0.3849 shares at $100/share.Positive Position Gamma: (1* 0.0181) + (0.3849* 0) = 0.0181.

Value ofStock Put 0.3849 Port. Pct. Chg.Price Price Shares Value (From S=100)

90 11.25 34.64 45.89 2.14100 6.44 38.49 44.93110 3.41 42.34 45.75 1.83


Using Calls:

Delta Neutral Strategy: Short 1 call at $10.30; long 0.6151 shares at $100/share. Note that this results in a Negative Position Gamma: (-1* 0.0181) + (0.6151* 0) = -0.0181.

Value of

Stock Call 0.6151 Port. Pct. Chg.

Price Price Shares Value (From S=100)

90 5.12 55.36 50.24 -1.89

100 10.30 61.51 51.21

110 17.28 67.66 50.38 -1.62


• 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.


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.


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 = K1 = $100, r = 8%, = 30%, T = 180

days, and K2=110. Then, by using the BSOPM, one can generate the

following information.

Call Option Call Option

Variable (K=100) (K=110) Stock

Price $10.30 $6.06 $100

Delta 0.6151 0.4365 1

Gamma 0.0181 0.0187 0


• 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, K110.

• 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) + (K110*100*0.0187)

• which yields K110 = 193.583 for any value of S.


• 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.


Stock K=100 Call K=110 Call Portfolio Pct. Chg. Price Price Price Value (S0 = 100)

99 9.6984 5.6310 $294,839 0.0007 100 10.3044 6.0580 $294,837

101 10.9285 6.5040 $294,841 0.0015

(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) = 0.10

@ S = 99, = (3834)(1) + (194)(100)(0.4178) + (-200)(100)(0.5967) = 5.32

@ S = 101, = (3834)(1) + (194)(100)(0.4552) + (-200)(100)(0.6330) = 4.88

©david dubofsky and 19-1 thomas w. miller, jr. chapter 19 using options for risk management i. the...

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