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**Review 1: Slope of Payoff Profile for Call**

Consider the Total Value of a Call Option ∆ (or ) is dc / dS = slope of payoff profile for a call. For Call, What is min value of ? For Put, … ? What is max value of ? What is of ATM call?

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**Review 2: Total, Intrinsic, and Extrinsic Value**

0 < dc/dS < 1 dc/dS = 1 Total Value If σ ↑ S dc/dS = 0 K Intrinsic Value S K Extrinsic Value S K

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**Chapter 17. The Greeks: Sensitivities & Delta Hedging**

I. Background: Risk, Sensitivities, Derivatives, & Hedge Ratios A. Risk: Sensitivity to changes in the underlying source of uncertainty. 1. Stock Market: Ri = αi + βi RMt + εi . Source of uncertainty = Market Return = RM . Risk = dRi / dRM = βi . [ Recall, N* = β(VA / Vf).] n 2. Bond Market: Bond Price = B = Σ ck / (1+r)k k=1 Bond Return = RB = ΔB / B. Source of uncertainty = Market Interest Rates = r. Risk = dRB / dr = Duration. [ Recall, N* = VA DA / Vf Df.]

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I.A. Risk Hedge Ratios: how many futures / options to trade to hedge spot asset. ΔF 3. Hedging with Futures: | | | | _____________________|______________________ ΔS . | . . | | | | Futures Hedge Ratio = h* = slope of regression line = sensitivity of F to changes in S = dF / dS.

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**I.A. Risk 4. Hedging with Options: a. Single-Period Binomial Model:**

Options Hedge Ratio: Δ = (cu - cd) / (Su - Sd) = dc / dS. b. N-Period Binomial Model: c = S * B[a,n,p’] - Ke -rT * B[a,n,p] Options Hedge Ratio: dc / dS = B[a,n,p’] c. Black / Scholes Model: c = S * N(d1) Ke -rT * N(d2) Options Hedge Ratio: dc / dS = N(d1). The Point: Hedge Ratios are derivatives -- measures of sensitivity, risk. They also show how many futures or options are needed to hedge.

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**I.B. Motivation behind the Greeks**

1. Banks sell products involving options. The bank is short options! Needs to hedge! 2. Examples: a. Currency Options. i. Corporate clients can buy FX options thru exchange or OTC. ii. Banks sell FX options to meet clients’ needs. b. Equity Options. i. Bank underwrites equity issue. Guarantees issue price for the firm’s stock. (Bank will buy if can’t K.) Effectively sells put option to firm’s owners. ii. Bank sells products that: a. Guarantee no loss of principal on a market investment; b. Guarantee return as some proportion of market return. c. These products have embedded options on the market.

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**I.B. Motivation behind the Greeks**

c. Interest Rate Options. i. Guarantee interest rate on 15-year mortgage for 3 months. If rate increases, client will exercise; If rate declines, client will insist on lower rate. ii. Guarantee floating rate cap at 10% for 5 years. Bank has sold portfolio of European options on r. If floating rate on any interest payment > 10%, exercise option. iii. Bank offers prepayment privileges on a loan. Consider loan as a bond sold by borrower to the bank. Prepayment privilege gives borrower right to buy back bond. Bank has sold American call on the bond. iv. Bank offers early redemption privilege on CD. Consider CD as bond bought by depositor from the bank. Early redemption privilege gives depositor right to sell back CD. Bank has sold American put on the bond. 3. Point: When Bank sells options, should be paid, & should hedge!

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**II. Example Firm sells European call on 100,000 shares:**

S = $49; K = $50; r = .05; = .20; T = .3846; CBS = $240,000; but suppose firm receives c = $300,000. A. Naked short position; Do not hedge ________│__________S K Bad If S to $60, call will be exercised. Must buy 100,000 shares to cover. Lose (100,000) x ($10) = $1,000,000. Get to keep c = $300,000. Total Loss = $700,000. Good 2. If S to $40, call OTM; Keep c = $300, ________│__________S B. Covered short position; Buy 100,000 shares. Good 1. If S > $50, deliver. Paid S = $49 for 100,000 shares (= $4,900,000). Exercise: receive K = $50/share (= $5,000,000), & Keep c = $300,000. Bad If S to $40, call OTM. Paid S = $49 for 100,000 shares (= $4,900,000); Shares worth only $40 (= $4,000,000). Loss on stock position = $900,000. Get to keep c = $300, Total Loss = $600,000.

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**II. Example Firm sells European call on 100,000 shares:**

S = $49; K = $50; r = .05; = .20; T = .3846; CBS = $240,000; but suppose firm receives c = $300,000. ** Summarizing A. & B.; If you bought this call, expected payoff is $240,000. Since you sold this call, expected cost is $240,000. NPV of expected cost is $240,000 for both A. & B. Value of any security in equil. = NPV of expected cash flows. Perfect hedge should guarantee expected cost = $240,000. However, for any one occasion, cost could range from $0 - $1,000,000 or more (as in A. & B.). A. & B. have expected costs = $240,000, but variances > 0. (σ unknown)

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**II.C. Hedging through the Cap**

C. Stop Loss Strategy; Buy when S above K; sell when S below K. 1. Objective: hold naked position when S < K (OTM); ← (like A) hold covered position when S > K ( ITM). ← (like B) 2. Cost of setting up initial hedge (per share) = $S if S > K; Cost = $0 if S < K. 3. All subsequent purchases & sales are at $K. 4. At expiration: if OTM, owe $0; if ITM, deliver, receive $K. 5. It appears that Total Cost of hedge is Q = max{ S - K, 0 }; Identical to call; Hedge seems to work perfectly.

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**II.C. Hedging through the Cap**

6. Stop Loss Strategy does not work so well in practice. a. Can’t ignore time value of money. Don’t know when you’ll be buying & selling @ K. Cash flows in future must be discounted. b. Ignoring Transactions Costs. Purchases & sales cannot be made exactly @ K. If S hits K, cannot know if it will continue to or . Will really buy when S above K, say, to K + . sell when S below K, say, to K - . Every round trip has TC = 2. 7. If S never reaches K, hedging scheme costs nothing. If S cross K many times, can be very costly. a. Can monitor S closely so that smaller. b. As toward 0, the # of trades & TC . 8. Simulations indicate: as time interval for monitoring (t) , this works better, but TC too; never good enough.

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**III. Delta Neutral Hedging**

B/S Model: c = S * N(d1) - Ke-rT * N(d2) A. To get hedge pf, how many calls should be sold for every share owned? (i.e., what is the right hedge ratio to eliminate risk?) 1. Define: = c / S = dc / dS = N(d1); Sensitivity of c to changes in S. a. When S $1, c by $. b. Example: If = .5, when S↓ $1, c↓ $.50 2. Hedge ratio = 1 / = 1 / N(d1). a. To hedge, sell 1 / calls for every share owned. b. Then, when S $1, short call position $1. c. Example: If = .5, sell 1 / = 2 calls; when S $1, the 2 short calls ↑ $1. Hedged. 3. In general, Hedge Portfolio has QS shares & QC calls sold. a. Hedge Portfolio = HP = QS (S) - QC (c). b. Let QS = 1 share; QC = 1 / = 1 / (c/S). c. Then HP = 1 * (S) - [ 1 / (c/S) ] * (c) = S - [ (S / c) * c] = S - S = It works!

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**III.B. Intuition behind Delta Neutral Hedging**

( & ) (theta) (vega) (rho) c = f (S, K, T, , r) B. Intuition: (or ∆) is the slope of the payoff profile for an option.

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**III.B. Intuition behind Delta Neutral Hedging**

( & ) (theta) (vega) (rho) c = f (S, K, T, , r) B. Intuition: can be thought of as P(ITM). < || < 1; ( can be negative); 2. ATM call should have 0.5; ________│_______ S ( = 50% chance to be ITM) K 3. ITM call should have .5 < < 1; ________│_______ S ( > 50% chance to be ITM) K 4. OTM call should have 0 < < .5; ________│_______ S ( < 50% chance to be ITM) K 5. Nearer to expiration, become more important.

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**III.B. Intuition behind Delta Neutral Hedging**

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**III.C. Other Characteristics of **

1. Put options have ’s the opposite sign; a. long position in ATM puts: -.5; b. long position in ITM puts: -1 < < -.5; c. long position in OTM puts: -.5 < < 0. 2. Sign of depends upon call or put, long or short: call put long short 3. = f( S, T, r, ); i.e., Hedge ratio changes! 4. As time passes, extrinsic value : C a. call ’s move away from +.5; | b. put ’s move away from | (toward 0 and 1 or -1). | | 5. As , opposite; extrinsic value : | a. call ’s move toward +.5; |______________________ S b. put ’s move toward K

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**III.D. Hedging Ownership of Underlying Asset (+S)**

1. Long position in stock (+S) has = +1. Short position has = -1. 2. Hedge ratio = 1 / (= how many options to trade), where = dc/dS. 3. Example: Suppose call option’s = .4; Hedge Ratio = 1 / = 1/(.4) = 2.5; Sell 2½ calls for each share owned. a. Long 1 share: = +1 b. Short 2½ calls: = 2.5*(-.4) = -1 c. Net delta: = 0 Hedged! 4. To establish -neutral hedge, ’s must sum to 0. a. As long as net delta 0, any combination of puts, calls, & stocks with different strike prices & expirations can be used. 5. While position may be -neutral when established, this will change quickly with changes in S, T, and . a. To remain -neutral, position needs continual adjustment. Dynamic hedging strategy! (Not hedge & forget.)

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**III.E. Hedging Short Position in Call Option (-c)**

Banks sell options; need to hedge! Recall example: sell call for $300,000; CBS = $240,000. 1. To hedge, create opposite position synthetically; Really want long call to offset short call. 2. Construct a synthetic call: Take long position in underlying asset, so that the of position is maintained to = of option. Value K S | (payoff on | short call) | 3. Suppose S = K; = -.5; If S $1, lose $.50 on short call. a. -neutral hedge: Buy shares for each call written. Same as hedge pf in Bin or B/S. b. Then if S $1, gain $.50 on hedge to offset loss (own ½ share). Delta neutral! 4. a. If S, closer to -1; Need to buy more shares to match new . (dynamic!) If S, closer to 0; Need to sell shares to match new . (monitor ) b. Buy in rising mkt; Sell in falling mkt; Buy high; Sell low!

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**III.E. Hedging Short Position in Call Option (-c)**

5. Delta Neutral Hedging is closely related to B / S analysis. a. B / S showed it is possible to set up a riskless hedge portfolio by selling one option and buying delta shares. **This hedge portfolio is just a delta-neutral position.** b. In this context, B / S valued options by setting up a delta-neutral position and arguing it should pay r.

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**III.F. Hedging Short Position in Put Option (-p)**

1. Put deltas are < 0. Thus, short put position has > 0. K Value S | (payoff on | short put) | 2. Example: Suppose S = K; = +.5; If S $1, lose $.50 on short put. -neutral hedge: Sell shares for each put written. Then if S $1, gain $.50 on hedge to offset loss. Delta neutral! 3. If S keeps ing, gets closer to +1; Need to sell more shares to match new . If S increases, gets closer to 0; Need to buy shares to match new . Buy in rising mkt; Sell in falling mkt; Buy high; Sell low.

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**III.G. Delta of Portfolio of Options**

1. Consider portfolio of options that all depend on S. a. Let i = delta of ith option. b. Let wi = weight of portfolio in ith option. 2. The delta of this portfolio = wi i . a. Delta of portfolio can be computed from deltas of options in portfolio. b. If bank has portfolio of short options, can calculate delta of portfolio, then buy this amount of asset, or futures on the asset, to get a delta neutral position (hedge!). (Why buy futures?) 3. If bank has many short option positions, too expensive & tedious to hedge all options individually. Only one trade in underlying asset or futures is necessary to hedge whole portfolio! Cheaper & easier to hedge whole portfolio.

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**III.H. Dynamic Aspects of Delta Hedging**

1. To avoid excessive transactions costs, allow net to deviate from 0, and set a limit on how long or short net can become before adjustment. 2. Another technique: make adjustments at set intervals (daily, weekly, …) As interval 0, hedge perfect. 3. Choose time interval or limit on net that make sense, Given transaction costs, market volatility, and stability of position’s .

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**IV. Other Option Sensitivities (Greeks)**

A. Gamma () = d / dS = d(dc/dS) / dS = d2c / dS2 . 1. Second derivative; rate of change of rate of change; curvature. a. measures speed of change in c as S changes; b. measures its acceleration or convexity. (curvature) 2. Gamma is a measure of stability of . a. Large , unstable ; Small , stable . b. Low positions are preferable for -neutrality. p c │ │ │_____________________s │_____________________s K K

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**IV.A. Gamma 3. Characteristics of :**

a. As S → K, more extrinsic value; (greatest curvature near K). b. As , more extrinsic value; (curve shifts up; less curvature at K). ’s of ITM and OTM options ↑, but ’s of ATM options ↓. c. As T → 0, less extrinsic value; (as expir. approaches, shifts down); ’s of ITM and OTM options , but ’s of ATM options . │ │ ______________________s │_____________________s K K

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**IV.A. Gamma 4. Summary of Gamma:**

a. Positive - If S by $1, value (c) by ; Negative - If S by $1, value (p) by ; Positive - If S by $1, by ; Negative - If S by $1, by . b. If portfolio is - neutral; when S , c does not . c. If portfolio is - neutral; when S , does not .

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**IV.B. Theta (T) = dc/dT (Time Decay)**

extrinsic value 1. Graph. | | |________________________________ T days remaining expiration 2. Characteristics of Theta: a. As , Theta increases; More volatility, more extrinsic value to decay. b. As S → K, Theta increases; ATM options have highest extrinsic value. c. As T decreases toward 0 (expiration approaches), Thetas of ATM options ; Thetas of ITM & OTM options . d. Theta is not same type of sensitivity as delta or gamma; There is uncertainty about future S, but not about time passing. Makes sense to hedge against moves in S! -- not against time! e. Theta is proxy for gamma in -neutral portfolio.

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**IV.C. Vega () = dc/dσ 1. Characteristics of Vega:**

a. As S → K, vegas increase. (ATM options have highest extrinsic value, and thus are most sensitive to changes in .) b. As T decreases toward 0, Vega’s . (As expiration approaches, extrinsic value .) 2. Summary of Vega implications: a. Positive Vega (e.g. long options) if , value ; Negative Vega (e.g. short options) if , value ; b. Make portfolio Vega-neutral; Then if changes, value of portfolio does not change.

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V. Hedging in Practice A. Financial institutions sell many options OTC to their clients. They have portfolios that are short many kinds of options. 1. In ideal world, banks should rebalance frequently to maintain neutrality (with respect to portfolio’s , , T, and ). In reality, too expensive. 2. In practice, traders usually adjust net to zero daily by trading in the underlying asset. 3. More difficult to maintain neutrality in gamma & vega; Difficult to find options or other nonlinear derivatives that can be traded in large volume at reasonable prices. a. So gamma & vega are usually just monitored. b. If they get too large, then adjustments are made.

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V. Hedging in Practice B. Banks have economies of scale trading derivatives. 1. Maintaining delta neutrality on an individual option is likely to be very expensive relative to the gains made. 2. But maintaining delta neutrality on a portfolio of options is good, since the high cost of rebalancing is likely to be offset by the profit on many different trades.

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**V. Hedging in Practice C. Banks are usually short options.**

1. They usually have negative gammas and vegas. 2. As time passes, their gamma & vegas may get more negative. 3. Their traders are always looking for ways to buy options cheaply (i.e., acquire positive gamma & vega - at reasonable prices). 4. One aspect of options helps reduce this problem. a. When sold, options are often ATM (high & ). b. Over time, often become deep ITM or OTM (low & ). c. If options stay ATM, & remain high & need attention. D. Banks use VaR, simulation, & sensitivity analysis to stay aware of the risks (good & bad possible outcomes). (See Hull, Ch. 20.)

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V. Hedging in Practice E. Creating Options Synthetically for Portfolio Insurance. 1. Portfolio managers are often long certain assets (e.g., stocks, bonds, foreign currency, … ). 2. Want a put on the underlying asset (protective put – maintain upside but hedge downside). 3. Creating a put synthetically involves maintaining a short position in the underlying asset (or a futures on it) so that of position = of the required option. If you short futures, rather than spot, have low margin. This delta hedging procedure is called “portfolio insurance.” 4. May be good to create this put synthetically: a. Fund managers often require large positions, and the options needed are often illiquid & costly to trade. b. Often require options with strike prices and maturities that are not available at exchanges, or are illiquid.

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V. Hedging in Practice 5. Synthetic put option can be created by trading the underlying asset or futures on the underlying asset. a. If long the asset, create a put by selling asset in proportional to delta. b. e.g., for a stock portfolio, make sure that at any time, a proportion () of the portfolio of stocks have been sold, and the proceeds invested @ r. 6. Remember, maintaining delta-neutrality means Buy high, Sell low. a. As the value of the portfolio , gets more negative; so the proportion of the portfolio of stocks sold es. Sell low. b. As the value of the portfolio , gets less negative; so the proportion of the portfolio of stocks sold es. Buy high. c. This portfolio insurance costs, because manager is always selling after a decline & buying after a rise. 7. Portfolio Insurance & the crash of October 1987. a. Selling as market crashed likely made crash worse.

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