Deutsche Bank Introduction to Exotic Options Alan L. Tucker, Ph. D

Deutsche Bank Introduction to Exotic Options Alan L. Tucker, Ph. D
Deutsche Bank Introduction to Exotic Options Alan L. Tucker, Ph.D (tel) (fax) Copyright © Marshall, Tucker & Associates,LLC All rights reserved 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

ALAN L. TUCKER, Ph.D. Alan L. Tucker is Associate Professor of Finance at the Lubin School of Business, Pace University, New York, NY and an Adjunct Professor at the Stern School of Business of New York University, where he teaches graduate courses in derivative instruments. Dr. Tucker is also a principal of Marshall, Tucker & Associates, LLC, a financial engineering and derivatives consulting firm with offices in New York, Chicago, Boston, San Francisco and Philadelphia. Dr. Tucker was the founding editor of the Journal of Financial Engineering, published by the International Association of Financial Engineers (IAFE). He presently serves on the editorial board of Journal of Derivatives and the Global Finance Journal and is a former associate editor of the Journal of Economics and Business. He is a former director of the Southern Finance Association and a former program co-director of the 1996 and 1997 Conferences on Computational Intelligence in Financial Engineering, co-sponsored by the IAFE and the Neural Networks Council of the IEEE. Dr. Tucker is the author of three books on financial products and markets: Financial Futures, Options & Swaps, International Financial Markets, and Contemporary Portfolio Theory and Risk Management (all published by West Publishing, a unit of International Thompson). He has also published more than fifty articles in academic journals and practitioner-oriented periodicals including the Journal of Finance, the Journal of Financial and Quantitative Analysis, the Review of Economics and Statistics, the Journal of Banking and Finance, and many others. Dr. Tucker has contributed to the development of the theory of derivative products including futures, options and swaps, and to the theory of international capital markets and trade. He has also contributed to the theory of technology adoption over the life-cycle. The Social Sciences Citation Index shows that his research has been cited in refereed journals on over one hundred occasions. As a consultant, Dr. Tucker has worked for The United States Treasury Department, the United States Justice Department, Morgan Stanley Dean Witter, Union Bank of Switzerland, LG Securities (Korea), and Chase Manhattan Bank. Dr. Tucker holds the B.A. in economics from LaSalle University (1982), and the MBA (1984) and Ph.D. (1986) in finance from Florida State University. He was born in Philadelphia in 1960, is married (Wendy) and has three children (Emily, 1993, Michael and Matthew, both 1995). 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Introduction to Exotic Options
Purpose: The purpose of this session is to introduce exotic options. Given our time constraints, we will focus on the following topics: The major types of exotic options and their payoffs A general discussion of the pricing of exotic options The use of exotic options to build structured equity products The added hedging complexities of exotic options for their dealers A relatively recent hedging technology known as static replication 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Major Types of Exotic Options. Path-Dependent Options Asian or average price options Max[0, Save - X] Look back options Max[0, Smax - X] Ladder options Max[0, S(T) - X, L - X] Cliquets/Clicks Max[0, S(t) - X, S(T) - X] Shouts Max[0, S(T) - X, S(shout) - X] Range options Max[0, (S(0) x k1/R + k2) - X] Barriers (8 types) Up-and-out call Max[0, S(T) - X] if S(t) < B Down-and-out call Max[0, S(T) - X] if S(t) > B et cetera 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Correlation Options Rainbow or out-performance Max[0, S1(T) - X1, S2(T) - X2] Triggers Max[0, S1(T) - X1] if S2(T) > X2 Spread options Max[0, S1(T) - S2(T)] x NP Ratio options Max[1, S1(T)/S2(T)] x NP Digital/Binary Options Cash-or-Nothing (CoN) $Q if S(T) > X Asset-or-Nothing (AoN) S(T) if S(T) > X 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Compound Options Calls on calls Calls on puts et cetera Others Chooser/as-you-like-it options Hybrids Partials Power options Power currency options 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Exotic Option Pricing. Pricing exotic options is in general no more difficult than the pricing of plain vanilla options. Indeed, if an exotic option is European-style and it is assumed that underlying asset prices are log-normal, then in the vast majority of cases an analytic or closed-form pricing solution exists. And even if there is a violation of the log-normal property, e.g., an Asian option with arithmetic averaging, then typically a good analytic approximation exists. If the exotic is American-style, then any one of a number of numerical methods, e.g., binomial trees with areas of finer meshing, are readily available. Still, market prices of exotic options can be far different than their theoretical model values as dealers will price off of what it cost to hedge the exotic. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Using Exotic Options to Build Structured Equity Products. We now examine five structured equity products: A capped equity floater A collared equity floater An equity range floater An equity-linked note Another equity-linked note Building and marketing these structures can involve several desks including the corporate underwriting desk, the private placement desk, the prime brokerage desk, various equity derivatives desks, and the institutional sales and trading desk. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Structure I: A Capped Equity Floater Assume that a corporation could issue a four-year fixed rate note if it agreed to pay a fixed rate of 2.00% each quarter (about 8% per annum). An investor is willing to hold this corporation’s debt, but wants to receive a floating coupon pegged to the total return of the SP500. Also, the investor is willing to accept a cap of 2.25%, i.e., under no circumstances will the quarterly coupon exceed 2.25%. After some discussions with Deutsche, it is understood that the investor will take the floater if the floater pays qu SP500 TR bps. Let’s build it. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Corporate issuer wants to pay fixed
Introduction to Exotic Options Corporate issuer wants to pay fixed Deutsche Bank equity (SP500) swap pricing bid offer 4-year plain vanilla % % (these are qu rates) SP500 option pricing bid offer 4-year 1.875% qu SP500 TR cap bps bps (premium here is stated on a qu basis) Fixed Rate Structured Equity Product qu coupon = min[qu SP500 TR bps, 2.25%] Investors want to hold capped floater 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Introduction to Exotic Options Note that by this structure the corporate has saved about 6.25 bps qu relative to issuing a straight fixed-rate note. Corporate issuer wants to pay fixed 1.9375% qu 1.6375% qu equity swap qu SP500 TR Deutsche Bank DPG Structured Equity Product 4 year max[qu SP500 TR %, 0] corporate sells 4-year equity cap to dealer Coupon = min[qu SP500 TR bps, 2.25%] 7.5 bps dealer agrees to pay corporate a premium equivalent to 7.5 bps qu Note: option premia are usually paid in full up-front, but this can be annuitized. Investors want to hold capped floater 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Structure II: A Collared Equity Floater Assume that a corporation could issue a four-year fixed rate note if it agreed to pay a fixed rate of 2.00% (again about 8% p.a.). An investor is willing to hold this corporation’s debt, but wants to receive a floating rate pegged to the quarterly total return on the SP500 and is willing to accept a cap of 2.25% qu provided that the note will also have a floor of 1.50% qu, i.e., under no circumstances will the quarterly coupon exceed 2.25% or be less than 1.50%). After some discussions with Deutsche, it is understood that the investor will take the floater if the floater pays qu SP500 TR + 25 bps. Let’s build it. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Deutsche Bank Corporate issuer wants to pay fixed equity (SP500) swap pricing bid offer 4-year plain vanilla % % (these are qu rates) SP500 option pricing bid offer 4-year 2.00% qu SP500 TR cap bps bps 4-year 1.25% qu SP500 TR floor bps bps (premiums here are stated on a qu basis) Fixed Rate Structured Equity Product Coupon = max[min[qu SP500 TR + 25 bps, 2.25%], 1.50%] Investors want to hold collared floater 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Introduction to Exotic Options Note that by this structure the corporate has saved 5 bps qu relative to issuing a straight fixed-rate note. Corporate issuer wants to pay fixed 1.95% qu 1.6375% equity swap qu SP500 TR Deutsche Bank DPG Structured Equity Product 4 year max[qu SP500 TR %, 0] corporate sells 4-year cap to dealer (dealer agrees to pay corporate 5 bps qu) max[1.25% - qu SP500 TR, 0] corporate buys 4-year floor from dealer (corporate agrees to pay dealer bps qu) Coupon = max[min[qu SP500 TR + 25 bps, 2.25%], 1.50%] 5 bps qu Investors want to hold collared floater 11.25 bps qu 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Structure III: An Equity Range Floater Assume that a corporation could issue a four-year fixed rate note if it agreed to pay a fixed rate of 8.00%. An investor is willing to hold this corporation’s debt, but wants to receive a floating rate pegged to the quarterly total return on theSP500, provided that it stays within a very well defined range of 1.25% to 2.00%. The investor is short vol. The investor is willing to hold the range floater provided it pays qu SP500 TR + 50 bps while the qu SP500 TR is within the range and the investor is willing to accept nothing if it is outside the range. This structure requires all-or-nothing options. Let’s build it. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Introduction to Exotic Options Deutsche Bank Corporate issuer wants to pay fixed equity (SP500) swap pricing bid offer 4-year plain vanilla % % (rates are qu) SP500 all-or-nothing option pricing bid offer 4-year 2.00% qu SP500 TR cap bps bps 4-year 1.25% qu SP500 TR floor bps bps (premiums here are stated on an annual basis) Fixed Rate Structured Equity Product Coupon = qu SP500 TR + 50 bps if 1.25%  SP500  2.00% = 0 qu SP500 TR < 1.25% or > 2.00% max[ SP %, 0] —————————— × (SP bps) SP500 – 2.00% Investors want to hold range floater max[1.25% - SP500, 0] —————————— × (SP bps) 1.25% – SP500 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Introduction to Exotic Options Note that by this structure the corporate has saved 8.75 bps qu relative to issuing a straight fixed-rate note. Corporate issuer wants to pay fixed 1.9125% 1.6375% equity swap qu SP500 TR Deutsche Bank DPG Structured Equity Product 4 year max[SP %, 0] —————————— × (SP bps) SP % corporate sells 4-yr AoN cap to dealer max[1.25% - SP500, 0] —————————— × (SP bps) 1.25% - SP500 corporate sells 4-yr AoN floor to dealer Coupon = qu SP500 TR + 50 bps if 1.25%  SP500  2.00% = 0 if qu SP500 TR < 1.25% or > 2.00% 12.5 bps Investors want to hold range floater 10 bps 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Structure IV: An Equity-Linked Note Assume that a corporation could issue a four-year fixed rate note if it agreed to pay a fixed rate of 8.00% in two semiannual installments. An investor is willing to hold this corporation’s debt, but wants to receive a coupon tied to the performance of the QQQ. At the same time, the investor wants the principal on the note protected so that he is assured of full repayment at maturity. Finally, it is important that the coupon never be negative. Suppose that the investor would be willing to take a coupon tied to the total return on the QQQ (TRQQQ) and that payments will be made quarterly. Specifically, the note would pay max[qu TRQQQ bps, 0] each quarter. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Introduction to Exotic Options Note that by this structure the corporate has saved 3.75 bps relative to issuing a straight fixed-rate note. Corporate issuer wants to pay fixed 1.9625% 1.7125% equity (QQQ) swap Deutsche Bank DPG Structured Equity Product 4 year qu TRQQQ [1% - (qu TRQQ)] if qu TRQQQ < 1% equity trigger option Coupon = max[qu TRQQQ bps, 0] Investors want to equity-linked note for the floor, the corporate pays DB a premium that annualizes to 600 bps at the rate of 125 bps a quarter. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Structure V: Another Equity-Linked Note Assume that a corporation could issue a four-year fixed rate note if it agreed to pay a fixed rate of 8.00% in two semiannual installments. An investor is willing to hold this corporation’s debt. The investor wants to receive a fixed coupon and would accept 3.00% p.a., provided that the investor would also receive that portion of the 4-year total return on a specified hedge fund that is in excess of 48%, provided that that is a positive sum. Here Deutsche Bank would sell a four-year call option whose pay out is as follows: Payout at end of four years = max[TRHF - 48%, 0] DB would charge 450 bps a year in semiannual installments of 225 bps each. DB would have to assume an equity position in the fund in order to delta hedge. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Introduction to Exotic Options Note that by this structure the corporate has saved 50 bps relative to issuing a straight fixed-rate note. Corporate issuer wants to pay fixed 7.50% 4.50% sa Deutsche Bank DPG Structured Equity Product 4 year max[TRHF - 48%, 0] (payable at the end of 4 years) Coupon = 3% Investors want to equity-linked note Par at maturity = max[TRHF - 48%, 0] × 100 measured over 4 years 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Hedging Complexities Occasioned by Exotic Options. Unfortunately for dealers, some exotic options occasion additional hedging complexities that tend to preclude the application of traditional price risk management approaches, such as dynamic delta-gamma-vega hedging. And so hedging price risk at the local or book level can be difficult for dealers of some exotic options. A relatively new hedging technology, called static replication, can overcome some of these complexities and is discussed shortly. For now, however, we turn our attention to the complexities themselves. The ones that we will discuss are illustrative, and include multiple greeks for correlation options, discontinuous greeks for barrier options, and large greeks for digital options. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Take a correlation option like an equity ratio option that has pay off MAX[1,S1/S2] x NP. This option will have two delta’s - one for S1 and one for S2. A dealer can delta hedge in each asset separately. But this option has three gamma’s: the second derivative with respect to S1, the second derivative with respect to S2, and a (transitive) cross partial derivative. It is impractical to hedge the cross-partial gamma at the local or book level, as the dealer would have to find a traded option that involved the same to state variables. The correlation option would also have correlation risk, which is sometimes called chi risk. In order to chi hedge, the dealer would again have to find a traded option involving the same two state variables and therefore the same source of correlation. This is not practical. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Now consider barrier options. A problem faced by dealers here is that said options have discontinuous greeks. Their greeks “jump” and that can make traditional dynamic delta-gamma-vega hedging impractical. For example, a down-and-in put will have a delta that jumps when the in-barrier is touched. A dealer who is short the down-and-in put will need to short the underlying stock in order to delta hedge at that time. But the dealer may not be able to short the stock as its price has been falling (thus hitting the barrier) and there are up-tick and similar trading regulations that frustrate the hedging strategy. As a final example of just some of the additional hedging complexities associated with exotic options, consider an all-or-nothing call with strike $100 and pay off $10 million that is one day from expiration and the reference asset price is say $99.90 and a vol of say 30%. This option’s greeks will be huge. Indeed, its delta will be about 250,000 per share, implying that the dealer would need to tie up a credit line of about $25 million in order to delta hedge this one option - one that may be worthless a day from today. That is a big problem as those of you who are familiar with balance sheet considerations know too well. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Static Replication: Static replication was developed in the early 1990’s as a technique to permit exotic option dealers to hedge price risk, that is, to handle some of the added complexities just discussed. The basic idea is to form a portfolio of plain vanilla options that replicates the pricing of the exotic all along its boundary conditions. The portfolio of plain vanilla options would be held until the hedge was to be lifted, e.g., upon knock-out, and thus the name “static replication”. Like other hedging techniques, e.g., dynamic delta-gamma-vega hedging, static replication has shortcomings and is not a panacea. Here we introduce the technique via an example involving an up-and-out call. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Example: Suppose that we write a European-style, nine-month (T - t) up-and-out call with strike price $50. The up barrier is $60, meaning that if the underlying asset price should touch $60 during the nine-month term, then the option would be knocked out and would be worthless. Thus in order to generate any pay off, this particular option would have to survive the entire term, i.e., the reference asset could never touch or exceed $60, and the terminal asset price would have to be greater than $50 per share (X). Assume that the asset’s implied volatility surface is flat at 40% and that the relevant interest rate term structure is flat at 5%. The theoretical/Black-Scholes-Merton price of this barrier option is about $0.15 per share, albeit the actual market price would likely be substantially different. Let us build a small replicating portfolio. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Continuing, we can illustrate the exotic option’s boundary conditions using a crude price grid. Discuss: S (60,0.00) . (60,0.25) . (60,0.50) . t 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Continuing, we start to build the replicating portfolio by buying the closest plain vanilla, so here we would buy 1 nine-month, X = 50 call. Call this Option A. Option A would have a value of $11.57 at coordinate (60,0.50). Obviously, this does not comport with the barrier option whose value is zero at the coordinate. So we need to introduce a second option, B. Let B we a nine-month, X = 60 call. It will have value $5.13 at coordinate (60,0.50). Therefore we need to short/write 2.26 of Option B: [(1 x $11.57) + (-2.26 x $5.13) = $0. All other boundary points are satisfied. Now roll back to coordinate (60,0.25). Option A will have value $13.27 and Option B will have value $7.43 here. And so we violate the boundary condition and must introduce a third option, C. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Continuing, let C be a six-month, X = 60 call and so it will have value $5.13 at (60,0.25). The amount of Option C required is given by (1)(13.27) + (-2.26)(7.43) + wC(5.13) = 0, and wC = So we buy 0.69 of Option C. We continue to match along all other boundary points. Finally, roll back to coordinate (60,0.00). Here the values of A, B, and C are $14.78, $9.25, and $7.43, respectively, and so we need to introduce a fourth option, D, to achieve a combined value of zero. Letting Option D be a three-month, X = 60 call insures a value for D of $5.13 at coordinate (60,0.00). We solve (1)(14.78) + (-2.26)(9.25) + (0.69)(7.43) + wD(5.13) = 0, and wD = We long 0.19 of D. Again, we continue to match along all other boundary points. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Continuing, thus our replicating portfolio of plain-vanilla call options is long 1 A, short 2.26 B, long 0.69 C and long 0.19 D (per share). Is this a good replicating portfolio? The answer is no, because the grid was too crude, the replicating portfolio was too small. Put another way, we only checked if the knock out price of $60 was hit at three times (t = 0.00, 0.25 and 0.50) over the barrier option’s life, whereas it could have been hit at any time. Another way to see this result is by inspecting the value of the replicating portfolio now, that is, at the coordinate (50,0.00). It is about $0.46, which is no where near the theoretical price of the barrier option, $ The two would converge as the grid was made finer. 01/01/02 Deutsche Bank: Exotic Options Copyright (c) by Marshall, Tucker & Associates, LLC

Deutsche Bank Introduction to Exotic Options Alan L. Tucker, Ph. D

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