Real Opions in Dynamic ricing and Revenue Managemen Chris Anderson Ivey School of Business, Univ. of Wesern Onario London, Onario, Canada canderson@ivey.ca Real Opions and Dynamic ricing 1
Agenda Financial Opions Real Opions Services & Dynamic ricing/ RM RO& RM Moivaion (car renals ricing and demand models RO and low price guaranees Real Opions and Dynamic ricing 2
Opions Derivaives A financial insrumen ha gives you he righ o buy or sell a share a a specified price Call opion - he righ o buy a a specific price (he exercise price u opion - he righ o sell a a specific price (he exercise price Real Opions and Dynamic ricing 3
Basic Types of Opions European call - gives he owner he righ o buy on a specific dae pu - gives he owner he righ o sell on a specific dae American gives he righ o buy or sell a any ime prior o a specific dae Real Opions and Dynamic ricing 4
Oher Terminology A long posiion - you acually own he securiy e.g. he way mos of us inves in socks or muual funds A shor posiion - you have sold a securiy ha you don own Noe: if you sell or wrie an opion conrac, you have a shor posiion on ha conrac Real Opions and Dynamic ricing 5
Why Buy Opions 1. Cheaper han underlying securiies - can ge a huge posiion on a securiy for a low price 2. Risk managemen - opion pays if sock rises or falls by a large amoun - can proec your porfolio from a very volaile marke 3. Regulaory reasons (e.g. some places don allow shor selling Real Opions and Dynamic ricing 6
Value of a European Call Gives you he righ o buy a sock for $K a some fuure dae, T When would you use i? If he sock price, S T, is bigger han K, hen you exercise your opion and buy a share for $K, hen immediaely sell i for $S T, and make a profi of S T -K If he fuure price is less han $K, you do nohing Thus, he value a ime T is max(s T -K,0 Real Opions and Dynamic ricing 7
Value of a European u Gives you he righ o sell a sock for $K a some fuure dae, T When would you use i? If he sock price, S T, is smaller han K, hen you buy a share on he marke for $S T, hen exercise your opion and sell i for $K, and make a profi of K-S T If he fuure price is more han $K, you do nohing Thus, he value a ime T is max(k-s T,0 Real Opions and Dynamic ricing 8
ayoff Diagrams Value Buy a share 10 10 Share rice Value Sell (shor a share 10 10 Share rice Real Opions and Dynamic ricing 9
ayoff Diagrams Buy a Call Value Value Buy a u K Share rice K Share rice Value Value K K Share rice Share rice Sell a Call Sell a u Real Opions and Dynamic ricing 10
Mehods for ricing Real Opions and Dynamic ricing 11
Valuing Opions by Arbirage Mehods If an invesmen has no risk, i should yield he risk-free rae of reurn (T-Bills, if no we can creae wealh money making machine Real Opions and Dynamic ricing 12
Example Sock rades a $40 European call has exercise price of $40 Risk free rae is 1/9% (per period A very simplified world... In one period, one of wo hings will happen: sock rades a $32 sock rades a $50 Form a orfolio x shares of sock, sell 1 call Real Opions and Dynamic ricing 13
Arbirage ricing Sock rice 50 32 orfolio 50x-10 32x-0 W/ no risk, porfolio mus be equal under boh sock price scenarios 50x-10=32x 18x=10 or x=5/9 A porfolio of 5/9 shares, shor a call no risk, gen. Risk free reurn Value in 1 period * 1/(1+r = iniial value 5/9*40 c =1/(1+1/9*32*5/9 c=56/9 Real Opions and Dynamic ricing 14
2 nd Example Sock rades a $20 European call has exercise price of $21 Risk free rae is 12% A very simplified world... In 3 monhs, one of wo hings will happen: sock rades a $18 sock rades a $22 Form a orfolio 0.25 shares of sock, sell 1 call Real Opions and Dynamic ricing 15
Example This consrucion gives us a risk-free porfolio whose value is he same no maer wha happens in he fuure! Good Value of socks =.25 22 = $5.50 Value of opions = -1 (22-21 = -$1.00 Value of porfolio = $5.50-$1.00 = $4.50 Bad Value of socks =.25 18 = $4.50 Value of opions = -1 (0 = 0 Value of porfolio = $4.50 Real Opions and Dynamic ricing 16
Risk Neural ricing If his porfolio is no subjec o risk, hen invesors mus be indifferen beween his porfolio and a risk free bond wih he same payoff ($4.50 in 3 monhs Why? If hey weren, you could buy one and sell he oher o creae a risk free money pump Real Opions and Dynamic ricing 17
Value of porfolio in hree monhs = $4.50 Value of porfolio now = 4.50 e -.12.25 = $4.37 4.37 =.25 20-1 (price of call oday price of call oday = 5.00-4.37 =.63 And if i wasn $0.63, we would have an arbirage opporuniy. Real Opions and Dynamic ricing 18
In a risk-neural world, invesors do no demand any premium o ake on exra risk (In he real world, risky invesmens have a higher average growh rae han safe ones - a risk-reurn radeoff. Thus, in a risk-neural world, all asses grow a he risk free rae. Why? If asse A grew faser han asse B, all invesors would prefer A since hey are neural o risk. We use his observaion o deermine he probabiliy ha he sock price rises or falls in a risk-neural world. Real Opions and Dynamic ricing 19
Since invesors are risk neural, he sock grows on average a he risk-free rae. rice in 3 monhs = 20 e.12.25 = $20.61 Then he expeced sock price afer 3 monhs mus equal $20.61. Le p = probabiliy ha he sock goes up in a risk-neural world. Good $22 $20 Bad p 1-p $18 EV sock price = p 22 + (1-p 18 = 18+4p Thus, 20.61 = 18+4p p=.65 Real Opions and Dynamic ricing 20
Deermining he Opion s Value Good $22; opion value = $1 $20.65 Bad.35 $18; opion value = $0 A 3 monhs: EV opion =.65 1 +.35 0 =.65 EV now =.65 e -.12.25 = $0.63 Real Opions and Dynamic ricing 21
Basic Approaches o ricing For vanilla European opions (pus and calls, a formula exiss For exoic European opions, can simulae For American-syle opions, need o use a decision ree approach Real Opions and Dynamic ricing 22
The Black-Scholes Formula Scholes, Meron received Nobel rize in Economics in 1997 Based on dynamic applicaion of risk neural pricing Real Opions and Dynamic ricing 23
The Black-Scholes Formula d 1 S0 1 2 ln + rf + σ T K 2 σ T = 2 1 d = d σ T rt 0 1 2 C = S N d Ke N d Value of a call: ( f ( rt = Ke N d S N d f Value of a pu: ( ( 2 0 1 Wha is N(d 1? Real Opions and Dynamic ricing 24
N(d 1 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0-4 -3-2 -1 0 1 d 2 3 4 1 Noe: N(-d 1 = 1-N(d 1 in Excel, N(d 1 = normsdis(d 1 or normdis(d 1,0,1,1 So, N(d 1 = he probabiliy ha he reurn is less han a cerain amoun Real Opions and Dynamic ricing 25
ricing via Simulaion Basic premise of finance an asse s value is derived from is fuure discouned (expeced cash flow Simulae he underlying value driver or asse (sock Calculae payoffs Replicae Average payous, discoun Real Opions and Dynamic ricing 26
Lognormal model of sock prices Over ime sock goes up More uncerainy δ farher ou ry o S esimae ln osiive values S S T ~ φ( u δ, σ δ 2 σ ST ~ φ(ln S0 + ( u T, σ 2 2 σ = S0 exp[ φ(( u T, σ T 2 T ] Real Opions and Dynamic ricing 27
ricing Opions Consider a (European call opion wih an exercise price of $120. In which cases does i have value a he end of 3 monhs? 140 130 120 110 100 90 80 Yes No No 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 Real Opions and Dynamic ricing 28
Real Opions Many invesmens, no jus hose involving socks, may be viewed as combinaions of pus and calls if we know he value of he pus/calls we can value many real invesmen opporuniies Real Opions and Dynamic ricing 29
Managerial View of Real Opions RO is a modern mehodology for economic evaluaion of projecs and invesmen decisions under uncerainy RO approach complemens he corporae ools RO considers he uncerainies and he opions (managerial flexibiliies, giving wo answers: The value of he invesmen opporuniy (value of he opion; and Theopimal decision rule (hreshold RO can be viewed as an opimizaion problem: Maximize he NV (ypical objecive funcion subjec o: (a Marke uncerainies (price; (b Technical uncerainies (volume and (c Relevan Opions (managerial flexibiliies Real Opions and Dynamic ricing 30
Value in he Real Opion Real opions increase in value as greaer he uncerainies and he flexibiliy o respond High Low Likelihood of receiving new informaion U n c e r a i n y High Abiliy o respond Low Room for Managerial Flexibiliy Moderae Flexibiliy Value Low Flexibiliy Value High Flexibiliy Value Moderae Flexibiliy Value Real Opions and Dynamic ricing 31
Opion o purchase an airplane 3 years from now for $20 million, =value(=3, uncerain (economic cycle ec.., cash flow max(-20,0 call opion, if you can value he call, you can value he opion o purchase Abandonmen Opion a R&D projec, in 5 years can sell devl for $80 million, = value(=5, value of opion max(80-,0 pu Expansion opion a o double invesmen Conracion opion a o cu scale osponemen opion o delay launch ill ime ioneer opion o ener new markes a ime, buy ve NV firm Flexibiliy build expensive plan ha can build hree ypes (cars versus one Licensing license a drug, such ha if sales > $50 million ge 20% gross sales (as developer Real Opions and Dynamic ricing 32
Complexiies of Service In manufacuring, we assume ha capaciy can be adjused over ime o mach supply w/ demand In services Capaciy is ofen fixed Oupus rarely sorable Sales opporuniy los if no me Demand ofen emporal Real Opions and Dynamic ricing 33
Coping Sraegies for ime-varying demand. Invenories, overime, backlogging, and many of he oher sraegies we use for producion planning aren available o us in service businesses. How do service operaions managers address he problem of maching capaciy o ime varying demand? Wih pricing acics Real Opions and Dynamic ricing 34
D in pracice Relaive rice Auomobile Apparel C Airline Ticke Time Real Opions and Dynamic ricing 35
D&RM - he basics Seing and updaing prices wih a wide variey of cusomers, producs, or channels. Aligning prices wih marke condiions Cusomer sensiiviy Compeiion s pricing Corporae objecives Airlines, hoels, renal cars, fashion goods, more each day Real Opions and Dynamic ricing 36
Curren Approaches Marginal analysis (inc. gain vs. inc. loss Mah rogramming (usually deerminisic Threshold curves (comparison o hisorical perf. Managerial experience Real Opions and Dynamic ricing 37
Focus of Car Renals 90 day planning horizon, relaively fixed capaciy (sunk coss, very low variable coss Decisions rice o pos, accep or deny a reques for renal LOR, upgrades, overbooking Real Opions and Dynamic ricing 38
Opionaliy View reservaion as an opion, exercised if booking allowed, held if capaciy reserved Tradeoff beween rae oday & poenial higher rae (uncerain demand laer Call opion w/limied demand Real Opions and Dynamic ricing 39
32 30 How many do I ren oday a (? 28 Average daily rae $ 26 24 22 20 3 Day Renals Daily Renals 18 16 Weekly Renals 14 0 2 4 6 8 10 12 14 Weeks prior o pickup Real Opions and Dynamic ricing 40
Real Opions and Dynamic ricing 42 A rice and Demand Model b u dx b d u d σ α = = + =, ( ( (, (, (, ( 1 0 1 0 β β β β D D = + = Marke price is key driver!
Real Opions and Dynamic ricing 43 ayous (,,..., 2,, max( 0 no revenue from unrened vehicles 1 1 2 1 1 1 1 1 1 j j k m j j m j j m j m j j k m j j m T m F k M k m k V V V V k V V V + + + + = + = + = = + + + + + + + + + + + θ θ θ M cars o ren over ime T subdivided ino periods (daily
Valuing he opion = V S porfolio of opion and fracion of sock = V V + 1 V 2 porfolio of one ype of car and anoher d = r d 1 2 σ 2 2 2 V 2 + ( µ λσ V rv = 0 Real Opions and Dynamic ricing 44
Soluion 2 V 1 2 2 V + σ + ( µ λσ 2 2 Requires numerical soluion. V rv = 0 Analyical under condiions of excess capaciy and large prices. Also under low volailiy in he price process. Real Opions and Dynamic ricing 45
12 10 N=12 (1 week M=50,r= 5% min =25, max =30 8 Number of Cars 6 4 2 0 0 10 20 30 40 50 60 Renal rice Real Opions and Dynamic ricing 46
Real Opions and Dynamic ricing 47 More general approaches ˆ( ( ( (, (, (, ( ˆ( (, ( ( (, (, (, ( u dx D b d u d d dd b u dx b d u d d p d d d d p = + = = = = + = α α α σ α
Decaying rice (erishables 2500 2000 1500 1000 500 0 0 5 10 15 20 25 30 Real Opions and Dynamic ricing 48
Decaying rice Fashion goods, elecronics dd = u(, d + b( D, dx ˆ( n u(, = α1( - subsiuion and own price effecs if u is a funcion of D hen have exponenial if no, linear demand. demand Real Opions and Dynamic ricing 49
Low rice Guaranees Moivaion DTAG pays 15% commissions on bookings hrough 3 rd pary websies (Expedia, Travelociy, Orbiz, ec versus allocaed coss of $0.75/renal for bookings on Dollar.com and Thrify.com Book now, if raes drop will rebae! Common in cruise indusry Mos Favoured Cusomer & Mee Or Release Reailers, big box sores Real Opions and Dynamic ricing 50
20 18 16 14 12 10 8 6 4 2 22 1 0.8 0.6 0.4 0.2 0 Days prior o pickup Real Opions and Dynamic ricing 51 24 26 28 30 % Bookings
Channel WALKUS 800 NUMBER Dollar.com, Thrify.com Travel agen bookings Inerne Sie A Inerne Sie B Inerne Sie C Oher Inerne sies % of oal 14.3% 16.3% 26.5% 12.9% 14.5% 7.5% 5.6% 2.2% Cos o DTAG (per renal 5% $6.00 $0.75 15% 15% 15% 15% 15% 29.8% Real Opions and Dynamic ricing 52
60 55 Daily Rae 50 45 40 35 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Days prior o pickup Real Opions and Dynamic ricing 53
Real Opions and Dynamic ricing 54
Goal reduce disribuion coss Move raffic from 3 rd pary sies o DTAG sies Two elemens in promo 1. DTAG sies have lowes DTAG raes 2. DTAG rebaes consumer if raes drop afer hey have booked 1 is simply spin as all channels use he same rae engine 2 migh be very cosly Cos? Break-even? 2 acually already exiss! Real Opions and Dynamic ricing 55
5/11/03 5/13/03 5/15/03 5/17/03 5/19/03 5/21/03 5/23/03 5/25/03 5/27/03 5/29/03 5/31/03 50 40 30 20 10 0 6/1/2003 Real Opions and Dynamic ricing 56 5/9/03 5/7/03 5/5/03 5/1/03 5/3/03
LG Opion Consumer ges a free opion ayou max[0, Reserved price - lowes from ime of T max[0, m ] e r( T max[0, E[ m T ]] reservaion o pickup] Real Opions and Dynamic ricing 57
Real Opions and Dynamic ricing 58 LG Opion assume LogN ] ln [ ~ ( ln ( ( ( ln ( 0 0 2 0 2 T T T m r T T m N m T T m N > + + σ µ σ µ σ µ 0 m 0 Res. ickup value T
Real Opions and Dynamic ricing 59 LG Opion price + + + + + = + + ( 2( ( 2 2( (,, ( 2 ( 2 2 2 2 2 2 ( 0 2 2 2 2 d N e d N e d N e T m p o r r τ µ σ σ µ σ τ τ µ σ σ τ µ σ µ σ σ σ τ µ σ
rice 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 30 28 26 24 Volailiy Impacs 22 20 18 16 14 12 10 8 6 4 2 Days prior Real Opions and Dynamic ricing 60
Inermediae cars, overnigh DFW 6.00 5.00 4.00 3.00 2.00 1.00 0.00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 DOLLAR ENTERRISE HERTZ ALAMO GBM Real Opions and Dynamic ricing 61
Res Build % of oal 0.25 0.2 0.15 0.1 0.05 0 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 Days ou DCA DFW IAD SFO Real Opions and Dynamic ricing 62
Impac Sum (Value * % booked ~$1.80 Implicaions Need o move a lo of raffic Marke share Cancellaion fee? Real Opions and Dynamic ricing 63
Oher Models Mean-reversion Exponenial curve Exponenial smoohing Moving average Real Opions and Dynamic ricing 64
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Quesions canderson@ivey.ca Real Opions and Dynamic ricing 67