real options in dynamic pricing and revenue management
TRANSCRIPT
Real Options and Dynamic Pricing 1
Real Options in Dynamic Pricing and Revenue Management
Chris AndersonIvey School of Business, Univ. of Western Ontario
London, Ontario, [email protected]
Real Options and Dynamic Pricing 2
Agenda• Financial Options• Real Options• Services & Dynamic Pricing/ RM• RO & RM
– Motivation (car rentals)– Pricing and demand models
• RO and low price guarantees
Real Options and Dynamic Pricing 3
Options• Derivatives• A financial instrument that gives you the
right to buy or sell a share at a specified price
• Call option - the right to buy at a specific price (the exercise price)
• Put option - the right to sell at a specific price (the exercise price)
Real Options and Dynamic Pricing 4
Basic Types of Options
• European– call - gives the owner the right to buy on a
specific date– put - gives the owner the right to sell on a
specific date• American
– gives the right to buy or sell at any time prior to a specific date
Real Options and Dynamic Pricing 5
Other Terminology
• A long position - you actually own the security– e.g. the way most of us invest in stocks or
mutual funds• A short position - you have sold a security
that you don’t own• Note: if you sell or write an option contract,
you have a short position on that contract
Real Options and Dynamic Pricing 6
Why Buy Options
1. Cheaper than underlying securities - can get a huge position on a security for a low price
2. Risk management - option pays if stock rises or falls by a large amount - can protect your portfolio from a very volatile market
3. Regulatory reasons (e.g. some places don’t allow short selling)
Real Options and Dynamic Pricing 7
Value of a European Call• Gives you the right to buy a stock for $K at
some future date, T• When would you use it?• If the stock price, ST, is bigger than K, then
you exercise your option and buy a share for $K, then immediately sell it for $ST, and make a profit of ST - K
• If the future price is less than $K, you do nothing
• Thus, the value at time T is max(ST-K,0)
Real Options and Dynamic Pricing 8
Value of a European Put• Gives you the right to sell a stock for $K at
some future date, T• When would you use it?• If the stock price, ST, is smaller than K, then
you buy a share on the market for $ST, then exercise your option and sell it for $K, and make a profit of K-ST
• If the future price is more than $K, you do nothing
• Thus, the value at time T is max(K-ST,0)
Real Options and Dynamic Pricing 9
Payoff Diagrams
Share Price
Share Price
Value
Value
Buy a share
Sell (short) a share 1010
10
10
Real Options and Dynamic Pricing 10
Payoff Diagrams
SharePrice
SharePrice
Value
Value
SharePrice
SharePrice
Value
Value
K
K
K
K
Buy a Call
Sell a Call
Buy a Put
Sell a Put
Real Options and Dynamic Pricing 11
Methods for Pricing
Real Options and Dynamic Pricing 12
Valuing Options by Arbitrage Methods
If an investment has no risk, it should yield the risk-free rate of return (T-Bills), if not we can create wealth –money making machine
Real Options and Dynamic Pricing 13
Example
• Stock trades at $40• European call has exercise price of $40• Risk free rate is 1/9% (per period)• A very simplified world...• In one period, one of two things will happen:
– stock trades at $32– stock trades at $50
• Form a Portfolio– x shares of stock, sell 1 call
Real Options and Dynamic Pricing 14
Arbitrage Pricing
32x-03250x-1050PortfolioStock Price
W/ no risk, portfolio must be equal under both stock price scenarios
50x-10=32x 18x=10 or x=5/9
A portfolio of 5/9 shares, short a call – no risk, gen. Risk free return
Value in 1 period * 1/(1+r) = initial value
5/9*40 –c =1/(1+1/9)*32*5/9 c=56/9
Real Options and Dynamic Pricing 15
2nd Example
• Stock trades at $20• European call has exercise price of $21• Risk free rate is 12% • A very simplified world...• In 3 months, one of two things will happen:
– stock trades at $18– stock trades at $22
• Form a Portfolio– 0.25 shares of stock, sell 1 call
Real Options and Dynamic Pricing 16
Example
Bad
GoodValue of stocks = .25×22 = $5.50Value of options = -1×(22-21) = -$1.00Value of portfolio = $5.50-$1.00 = $4.50
Value of stocks = .25×18 = $4.50Value of options = -1×(0) = 0Value of portfolio = $4.50
This construction gives us a risk-free portfolio whose value is the sameno matter what happens in the future!
Real Options and Dynamic Pricing 17
Risk Neutral Pricing
• If this portfolio is not subject to risk, then investors must be indifferent between this portfolio and a risk free bond with the same payoff ($4.50) in 3 months
• Why? If they weren’t, you could buy one and sell the other to create a risk free “money pump”
Real Options and Dynamic Pricing 18
Value of portfolio in three months = $4.50
Value of portfolio now = 4.50 × e-.12 ×.25 = $4.37
4.37 = .25 × 20 - 1 × (price of call today)
price of call today = 5.00 - 4.37 = .63
And if it wasn’t $0.63, we would have an arbitrage opportunity.
Real Options and Dynamic Pricing 19
• In a risk-neutral world, investors do not demand any premium to take on extra risk
• (In the real world, risky investments have a higher average growth rate than safe ones - a risk-return tradeoff.)
• Thus, in a risk-neutral world, all assets grow at the risk free rate.
• Why? If asset A grew faster than asset B, all investors would prefer A since they are neutral to risk.
• We use this observation to determine the probability that the stock price rises or falls in a risk-neutral world.
Real Options and Dynamic Pricing 20
Bad
Good
p
1-p
$20
$22
$18
EV stock price= p×22 + (1-p)×18= 18+4p
Let p = probability that the stock goes up in a risk-neutral world.
Since investors are risk neutral, the stock grows on average at the risk-free rate.
Price in 3 months = 20 × e.12×.25 = $20.61
Then the expected stock price after 3 months must equal $20.61.
Thus, 20.61 = 18+4pp=.65
Real Options and Dynamic Pricing 21
Bad
Good
.65
.35
$20
$22; option value = $1
$18; option value = $0
Determining the Option’s Value
At 3 months:EV option = .65×1 + .35×0 = .65
EV now = .65×e-.12×.25 = $0.63
Real Options and Dynamic Pricing 22
Basic Approaches to Pricing
• For “vanilla” European options (puts and calls), a formula exists
• For exotic European options, can simulate• For American-style options, need to use a
decision tree approach
Real Options and Dynamic Pricing 23
The Black-Scholes Formula
• Scholes, Merton received Nobel Prize in Economics in 1997
• Based on dynamic application of risk neutral pricing
Real Options and Dynamic Pricing 24
The Black-Scholes Formula20
1
1ln2f
S r TKd
T
σ
σ
+ + = 2 1d d Tσ= −
Value of a call: ( ) ( )0 1 2fr TC S N d Ke N d−= −
Value of a put: ( ) ( )2 0 1fr TP Ke N d S N d−= − − −
What is N(d1)?
Real Options and Dynamic Pricing 25
N(d1)
Note: N(-d1) = 1-N(d1)in Excel, N(d1) = normsdist(d1) or normdist(d1,0,1,1)
0
0.05
0.10.15
0.2
0.25
0.30.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 4d1
So, N(d1) = the probability that the return is less than a certain amount
Real Options and Dynamic Pricing 26
Pricing via Simulation
• Basic premise of finance – an asset’s value is derived from its future discounted (expected) cash flow
• Simulate the underlying value driver or asset (stock)
• Calculate payoffs• Replicate• Average payouts, discount
Real Options and Dynamic Pricing 27
Lognormal model of stock prices
• Over time stock goes up
• More uncertainty farther out try to estimate
• Positive values
] ) )T,2
((exp[
) )T,2
((ln~ln
) , (~
2
0
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TuSS
TuSS
ttuSS
T
T
σσφ
σσφ
δσδφδ
−=
−+
Real Options and Dynamic Pricing 28
8090
100110120130140
1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91
Yes
No
No
Consider a (European) call option with an exercise price of $120.In which cases does it have value at the end of 3 months?
Pricing Options
Real Options and Dynamic Pricing 29
Real Options
Many investments, not just those involving stocks, may be viewed as combinations of puts and calls – if we know the value of the puts/calls we can value many real investment opportunities
Real Options and Dynamic Pricing 30
Managerial View of Real Options • RO is a modern methodology for economic evaluation of
projects and investment decisions under uncertainty– RO approach complements the corporate tools
• RO considers the uncertainties and the options (managerial flexibilities), giving two answers: – The value of the investment opportunity (value of the option); and – The optimal decision rule (threshold)
• RO can be viewed as an optimization problem:– Maximize the NPV (typical objective function) subject to: – (a) Market uncertainties (price);– (b) Technical uncertainties (volume) and– (c) Relevant Options (managerial flexibilities)
Real Options and Dynamic Pricing 31
Value in the Real OptionReal options increase in value as greater the uncertainties and the
flexibility to respondA
bilit
y to
res
pond
Low
HighLikelihood of receiving new informationLow High
U n c e r t a i n t y
Roo
m fo
r M
anag
eria
l Fle
xibi
lity
Moderate Flexibility Value
Moderate Flexibility Value
Low Flexibility Value
High Flexibility Value
Real Options and Dynamic Pricing 32
•Option to purchase an airplane 3 years from now for $20 million, P=value(t=3), P uncertain (economic cycle etc..), cash flow max(P-20,0) – call option, if you can value the call, you can value the option to purchase
•Abandonment Option a R&D project, in 5 years can sell devl’t for $80 million, P= value(t=5), value of option max(80-P,0) – put
•Expansion – option at t to double investment
•Contraction – option at t to cut scale
•Postponement – option to delay launch till time t
•Pioneer – option to enter new markets at time t, buy –ve NPV firm
•Flexibility – build expensive plant that can build three types (cars) versus one
•Licensing – license a drug, such that if sales > $50 million get 20% gross sales (as developer)
Real Options and Dynamic Pricing 33
Complexities of Service
• In manufacturing, we assume that capacity can be adjusted over time to match supply w/ demand
• In services– Capacity is often fixed– Outputs rarely storable– Sales opportunity lost if not met– Demand often temporal
Real Options and Dynamic Pricing 34
Coping Strategies for time-varying demand.
•Inventories, overtime, backlogging, and many of the other strategies we use for production planning aren’t available to us in service businesses.
•How do service operations managers address the problem of matching capacity to time varying demand?
•With pricing tactics
Real Options and Dynamic Pricing 35
DP in practice
Time
Automobile
Apparel
PC
Airline TicketRel
ativ
e Pr
ice
Real Options and Dynamic Pricing 36
DP&RM - the basics
• Setting and updating prices with a wide variety of customers, products, or channels.
• Aligning prices with market conditions– Customer sensitivity– Competition’s pricing– Corporate objectives
• Airlines, hotels, rental cars, fashion goods, more each day
Real Options and Dynamic Pricing 37
Current Approaches
• Marginal analysis (inc. gain vs. inc. loss)
• Math Programming (usually deterministic)
• Threshold curves (comparison to historical perf.)
• “Managerial experience”
Real Options and Dynamic Pricing 38
Focus of Car Rentals
• 90 day planning horizon, relatively fixed capacity (sunk costs), very low variable costs
• Decisions– Price to post, accept or deny a request for rental– LOR, upgrades, overbooking
Real Options and Dynamic Pricing 39
Optionality
• View reservation as an option, exercised if booking allowed, held if capacity reserved
• Tradeoff between rate today & potential higher rate (uncertain demand) later
• Call option w/limited demand
Real Options and Dynamic Pricing 40
0 2 4 6 8 10 12 1414
16
18
20
22
24
26
28
30
32
Weeks prior to pickup
Ave
rage
dai
ly r
ate
$
Weekly Rentals
3 Day Rentals
Daily Rentals
How many do I rent today at P(t)?
Real Options and Dynamic Pricing 42
A Price and Demand Model
PtPbPtPtPu
dXtPbdttPudP
σα
=−=
+=
),())((),(
),(),(
10
10ββ
ββ
PD
PD
=
+=
Market price is key driver!
Real Options and Dynamic Pricing 43
Payouts
)(,
),...,2,,max(
vehiclesunrented from revenue no 0
112
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t
jj
kmjj
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θ
θ
θ
M cars to rent over time T subdivided into periods (daily)
Real Options and Dynamic Pricing 44
Valuing the option
0)(21
another andcar of typeone of portfolio
stock offraction andoption of portfolio
2
222
21
=−∂∂
−+∂∂
+∂∂
∏=∏
∆−=∏
∆−=∏
rVPVP
PVP
tV
dtrd
VV
SV
λσµσ
Real Options and Dynamic Pricing 45
Solution
process. price in they volatilitlowunder Alsoprices. large
andcapacity excess of conditionsunder Analyticalsolution. numerical Requires
0)(21
2
222 =−
∂∂
−+∂∂
+∂∂ rV
PVP
PVP
tV λσµσ
Real Options and Dynamic Pricing 46
0
2
4
6
8
10
12
0 10 20 30 40 50 60
Num
ber
of C
ars
Rental Price
N=12 (1 week)
M=50,r= 5%
Pmin=25, Pmax=30
Real Options and Dynamic Pricing 47
More general approaches
))(ˆ())((),(
),(),(
))(ˆ(
),(
))((),(),(),(
PtPPtPtPu
dXtDbdttPudP
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−−=
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αα
α
σ
α
Real Options and Dynamic Pricing 48
Decaying Price (Perishables)
0
500
1000
1500
2000
2500
0 5 10 15 20 25 30
Real Options and Dynamic Pricing 49
Decaying PriceFashion goods, electronics
demand.linear not, ifdemand lexponentia have then offunction a is if
effects priceown andon substituti -
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),(),(
1
Du
PPtPtPu
dXtDbdttPudD
n−=
+=
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Real Options and Dynamic Pricing 50
Low Price Guarantees• Motivation
– DTAG pays 15% commissions on bookings through 3rd party websites (Expedia, Travelocity, Orbitz, etc…) versus allocated costs of $0.75/rental for bookings on Dollar.com and Thrifty.com
• Book now, if rates drop will rebate!– Common in cruise industry
• Most Favoured Customer & Meet Or Release– Retailers, big box stores
Real Options and Dynamic Pricing 51
0
0.2
0.4
0.6
0.8
1
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2
Days prior to pickup
% B
ooki
ngs
Real Options and Dynamic Pricing 52
15%2.2%Other Internet sites
15%5.6%Internet Site C
15%7.5%Internet Site B
15%14.5%Internet Site A
15%12.9%Travel agent bookings
$0.7526.5%Dollar.com, Thrifty.com
$6.0016.3%800 NUMBER
5%14.3%WALKUPS
Cost to DTAG(per rental)
% of totalChannel
29.8%
Real Options and Dynamic Pricing 53
35
40
45
50
55
60
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Days prior to pickup
Daily
Rat
e
Real Options and Dynamic Pricing 54
Real Options and Dynamic Pricing 55
Goal – reduce distribution costs
• Move traffic from 3rd party sites to DTAG sites• Two elements in promo
1. DTAG sites have lowest DTAG rates2. DTAG rebates consumer if rates drop after they have booked
• 1 is simply spin as all channels use the same rate engine
• 2 might be very costly• Cost?• Break-even?
• 2 actually already exists!
Real Options and Dynamic Pricing 56
0
10
20
30
40
50
5/1/03
5/3/03
5/5/03
5/7/03
5/9/03
5/11/0
35/1
3/03
5/15/0
35/1
7/03
5/19/0
35/2
1/03
5/23/0
35/2
5/03
5/27/0
35/2
9/03
5/31/0
3
6/1/2003
Real Options and Dynamic Pricing 57
LPG Option
• Consumer gets a free option• Payout
]]|[,0max[
],0max[pickup]n toreservatioof timefromlowest -priceReserved ,0max[
)(t
Ttt
tTr
Ttt
PmEPe
mP
−
−−−
Real Options and Dynamic Pricing 58
LPG Option – assume LogN
]ln[~
))(ln
()())(ln
(
0
02
0 2
Tt
t
t
Tt
tTt
mPrP
tT
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tT
tTmP
N
>
−
−+−
−
−+
σ
µ
σ
µσµ
P0
mt0
Pt
Res. PickupvalueT
Real Options and Dynamic Pricing 59
LPG Option price
−−
++−
+
++
−−=
+
+−
−−
)())(2(2
))(2(),,(
)2
(22
2
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2
dNedNPPPe
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t
ot
r
r
tt
t
τµσσµσ
τ
τµσ
στµσ
µσσ
στµσ
Real Options and Dynamic Pricing 60
00.5
11.5
22.5
33.5
44.5
5
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2
Days prior
Pric
e
Volatility Impacts
Real Options and Dynamic Pricing 61
0.00
1.00
2.00
3.00
4.00
5.00
6.00
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
DOLLAR ENTERPRISE HERTZ ALAMO GBM
Intermediate cars, overnight DFW
Real Options and Dynamic Pricing 62
Res Build
0
0.050.1
0.150.2
0.25
30 28 26 24 22 20 18 16 14 12 10 8 6 4 2
Days out
% o
f tot
al
DCA DFW IAD SFO
Real Options and Dynamic Pricing 63
Impact• Sumt (Value * % booked)• ~$1.80
• Implications• Need to move a lot of traffic• Market share
• Cancellation fee?
Real Options and Dynamic Pricing 64
Other Models
• Mean-reversion• Exponential curve
• Exponential smoothing• Moving average
Real Options and Dynamic Pricing 65
Real Options and Dynamic Pricing 66