Transcript No Slide Title
Arbitrage Enforced Valuation + Black-Scholes
Richard de Neufville Professor of Engineering Systems and of Civil and Environmental Engineering MIT Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 1 of 25 Arbitrage Enforced Valuation Black Scholes ©
Outline
Background
Application of Arbitrage-Enforced Valuation to Binomial
The Formula
–
Applicability ; Interpretation ; Intuition about form
–
Derivation principles
Applicability of this material to design Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 2 of 25 Arbitrage Enforced Valuation Black Scholes ©
Basis for Options analysis
The valuation of very simple option of last class has fundamentally important lessons
Surprisingly, when replication possible: value of option does NOT depend on probability of payoffs!
–
Contrary to intuition associated with probabilistic nature of process
–
This surprising insight is important basis for options analysis Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 3 of 25 Arbitrage Enforced Valuation Black Scholes ©
Application to Binomial Lattice
How does arbitrage-enforced pricing apply to the binomial lattice?
It replaces actual binomial probabilities
–
(as set by growth rate, v, and standard deviation, σ)
by relative weights derived from replicating portfolio
These relative weights reflect the proportion ratio of asset and loan (as in example) – but look like probabilities: they are called the risk neutral “probabilities” Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 4 of 25 Arbitrage Enforced Valuation Black Scholes ©
Single Period Binomial Model Set-up
Apply to generalized form of example Value of Asset is up or down S uS dS Value of call option Is up or down C Max(Su - K, 0) =Cu Max(Sd - K, 0) =Cd Value of loan rises by Rf (no risk) to R = 1 + Rf (for 1 year) Engineering Systems Analysis for Design Massachusetts Institute of Technology 1 R R Richard de Neufville Slide 5 of 25 Arbitrage Enforced Valuation Black Scholes ©
Single Period Binomial Model Solution
The issue is to find what proportion of asset and loan to have to establish replicating portfolio
Set: asset share = “x” loan share = “y”
then solve: xuS + yR = Cu and xdS + yR = Cd => x = (Cu - Cd) / S(u - d) from Cu - Cd => y = (1/R) [uCd - dCu] / (u - d) by substitution Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 6 of 25 Arbitrage Enforced Valuation Black Scholes ©
Single Period Binomial Model Solution
Now to find out the value of the option
Portfolio Value = Option Price = xS + y(1) = (Cu – Cd) / (u-d) + (uCd – dCu)/ R(u-d) = {(R - d)Cu + (u - R)Cd} / {R(u-d)} Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 7 of 25 Arbitrage Enforced Valuation Black Scholes ©
Application to Example
For Example Problem from previous class: R = 1 + Rf = 1.1 (Rf assumed = 10% for simplicity) Cu = value of option in up state = 15 Cd = value of option in down state = 0 u = ratio of up movement of S = 1.25
d = ratio of down movement of S = 0.8
Portfolio Value = Option Price = [(R - d)Cu + (u - R)Cd] / R(u-d) = [ (1.1- 0.8) (15) + (1.25 - 1.1)(0)] / 1.1(1.25 - 0.8) = [ 0.3(15)] / 1.1(.45) = 10 / 1.1 = 9.09 as before Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 8 of 25 Arbitrage Enforced Valuation Black Scholes ©
Reformulation of Binomial Formulation
Option Price = {(R - d)Cu + (u - R)Cd} / {R(u-d)}
We simplify writing of formula by substituting a single variable for a complex one: “q” ≡ (R - d) / (u - d)
Option Price = {(R - d)Cu + (u - R)Cd} / {R(u-d)} = (1/R) [{(R-d)/(u-d)}Cu + {(u-R)/(u-d)}Cd]
= (1/R) [q Cu + (1- q) Cd)]
Option Value is weighted average of q, (1 – q) Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 9 of 25 Arbitrage Enforced Valuation Black Scholes ©
q factor = risk neutral “probability”
Option Price = (1/R) [q Cu + (1- q) Cd)]
This leads to an extraordinary interpretation! Value of option = “expected value” with binomial probabilities q and (1 - q)
These called: “risk- neutral probabilities”
Yet “q” defined by spread: q ≡ (R - d) / (u - d) actual probabilities do not enter into calculation!
q C (1-q) Engineering Systems Analysis for Design Massachusetts Institute of Technology Max(Su - K, 0) =Cu Max(Sd - K, 0) =Cd Richard de Neufville Slide 10 of 25 Arbitrage Enforced Valuation Black Scholes ©
Binomial Procedure using q
“Arbitrage-enforced” pricing of options in binomial lattice proceeds as with “decision analysis” based valuation covered earlier
Difference is that probabilities are no longer (p, 1 - p) but (q, 1 - q)
From the perspective of calculation, (q, 1 - q) are exactly like probabilities
However, never observed as frequencies, etc.
Said to be “risk-neutral”, because derived from assumption of risk-free arbitrage Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 11 of 25 Arbitrage Enforced Valuation Black Scholes ©
Meaning of “options analysis”
Need to clarify the meaning of this term
Methods presented for valuing options so far (lattice, etc) are all analyzing options. In that sense, they all constitute “options analysis”
HOWEVER, in most literature “options analysis” means specific methods – based on replicating portfolios and random probability – epitomized by Black-Scholes Keep this distinction in mind!
Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 12 of 25 Arbitrage Enforced Valuation Black Scholes ©
Background
Development of “Options Analysis” Recent
Depends on insights, solutions of
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Black and Scholes ; Merton
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Cox, Ross, Rubinstein
This work has had tremendous impact
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Development of huge markets for financial options, options “on” products (example: electric power)
Presentation on options needs to discuss this – although much not applicable to engineering systems design Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 13 of 25 Arbitrage Enforced Valuation Black Scholes ©
Key Papers and Events
Foundation papers:
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Black and Scholes (1973) “The Pricing of Options and Corporate Liabilities,” J. of Political Economy, Vol. 81, pp. 637 - 654
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Merton (1973) “Theory of Rational Option Pricing,” Bell J. of Econ. and Mgt. Sci., Vol. 4, pp. 141 - 183
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Cox, Ross and Rubinstein (1979) “Option Pricing: a Simplified Approach,” J. of Financial Econ., Vol. 7, pp. 229-263. [The lattice valuation]
Events
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“Real Options” term coined by MIT Prof Myers ~ 1990
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Nobel Prize in 1997 to Merton and Scholes (Black had died and was no longer eligible) Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 14 of 25 Arbitrage Enforced Valuation Black Scholes ©
Black-Scholes Options Pricing Formula
C = S
*
N(d
1
) - [K
*
e
- rt
*
N(d
2
)]
It applies in a very special situation: ─ a European call ─ on a non-dividend paying asset “European” ≡ only usable on a specific date “American” ≡ usable any time in a period (usual situation for options “in” systems) “no dividends” -- so asset does not change over period Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 15 of 25 Arbitrage Enforced Valuation Black Scholes ©
Black-Scholes Formula -- Terms
C = S
*
N(d
1
) - [K
*
e
- rt
*
N(d
2
)]
S , K r = current price, strike price of asset = Rf = risk- free rate of interest t
N(x) = time to expiration = standard deviation of returns on asset = cumulative pdf up to x of normal distribution with average = 0, standard deviation = 1 d d 1 2 = [Ln (S/K) + (r + 0. 5
2
= d 1 - (
t) ) t ] / (
t) Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 16 of 25 Arbitrage Enforced Valuation Black Scholes ©
Black-Scholes Formula -- Intuition
C = S
*
N(d
1
) - [K
*
e
- rt
]
*
N(d
2
)
Note that, since N(x) < 1.0, the B-S formula expresses option value, C, as ● a fraction of the asset price, S, less ● a fraction of discounted amount, (K * e - rt ) These are elements needed to create a replicating portfolio (see “Arbitrage-enforced pricing” slides). Indeed, B-S embodies this principle with a continuous pdf. Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 17 of 25 Arbitrage Enforced Valuation Black Scholes ©
Why does Black-Scholes matter?
Development of Formula showed the way for financial analysts
Essentially “no” other significant closed form solutions…
But solutions worked out numerically through lattice (and more sophisticated) analyses
Led to immense development of use of all kinds of “derivatives” (an alternative jargon word that refers to various options) Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 18 of 25 Arbitrage Enforced Valuation Black Scholes ©
Why does B-S matter TO US?
What does B-S mean to designers of technological systems?
Important to understand the assumptions behind Black-Scholes equation and approach
Extent these assumptions are applicable to us, determines the applicability of the approach
Much research needed to
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address this issue
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Develop alternative approaches to valuing flexibility Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 19 of 25 Arbitrage Enforced Valuation Black Scholes ©
Price Assumption
B S approach assumes Asset has a “price”
When is this true?
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System produces a commodity (oil, copper) that has quoted prices set by world market
When this may be true
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System produces goods (cars, CDs) that lead to revenues and thus value – HOWEVER, product prices depend on both design and management decisions
When this is not true
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System delivers services that are not marketable, for example, national defense… Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 20 of 25 Arbitrage Enforced Valuation Black Scholes ©
Replicating Portfolio Assumption
B-S analysis assumes that it is possible to set up replicating portfolio for the asset
When is this true
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Product is a commodity
When this might assumed to be true
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Even if market does not exist, we might assume that a reasonable approximation might be constructed (using shares in company instead of product price)
When this is probably a stretch too far
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Private concern, owners unconcerned with arbitrage against them, who may want to use actual probabilities… Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 21 of 25 Arbitrage Enforced Valuation Black Scholes ©
Volatility Assumption
B-S approach assumes that we can determine volatility of asset price
When this is true
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There is an established market with a long history of trades that generates good statistics
When this is questionable
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The market is not observable (for example, because data are privately held or negotiated)
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Assets are unique (a prestige or special purpose building or special location)
When this is not true
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New technology or enterprise with no data Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 22 of 25 Arbitrage Enforced Valuation Black Scholes ©
Duration Assumption
B-S approach assumes volatility of asset price is stable over duration of option
When this is true
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Short-term options (3 months, a year?) in a stable industry or activity
When this is questionable
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Industries that are in transition – technologically, in structure, in regulation – such as communications
When this is not true
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Long-duration projects in which – major changes in states of markets, regulations or technologies are highly uncertain (Exactly where we want flexibility!) Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 23 of 25 Arbitrage Enforced Valuation Black Scholes ©
Take-Away from this Discussion
In many situations the basic premises of “options analysis” – as understood in finance – are unlikely to apply to the design and management of engineering systems
Yet these systems, typically being long-life, are likely to be especially uncertain – and thus most in need of flexibility – of “real options”
We thus need to develop pragmatic ways to value options for engineering systems TOPIC OF PAST PRESENTATIONS Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 24 of 25 Arbitrage Enforced Valuation Black Scholes ©
Summary
Black-Scholes formula elegant and historically most important
Its derivation based on some fundamental developments in Stochastic processes
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Random walk; Wiener and Ito Processes ; GBM
Underlying assumptions limit use of approach
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Price; Replicating Portfolio; Volatility; Duration
Developing useful, effective approaches for design is an urgent, important task Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Slide 25 of 25 Arbitrage Enforced Valuation Black Scholes ©