Transcript The Theory of Dynamic Hedging

The Theory of Dynamic Hedging
Nassim Nicholas Taleb
Courant Institute of Mathematical Science
Sept 4, 2003
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About this part of the course
– This part is Clinical Finance, which will be further
defined in the next lecture.
– It is not the marriage of theory & practice. Practice
comes first & last. This is best called theoretically
inspired & enhanced practice.
– No (or minimum) theorems, no proofs. The important
matter is to be convinced. Why? Because theorems are
only as good as the assumptions on which they are
built.
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Some Holes With Existing Theory
– According to strict theoretical considerations, derivatives do not
exist. Markets are fundamentally incomplete, and we have to live
with it.
– Hakansson’s paradox: if markets are complete we do not need
options; if they are incomplete then according to financial theory
we cannot price options…
– The paradox has not been solved so far in finance theory 
Finance theory may be total nonsense.
– The objective of this course is to make you live with it as well so
you do not get a shock when you get out of here.
– Ask questions --the real world does not have an owner’s manual.
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– Clinical finance will be further discussed after a brief
presentation of the existing theories --so we have
enough material to engage in a critique of the current
framework.
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The Theoretical Backbone of
Modern Finance
– This first lecture will focus on the theoretical backbone
of modern finance, particularly in what applies to asset
pricing
– We will explore the origin of the thinking in financial
economics
– If so little in successful quantitative Wall Street is
linked to the financial economics aspect of finance, it is
not quite without a reason
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Neoclassical Economics
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Adam Smith’s invisible hand
Walras’ auctioneer
Marshall’s partial equilibrium
Arrow & Debreu’s proof of the existence and
uniqueness
– The central conclusion is the idea of laisser-faire: the
government should not interfere with the system of
markets that allocates resources in the private sector of
the economy
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Arrow-Debreu General Equilibrium
– A competitive system with market prices coordinates
the otherwise independent activities of consumers and
producers acting purely in their self-interest.
– Stands on shaky empirical foundations
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no information
no adverse selection
no intermediation, no transaction costs
the model might have been reverse engineered, i.e. the correct
assumptions were chosen because they led to the adequate
solution.
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Some more attributes
• “Theoretically Elegant”
• “Idealized”
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Enters Uncertainty
– Arrow is credited with the introduction of uncertainty
in the model, thanks to a contraption now called
“Arrow” (now “Arrow-Debreu”) securities.
– In a 2-period model, it delivers 1 unit of numeraire in a
given state of the world, 0 otherwise.
– These securities complete the market, i.e. eliminate
uncertainty as agents can buy them as insurance.
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Completeness
– Definition of a complete market
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State Prices
– a.k.a.state contingent claims, elementary securities,
building blocks,
– These securities, by arbitrage, sum up to 1. Like
probabilities, they are exhaustive and mutually
exhaustive.
– It is important to see that they are not quite
probabilities, even when translated into their continuous
price & time limit.
– This leads to the analog state price density for one
period models.
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Lexicon
• State price: a security that pays 1 in a state of the
world, 0 elsewhere
• The price paid today for a state price resembles a
density.
• Why resemble? Because of the difference between
probability and pseudoprobability.
• Why pseudoprobability? Something called
arbitrage
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Credits
• Note: I credit for the exposition of the next three
theorems, Rubinstein’s e-textbook (1999),
www.in-the-money.com
• Note: a brief discussion of the “inverse problem”,
i.e. the ability to pull out the state prices from
derivatives (Breeden-Litzenberger, 1978)
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First “Theorem”: Existence
– Risk-neutral probabilities exist if and only if there are
no riskless arbitrage opportunities.
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Arbitrage opportunities
• an arbitrage exists if and only if either:
• (1) two portfolios can be created that have identical payoffs in
every state but have different costs; or
• (2) two portfolios can be created with equal costs, but where
the first portfolio has at least the same payoff as the second in
all states, but has a higher payoff in at least one state; or
• (3) a portfolio can be created with zero cost, but which has a
non-negative payoff in all states and a positive payoff in at
least one state.
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Second Major Theorem: Uniqueness
– The risk-neutral probabilities are unique if and only if
the market is complete.
– Hint: an incomplete market provides many solutions
under this framework.
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Third Major Theorem: Dynamic
Completeness
– Arrow, in 1953, (tr. 1964) showed that, under some
conditions, the ability to buy and sell securities can
effectively make up for the missing securities and
complete the market.
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Bachelier
– Aside from minor problems concerning the returns
(arithmetic v/s logarithmic, which constitute a very
small difference in common practice), Bachelier
presented an option pricing tool that reposes on the
actuarial distribution. In essence we are using his
pricing method supplemented with arbitrage arguments.
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Keynes’ Arbitrage argument
– In 1923, Keynes effectively showed that by arbitrage
argument, the forward needs to be equal to its arbitrage
value, when lending & borrowing are possible.
– Covered Interest Parity Theorem:applied to the forward
for a currency pair and, by extension, to any security
that can be lent and borrowed.
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Keyne’s Argument
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Currency 1 has a rate r1
Currency2 has a rate r2
Spot rate S
Forward rate F
F = S (1+r1)/(1+r2)
The Forward has nothing to do with
expectations!!!
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The Importance of Keynes’ Intuition
• Keynes was the first person in modern times to
express the notion that the forward is not an
expected future price, but an arbitrage-derived
pseudo-expectation.
• However it is of required use as an equivalent
mean return in an arbitrage framework
• Arrow’s state prices are the equivalent
probabilitites pseudoprobabilities
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The Essence of Black-ScholesMerton
• What Black-Scholes did was not “price” options as we do
it today. It merely made option pricing compatible with
financial economics.
• There are two aspects to BSM
• First aspect: the mean of the probability distribution used
in their framework is that of the risk-neutral one (the
µbecomes the difference between the carry and the
financing).
• Second aspect: There is no risk premium involved in the
process --the package is deemed to be riskless.
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Assumptions Behind BSM
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no riskless arbitrage opportunities
perfect markets
constant r
constant and known volatility (comment on the
known)
• no jumps
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The Intuition
• Assuming an economy with no interest rates, the
operator has two packages:
– L1: short the European call option
– L2: long the local sensitivity of the option worth of
stock
– L1: +C(St,t) -C(St+t,t+t)
– L2:  (St+t -St)
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– It is important to see that L1 and L2 are negatively
correlated, that the negative correlation increases as t
goes to 0
– It is important to see that, since the portfolio is delta
neutral, there is no corresponding sensitivity of the total
package to the asset price returns --therefore the return
of the asset price becomes sensitive to the square
variations .
– Pricing by replication allows the option to be a
redundant security.
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Dynamic Hedging Effect
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Risk and Insurance
• Why dynamic hedging separates finance from
other disciplines of risk bearing
• The option value is no longer the actuarial value
of the payoff
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Food for Thought
• As an option trader I deem dynamic hedging
unattainable --most of the package variance comes
from the jumps. We cannot ignore the actuarial
aspect of things.
• Black-Scholes reposes on a “known” distribution,
with known parameters --I do not know much
about the distribution
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The problem of the
Normal/Lognormal
• Blaming Bachelier for having a “normal” not
“lognormal” distribution may be unfair since in
the real world we use a distribution of some
uncharacterized shape
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Butterfly
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Buy call Struck at K +
Buy call Struck at K - 
Sell 2 calls struck at K
What do we get at expiration?
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Breeden-Litzenberger
• The “infinitely small butterfly” scaled by 1/ at
the limit delivers 1 at expiration at K and 0
elsewhere
• More on that with JG’s part: the butterfly, called
elementary securities, are the building block of
everything
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