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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

– 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 • 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

• 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-Scholes-Merton

• 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

• 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

• 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