The Theory of Dynamic Hedging

1. The Theory of Dynamic Hedging Nassim Nicholas Taleb Courant Institute of Mathematical Science Sept 4, 2003

2. 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.

3. 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.

4. 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.

5. 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

6. 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

7. 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.

8. Some more attributes • “Theoretically Elegant” • “Idealized”

9. 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.

10. Completeness • Definition of a complete market

11. State Prices • a.k.a.state contingent claims, elementary securities, building blocks, • These securities, byarbitrage, 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.

12. 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

13. 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)

14. First “Theorem”: Existence • Risk-neutral probabilities exist if and only if there are no riskless arbitrage opportunities.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. 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!!!

21. 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

22. 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 • Firstaspect: 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). • Secondaspect: There is no risk premium involved in the process --the package is deemed to be riskless.

23. Assumptions Behind BSM • no riskless arbitrage opportunities • perfect markets • constant r • constant and known volatility (comment on the known) • no jumps

24. 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)

25. 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.

26. Dynamic Hedging Effect

27. 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

28. 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

29. 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

30. Butterfly • Buy call Struck at K + • Buy call Struck at K -  • Sell 2 calls struck at K • What do we get at expiration?

31. 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