1 Derivatives & Risk Management: Part II Models, valuation and risk management

2 What we are going to do –Value a derivative in a one period binomial model: classical approach vs. risk neutral valuation –The binomial and the random walk models for the stock price.

3 What we are going to do II –The Black & Scholes model –Risk neutral vs classical valuation in continuous time –Applications of the Black Scholes model Portfolio insurance

4 What we’re going to do III –When B&S is inapplicable Monte Carlo binomial trees –Risk management The partial derivatives Value at Risk

5 This lecture More abstract than earlier lectures but will quickly lead to practical applications Concepts of valuation for derivatives on traded assets Illustrative tool: –1 period binomial –2 period binomial model

6 More on this lecture Goal: to provide the insights necessary to understand valuation theory Contrast the “Classic” approach to that of “Risk Neutral Valuation” Novelty: introduce assumptions about the statistical properties of security prices (binomial, GBM)

7 Relative pricing A derivatives is a financial instrument whose value can be derived from that of one or more underlying securities The derivative will be priced RELATIVE the underlying. –Information about market conditions expectations, risk preferences, etc... will be incorporated into the value of the derivative insofar as they are compounded in the current prices of the underlying.

8 Arbitrage and risk free portfolios Derivative valuation relies on the assumption of no arbitrage –the idea is to construct a portfolio of the derivative and the underlying which yields a known risk free payoff regardless of future states of the world. –Given that this is the case we can value the portfolio by discounting at the risk free rate –and then we can solve for the price of the derivative

9 More concretely To make matters more concrete we will think of the derivative as the a call option and the underlying as a non-dividend paying stock The idea is however applicable to any derivative security. Most complications will be purely technical

10 Preliminaries Basic assumptions –frictionless markets –borrowing and lending –investors prefer more to less –no arbitrage Then we add the binomial stock dynamics - unrealistic simple one-period model.

11 Preliminaries II

12 The classical approach 70 The stock price tree

13 The classical approach II Option price tree

14 The classical approach III We will now construct a portfolio of  stocks and one short call option. The value of this portfolio in 6 months will be the (random) value

15 The classical approach IV The trick is to design this portfolio so that its value turns out to be non-random. In other words we want to select  so that

16 The classical approach V More precisely

17 The argument Now since with certainty we must have that

18 The result And since

19 The hedge ratio Binomial equivalent to the “delta” which we will talk about in more detail later on when we’ve introduced the B&S model.

20 A summary of the argument Set up a portfolio in the call and the stock (long in one and short in the other) which has a certain payoff in both states of the world Argue that since its payoff is certain the appropriate discounting rate is the risk free rate Solve for the option price using today’s stock price, the hedge ratio  and r

21 A slightly different way of seeing it Instead of setting up risk free portfolios in the asset and the option we could replicate the derivative by borrowing and investing in the stock

22 A different way

23 A different way II

24 IMPORTANT This approach is by no means limited to options. It is the basic approach (which from case to case may require some alteration) to the valuation of any derivative security on a traded financial security

25 An alternative approach: “Risk-neutral” valuation In principle we could value an option by discounting the expected future cash flow at the appropriate rate Problem: the discount rate is extremely hard to estimate

26 “Risk-neutral” valuation Suppose that we turn the question around: –suppose we insist on discounting the expected payoff at the risk free rate, then what probabilities should we use to obtain the arbitrage free (correct) option price Answer:

27 A sketch of the proof Start with the composition of the initial “classical portfolio” Substitute

28 Sketching on Then after some reshuffling you will find that: where

29 Interpretation One way of interpreting this result is that it is the present value of the expected payoff in a risk neutral world p are the risk free probabilities and since there is no compensation for risk the appropriate discounting rate is r

30 Interpretation II This interpretation is dangerous since it may lead to the misconception that in order to use the result one implicitly assumes that investors are risk neutral But we have seen that this is not so, all we require are frictionless markets and agents that prefer more to less

31 Interpretation III Furthermore this interpretation is not fully accurate since it neglects the fact that the volatility that we will use is the one estimated on actual markets and this quantity may reflect market preferences

32 Bottom line The risk neutral valuation result should be considered a purely mathematical tool that will prove extremely useful for computing derivative prices in more complex situations It does not require any assumptions about investor risk preferences.

33 Using it Back to our example

34 Constraints on probabilities For p to qualify as a probability we require that

35 A general statement of the result For an arbitrary derivative f we can write its price as And in the special case of non random interest rates

36 Comments In practice the question is how to compute the expectation under risk neutral probabilities. This task will differ in degrees of difficulty that depend on assumptions about –the dynamics of underlying security, interest rates –features of the derivative security (path dependence...)

37 Example Consider a forward on a security paying no dividends What is the “risk neutral” expectation of the underlying security at expiration?

38 Example II What is the value of a long forward?

39 A 2 period example

40 Novelties On the path to more realistic models The hedge ratio will have to be adjusted dynamically as time goes by and the stock price moves. This in contrast to the portfolio we have to set up to value e.g. a forward contract.

41 The binomial tree

42 Computations We begin at t=0.25, after an up move

43 Computations II Then at t=0.25, after a down move

44 The binomial tree 70

45 The final step

46 Risk Neutral valuation 2 periods Begin by computing the risk-neutral probability

47 The binomial tree p p p (1-p)

48 An even quicker alternative

49 Replicating portfolios and dynamic arbitrage valuation The idea that the “hedge ratio”  must be changed as the stock price evolves brings us to the concept of dynamic replication which is the underlying concept for the development of the Black & Scholes model

50 Replication This should be contrasted to “static” replication –put-call-parity –coupon bonds as portfolios of discount bonds

51 Models for the behavior of security prices Discrete time - discrete variable –binomial model

52 Another model Discrete time - continuous variable 

53 Other models Continuous time - discrete variable

54 The geometric Brownian motion Continuous time; continuous variable

Markov Processes In a Markov process future movements in a variable depend only on where we are, not the history of how we got where we are In the developent of the B&S model, we will assume that stock prices follow Markov processes

Weak-Form Market Efficiency The assertion is that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work. A Markov process for stock prices is clearly consistent with weak-form market efficiency

Variances & Standard Deviations In Markov processes changes in successive periods of time are independent This means that variances (not standard deviations) are additive