Chapter 3

1. Chapter 3 Binomial Tree Methods ------ Discrete Models of Option Pricing

2. An Example • Question: When t=0, buying a call option of the stock at with strike price $40 and 1 month maturity. If the risk-free annual interest rate is 12% throughout the period [0, T], how much should the premium for the call option be?

3. Example cont.1 • payoff = • Consider a portfolio

4. Example cont.2 • When t=T, • has fixed value $35, no matter S is up or down

5. Example cont.3 • If risk free interest r =12%, a bank deposit of B=35/(1+0.01) after 1 month • By arbitrage-free principle

6. Example cont.4 • That is • Then • This is the investor should pay $2.695 for this stock option.

7. Analysis of the Example • the idea of hedging: it is possible to construct an investment portfolio with S and c such that it is risk-free. • The option price thus determined (c_0=$2.695) has nothing to do with any individual investor's expectation on the future stock price.

8. One-Period & Two-State • One-period: assets are traded at t=0 & t=T only, hence the term one period. • Two-state: at t=T the risky asset S has two possible values (states): , with their probabilities satisfying

9. One-Period & Two-State Model • The model is the simplest model. • Consider a market consisting of two assets: a risky S and a risk-free B • If: risky asset and risk free asset • known , when t=0, • t=T, 2 possibilities • Option Price at t=0? (for strike price K, expired time T)

10. Analysis of the Model - Stock Price, is a stochastic variable Up, with probability p Down, with probability 1-p where is a stochastic variable.

11. Question & Analysis • If known at t=T, • how to find out when t=0? • Assume the risky asset to be a stock. Since the stock option price is a random variable, the seller of the option is faced with a risk in selling it. However, the seller can manage the risk by buying certain shares (denoted asΔ) of the stocks to hedge the risk in the option. • This is the idea!

12. Δ- Hedging Definition • Definition: for a given option V, trade Δshares of the underlying asset S in the opposite direction, so that the portfolio is risk-free.

13. Analysis of Δ- Hedging • risk free asset • If Π is risk free, then, on t=T, is risk free. i.e. so that

14. Analysis of Δ- Hedging cont. • are random variables, when t=T, both of them have 2 possible values where are unknown, solve them:

15. Analysis of Δ- Hedging (Probability Measure) • Define a new Probability Measure • Obviously

16. Solution of Premium • From the discussion above, where denotes the expectation of the random variable under the probability measure Q.

17. Definition of Discounted Price • Let U be a certain risky asset, and B a risk-free asset, then is called the discounted price (also known as the relative price) of the risky asset U at time t.

18. Theorem 3.1 • Under the probability measure Q, an option's discounted price is its expectation on the expiration date. i.e.

19. Remark • In order to examine the meaning of the probability measure Q, consider S is an underlying risky asset. Calculate

20. Risk-Neutral World • Under the probability measure Q, the expected return of a risky asset S at t=T is the same as the return of a risk-free bond. A financial market possessing this property is called a Risk-Neutral World • In a risk-neutral world, no investor demands any compensation for risks, and the expected return of any security is the risk-free interest rate.

21. Definitions • the probability measure Q defined by is called by risk-neutral measure. • The option price given under the risk-neutral measure is called the risk-neutral price.

22. Definition of Replication • In a market consisted of a risky asset S and a risk-free asset B, if there exists a portfolio (where α,β are constants, Φ, V are both random variables) such that the value of the portfolio Φ is equal to the value of the option V at t=T, then Φ is called a replicating portfolio of the option V, then option price

23. Theorem 3.2 • In a market consisted of • a risky asset S and • a risk-free asset B, • d<ρ<u is true if and only if the market is arbitrage-free.

24. Proof of Theorem 3.2 (1st dir.) • 1) arbitrage-free d<ρ<u • Suppose ρ>= u, consider the following portfolio: Its values at t=0 and at t=T are:

25. Proof of Theorem 3.2 (1st dir.) cont. • is a random variable with two possible values:

26. Proof of Theorem 3.2 (1st dir.) cont. - • So that, for the portfolio Φ • That shows that there exists arbitrage opportunity for portfolio Φ, contradiction! to that the market is arbitrage-free. • Same to ρ<=d.

27. Proof of Theorem 3.2 (2nd dir.) • If market is arbitrage-free, for any portfolio • If • then • In fact, define a risk-neutral measure Q

28. Proof of Theorem 3.2 (2nd dir.) cont. • Then • Consider the expectation of the random variable • According to the definition of the risk-neutral measure Q,

29. Proof of Theorem 3.2 (2nd dir.) cont.- • That is, • But • Then • i.e. • There exists no arbitrage opportunity.

30. Theorem 3.2 • If the market is arbitrage-free, then there exists a risk-neutral measure Q defined by such that

31. Binomial Tree Method • Divide the option lifetime [0, T] into N intervals: • Suppose the price change of the underlying asset S in eachinterval can be described bythe one-period two-state model, then the random movement of Sin [0, T] forms a binomial tree

32. Binomial Tree Method cont. • This means that if at the initial time the price of the underlying asset is , then at t=T, will have N+1 possible values • Take call option as example, the option value at t=T, is also a random variable, with corresponding possible values

33. Binomial Tree Method Notation • Denote

34. Binomial Tree …… … …… …… … ……

35. Possible Values of Option at t=T

36. Problem Option Pricing by BTM • If are given, how can we determine • in particular

37. Answer to the Problem • With the one period and two-state model, and using backward induction, we can determine step by step.

38. Induction Steps • When are given, • to find • consider the following one period and two-state model. and

39. Induction Steps cont. • Define a risk-neutral measure Q • Then, • So that for any

40. Induction Steps cont.- • But • Denote • Thus

41. European call option valuation formula • Denote • Then the European call option valuation formula is • Especially, h=N, α=0,

42. Discount Factor • Discount Factor satisfies • The financial meaning of the discount factor: to have $1 at t=T (including continuous compound interests), one needs to deposit in bank at t (t<T).

43. Discount Factor in BTM • in the binomial tree method, trading occurs at discrete times • the compound interests should also be calculated for the discrete case. • Let denote the discrete discount factors. They satisfy the difference equations

44. Discount Factor in BTM cont. • That is where ρis the growth of the risk-free bond in i.e.

45. Call---Put Parity in discrete form • for the binomial tree method, the call---put parity (in discrete form) becomes

46. European put option valuation formula • Using European call option valuation formula and put---call party, we have

47. Investment vs. Gambling Game • Investing in options can be compared to a gambling game. • Initial stake be . After one game, the stake becomes . • is a random variable. • If the expectation then the gamble is said to be fair

48. Fair Gambling Game • - the bet at n-th game, - the next bet. • If under the condition that complete information of all the previous n-games are available, the expectation of equals the previous stake i.e., then we say the gamble is fair.

49. σ-Algebra • denotes complete information of the bets up to n-th game, and denotes the conditional expectation of X under condition Y. • In mathematics, is called σ- algebra in stochastic theory

50. Martingale • Martingale is often used to refer to a fair gamble. • The bet sequence that satisfies condition as a discrete random process, is called a Martingale.