Help Session 4. David Sovich. Washington University in St. Louis

Transcription

1 Help Session 4 David Sovich Washington University in St. Louis

2 TODAY S AGENDA More on no-arbitrage bounds for calls and puts Some discussion of American options Replicating complex payoffs Pricing in the binomial tree

3 A PRELIMINARY SURVEY Aside: Link to help session slides and protocol Do you find the lectures difficult? What s your opinion on Jason s teaching? What would you like (a) Jason and (b) Me to do differently in (a) the lectures and (b) the help sessions?

4 BOUNDING OPTION PRICES (AGAIN) To derive bounds on option prices, we use the principal of no-arbitrage If security A has strictly greater payoffs (cash flows) than security B in every state of the world ω, then security A must have a strictly greater price Therefore, option prices can be bounded by portfolios with strictly greater payoffs (or with equal payoffs)

5 BOUNDING THE CALL PRICE Last week we showed that c t S t for a non-dividend paying stock What if the stock pays a known dividend D at time τ (t,t)? We will show that the no-arbitrage bound is now lower: c t S t De r(τ t) Why? Think about the advantages of owning the stock versus the call In general, c t S t PV(DIVS) for any dividends paid before T

6 BOUNDING THE CALL PRICE Claim: c t S t De r(τ t) Proof: Suppose otherwise that c t > S t De r(τ t) Consider the following strategy: Long the stock, short the call, borrow De r(τ t) at rate r for τ t length, and repay the loan with any dividends The time t cash flow is CF t = c t (S t De r(τ t) ) > 0 The time τ cash flow is CF τ = D De r(τ t) e r(τ t) = 0 And the time T cash flow is CF T = S T (S T K) + 0 for any S T

7 BOUNDING THE CALL PRICE Note that this was our first example where a security had intermediate payments (e.g. the dividend) Constructing arbitrage strategies is not different though Suppose there are multiple time periods {t,τ 1,..,τ N } and there are Ω(τ) = {ω 1,...,ω n(τ) } possible states at each τ > t A portfolio V is an arbitrage if either (1) V }{{} t < 0 and V τ (ω) 0 for every τ and ω Ω(τ) }{{} PRICE PAYOFF (2) V t = 0 and V τ (ω) 0 for every τ and every ω Ω(τ) and V τ (ω ) > 0 for at least one τ and ω Ω(τ )

8 BOUNDING PRICES VIA PUT-CALL PARITY The put-call parity without dividends states that the price of a fiduciary call = price of a protective put c t + Ke r(t t) = p t + S t Why? Consider the payoffs of the two portfolios (on board) With dividends, the put-call parity is quite similar c t + Ke r(t t) + PV(D) = p t + S t

9 BOUNDING PRICES VIA PUT-CALL PARITY We can always exploit the fact that c t and p t are non-negative to generate lower bounds on option prices c t S t Ke r(t t) PV(D) p t Ke r(t t) + PV(D) S t Thus, combining with the non-negativity constraint, we know that the lower bounds are: ( ) c t max S t Ke r(t t) PV(D),0 ( ) p t max Ke r(t t) + PV(D) S t,0

10 BOUNDING PRICES VIA PUT-CALL PARITY We can also use the put-call parity relation to derive upper bounds For example, what happens to the upper bound on the put if the stock pays dividends? By Put-Call Parity p t = Ke r(t t) + c t (S t PV(D)) and by the upper bound on call prices p t = Ke r(t t) + c t (S t PV(D)) Ke r(t t) }{{} 0

11 SOME NOTES ON AMERICAN OPTIONS American options can be exercised early This implies that some of our European option bounds may not hold anymore Let p E t denote the price of a European put, and p A t denote the price of an American put When would be expect p E t Ke r(t t) p A t K?

12 REPLICATING COMPLEX PAYOFFS In the homework you are asked to replicate the payoff of a collar using bonds, stocks, and options, and then price the collar A nice result in finance states that any payoff can be replicated by liquid bonds, calls, and puts A nice algorithm for replicating complex payoffs is as follows: 1. Draw the payoff diagram 2. Use bonds for level shifts upwards 3. Use calls and puts with varying strikes to add slope or subtract slope from the payoff

13 PRICING ON THE BINOMIAL TREE Pricing options via the binomial tree is an example of an economic model To get the price of the option (if possible) we simply apply the concepts of no-arbitrage and the Law of One Price In particular, our cookbook for a security on the single-period binomial tree is 1. Replicate the terminal payoffs of the security using our basis assets 2. Check if no-arbitrage holds and then apply the Law of One Price to find the price Price Of Security = Price Of Replicating Portfolio

14 REPLICATION ON THE BINOMIAL TREE The basis assets are the assets we are given exogenously with known prices and payoffs For example, one stock and one bond will often be the basis assets in a binomial model As long as the payoffs of the stock and the bond are linearly independent, then we can replicate any asset with payoffs in R 2 This is because the payoff vectors S T = [S T (u),s T (d)] and B T = [1,1] span all of R 2

15 BINOMIAL TREE EXAMPLE Suppose that there are two-time periods t = 0 and T = 1 and two possible states, u and d, at time T Suppose there exists a stock with S 0 = 1 and S T = [2, 0] Suppose there also exists a bond with B 0 = 1 and r = 0 Problem: What is (a) the price of a call on S with K = 1, and (b) the price of a put on S with K = 1?

16 BINOMIAL TREE EXAMPLE Solution: Show on the board - replicating portfolio approach (no FTAP) We should have solved the replicating portfolios, applied the LoP, and arrived at c 0 = p 0 = 1 2 Note that the call and put payoffs mimic the payoffs of Arrow-Debrue securities, and hence c 0 and p 0 reflect state prices