Help Session 2 David Sovich Washington University in St. Louis
TODAY S AGENDA 1. Refresh the concept of no arbitrage and how to bound option prices using just the principle of no arbitrage 2. Work on applying the Law of One Price in the binomial tree: replication and (if time permits) the FTAP * Forewarning: Builds on concepts covered in the Help Session last week
NO ARBITRAGE REFRESHER An arbitrage is either: (1) getting paid to never pay anything out or (2) paying nothing for potentially something - e.g free lunch Suppose there are two time periods t (today) and T > t and there are Ω = {ω 1,...,ω n } possible states of the world at time T A portfolio V is an arbitrage if either 1. V }{{} t < 0, and V T (ω) 0 for every ω Ω }{{} PRICE PAYOFF 2. V t = 0, and V T (ω) 0 for every ω Ω and V T (ω ) > 0 for at least one ω Ω
NO ARBITRAGE REFRESHER No arbitrage is the absence of arbitrage - and no arbitrage is required for prices to make sense This is the main concept in option pricing - all the key theorems follow from assumptions about NA Recall that Futures (linear contracts) can be priced using only NA, for options we can only bound prices via NA
NO ARBITRAGE PRACTICE Suppose there are two dates, t and T, and a stock with price S t. Denote the continuously compounded risk-free rate by r. Assume there are no dividends and all options have strike K Practice: Show that the European call (c t ) and put prices (p t ) must satisfy: ( max 0,S t Ke r(t t)) < c t < S t ) max (0,Ke r(t t) S t < p t < Ke r(t t)
NO ARBITRAGE COOKBOOK To show (for example) that c t < S t, recall the cookbook procedure that we learned: 1. Suppose NA holds. But, to the contrary, that c t S t 2. Construct a candidate arbitrage strategy. Remember the trick of buy low and sell high - e.g. long the stock short the call here 3. Show that the candidate arbitrage is indeed an arbitrage using the definition from two slides ago 4. Finally, since we assume NA and we found an arbitrage, we have arrived at a logical contradiction and hence the claim must be true
NO ARBITRAGE PRACTICE Practice: Assume our previous set-up. Show that if NA holds, then the put-call parity holds for strike K c t + Ke r(t t) = p t + S t Solution: Notice that the time T payoffs of the put-call parity are always equal: c T + K = max(s T K,0) + K = max(s T,K) = S T + max(0,k S T ) = S T + p T Hence, if the equality does not hold then we can long one side and short the other and construct an arbitrage
BOUNDING THE CALL PRICE WITH DIVIDENDS Two slides ago 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
BOUNDING THE CALL PRICE WITH DIVIDENDS 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
NO ARBITRAGE WITH DIVIDENDS 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 )
OPTIONS PRICING AND THE LOP Getting the exact price of call and put options requires more assumptions about the economy than just NA However, pricing is still done with just NA and the Law of One Price The Law of One Price states that portfolios with the same payoffs must have the same price If the Law of One Price fails (in complete markets), then we can construct an arbitrage
OPTIONS PRICING VIA REPLICATION One way to price options is to assume an economic model and price options via replication That is, suppose we want to price an option with a random payoff of Y t Suppose there are several assets 1,...,N with payoffs,x(2),...,x(n) and known prices X (1) T T T Then we can price the option Y t by: 1. Replicating the payoffs Y T in each state using a portfolio of traded (X (i) ) securities with known prices 2. Apply the Law of One Price - the price of security Y must equal the price of the portfolio of replicating securities
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 (u) = 2 and S T (d) = 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?
BINOMIAL TREE EXAMPLE Solution: Consider first the case of the put option. To price the put, we first need to calculate the payoffs. We have that p T (u) = (1 2) + = 0, and p T (d) = (1 0) + = 1 Now let us find a portfolio of α shares of the stock and β shares of the bond to replicate the payoff of p T in both states This is equivalent to solving the system of two linear equations: αs T (u) + βb T (u) = p T (u) αs T (d) + βb T (d) = p T (d)
BINOMIAL TREE EXAMPLE Plugging in the values for the stock and bond payoffs, we have: 2α + 1β = 0 0α + β = 1 Which implies that β = 1 units of the bond and α = 0.5 shares of the stock replicate the payoff of the put option By the Law of One Price, the price of the replicating portfolio must equal the price of the put option: αs 0 + βb 0 = p 0 And hence, p 0 = 1 2 1 + 1 1 = 1 2
BINOMIAL TREE EXAMPLE Now let us solve for the call option price. First, we calculate the payoffs We have that c T (u) = (2 1) + = 1, and c T (d) = (0 1) + = 0 Now let us find a portfolio of α shares of the stock and β shares of the bond to replicate the payoff of p T in both states This is equivalent to solving the system of two linear equations: 2α + 1β = 1 0α + 1β = 0
BINOMIAL TREE EXAMPLE Thus, a portfolio of long α = 1 2 shares and no position in the bond replicates the call option payoff By the Law of One Price, the time t = 0 price of the option must equal the time t = 0 price of the replicating portfolio And hence we have that c 0 = 1 2 1 + 1 0 = 1 2
REPLICATION COOKBOOK Pricing options via replication always follows the same steps 1. Calculate the payoffs of the option. If this is already done for you, then do nothing. 2. Replicate the payoffs of the option using a portfolio of stocks and bonds. Solve the system of equations. 3. State (in words and numbers) what the replicating portfolio is comprised of 4. Apply the Law of One Price, the time t price of option must be equal to the time t price of the replicating portfolio
OPTION PRICING WITH THE FTAP Another way to price options is via the Fundamental Theorem of Asset Pricing (FTAP) The FTAP states that if there is no arbitrage, then the price of any asset is equal to its expected discounted value under the risk neutral probability measure In a two state two period world with risk-free rate r, this is equivalent to saying that the t price of any asset X equals X t = e r(t t) [q u X T (u) + q d X T (d)] where q u + q d = 1 are the risk-neutral probabilities and X T are payoffs (or (1 + r) T for discrete compounding)
FTAP PRICING EXAMPLE Practice: Using our previous example, compute the call price using the FTAP Solution: Recall that r = 0 in the previous example. The call price is given by the FTAP equation c t = (1 + r) T [q u c T (u) + (1 q u )c T (d)] We know that c u = 1 and c d = 0. But what is q u?
FTAP PRICING EXAMPLE One way is to compute q u using the formula in Jason s notes Another way is to notice that since the FTAP must hold for any security, it must hold in particular for the stock 1 = S t = (1 + r) 1 (q u S u + (1 q u )S d ) = 1 (2q u + 0(1 q u )) = q u = 1 2 Plugging this in yields c 0 = 1 [ 1 2 1 + 1 2 0] = 1 2
FTAP PRICING EXAMPLE Now that we have q u pricing any asset is easy! For example, the put price is [ 1 p t = 1 2 p T(u) + (1 1 ] 2 )p T(d) = 1 2 And a call with strike K = 1 2 has a price of [ 1 c t = 1 2 (2 0.5)+ + 1 ] (0 0.5)+ = 0.75 2 And a put with strike K = 3 has a price of [ 1 p t = 1 2 (3 2)+ + 1 ] (3 0)+ = 2 2
FTAP PRICING COOKBOOK Pricing options via the FTAP always follows the same steps 1. Calculate the payoffs of the option. If this is already done for you, then do nothing. 2. Calculate the risk-neutral probabilities. Either use the formula in the notes, or use the stock price and payoffs to extract q u 3. Write the FTAP equation for the option out explicitly 4. Plug in q u, the option prices, and the risk-free rate and compute the option price