CFA QA

CFA Q & A

CFAQ & A14-06-2015Ex pp59: A 1-year call option on Cross Reef Inc. with an exercise price of $60 is trading for $8.The current stock price is $62. The risk-free rate is 4%. Calculate the price of the synthetic put option implied by put-call parity.

C0 + [ X / (1+r) t ] = P0 + S0 Fiduciary call = Protective putAccording to put-call parity, the price of the synthetic put option is: Synthetic put: p = c S + [x/(1+r)t] = 8 - 62 + [60/ (1+.04)] = $ 3.69

Example: pp59/60: Exploiting violations of put-call parity The stock of ArbCity Inc. is trading for $75. A 3-month call option with an exercise price of $75 is selling for $4.50, and a 3-month put at $75 is selling for $3.80. The risk-free rate is 5%. Calculate the no-arbitrage price of the put option, and illustrate how the violation of put-call parity can be exploited to earn arbitrage profits.

First, calculate what the put option should be selling for, given the other prices:P = c S + [x/(1+r)t] = 4.50-75+[75/(1.05)0.25] = $3.59Put-call parity doesnt hold because the actual market price of the put is $3.80, not the no-arbitrage price of $3.59: c + [x/(1+r)t] P + S , 4.50+74.09 3.80+75 , 78.59 78.80The fiduciary call (the left side of the equation) is relatively underpriced, and theprotective put (the right side) is relatively overpriced. Therefore, the arbitrage strategy is to buy the fiduciary call (and pay $78.59) and short-sell the protective put (and receive $78.80). TThe arbitrage profits from this trade are $78.80 $78.59, or $0.21 per share.The net cash flow at maturity will be zero, so weve produced cash flow of $0.21 today with no cash outflow obligation in the futurethats what we call an arbitrage profit.

One period Binomial ModelShare of stock current price = $30Size of possible price change & probability of these change

U = Size of up move = 1.333 (33.3%)D = size of down move = 1/U = 1/1.333 = 0.75 (-25%) U = probability of up-move = 0.55 D = probability of down move= 1- U = 1-0.55= 0.45

The one-period binomial tree for the stock is shown asOne period Binomial Model Today One YearThe probabilities of an up-move and a down-move are calculated based on the size of the moves and the risk-free rate as: u = risk-neutral probability of an up-move= 1+ Rf D/ U D D = risk -neutral probability of a down-move = 1 D

P62 Example: Calculating call option value with a one-period binomial tree Use the binomial tree in Figure 3 to calculate the value today of a one-period call option on the stock with an exercise price of $30. Assume the risk-free rate is 7%, the current value of the stock is $30, and the size of an up-move is 1.333. D = size of downward move =1/U = 1/1.333 = 0.75u = risk-neutral probability of an up move = 1 + 0.07 0.75 / 1.333 0.75 = 0.55

D = risk neutral probability of adown move = 1 0.55 = 0.45

Next, determine the payoffs to the option in each state. If the stock moves up to $40, a call option with an exercise price of $30 will pay off $10. If the stock moves down to $22.50, the call option will be worthless. The option payoffs are illustrated in the following figure.

Let the stock values for the up-move and down-move be S1+ and S1 and for the call values, c1 + and c1- One period call option with X= $30 Today One YearThe expected value of the option in one period is:E(call option value in 1 year) = ($10 0.55) + ($0 0.45) = $5.50

The value of option today, discounted at Rf of 7% C0 = $5.50 /1.07 = $5.14 u = 0.55 D = 0.45P63:Example: Valuing a one-period put option on a stock Use the information in the previous two examples to calculate the value of a put option on the stock with an exercise price of $30.If the stock moves up to $40, a put option with an exercise price of $30 will be worthless. If the stock moves down to $22.50, the put option will be worth $7.50.The risk-neutral probabilities are 0.55 and 0.45 for an up- and down-move, respectively. The expected value of the put option in one period is:

E(put option value in 1 year) = ($0 0.55) + ($7.50 0.45) = $3.375

The value of the option today, discounted at the risk-free rate of 7%, is: P0 = $3.375/ 1.07 = $ 3.154 Arbitrage with a One-Period Binomial Model Example P64: Calculating arbitrage profitAssume the option in the previous example is actually selling for $6.50. Illustrate how this arbitrage opportunity can be exploited to earn an arbitrage profit. Assume we trade 100 call options.AnswerBecause the option is overpriced, we will short 100 options and purchase a certain number of shares of stock determined by the hedge ratio: Delta = $10-$0/$40-$22.5 =0.5714 shares per option total shares of stock to purchase =1000.5714 =57.14

A portfolio that is long 57.14 shares of stock at $30 per share and short 100 calls at $6.50 per call has a net cost of: net portfolio cost = (57.14 $30) (100 $6.50) = $1,064 Arbitrage with a One-Period Binomial Model Example P64: Calculating arbitrage profitThe values of this portfolio at maturity if the stock moves up to $40 or down to $22.50 are:portfolio value in up-move = (57.14 $40) (100 $10) = $1,286portfolio value in down-move = (57.14 $22.50) (100 $0) = $1,286

The return on the portfolio in either state is: Portfolio return = $1286 / $1064 - 1 = 0.209 =20.9% Arbitrage with a One-Period Binomial Model Example P64: Calculating arbitrage profitThis is a true arbitrage opportunity! Weve created a portfolio that earns a return of 20.9% no matter what happens next period. That means its risk free. The actual risk-free rate in the market is 7%. The profitable arbitrage trades are to:

Borrow $1,064 at 7% for one year. In one year, well owe $1,064 1.07 = $1,138.48.

Buy the hedged portfolio for $1,064.

In one year, collect the $1,286, repay the loan at $1,138.48, and keep the arbitrage profits of $147.52.Calculating deltaDelta, the change in the price of an option for a one-unit change in the price of the underlying security:

Delta call = C1 C0 / S1 S0 = C / SC = change in the price of the call over a short time intervalS = change in the price of the underlying stock over a short time interval

Example: Calculating deltaDuring the last ten minutes of trading, call options on Commart Inc. common stock have risen from $1.20 to $1.60. Shares of the underlying stock have risen from $51.30 to $52.05 during the same time interval. Calculate the delta of the call option

Delta call = $1.60 - $1.20 /$52.05 - $51.30 = 0.40/0.75 =0.5331. C ompare the call and put prices on a stock that doesnt pay a dividend (NODIV) with comparable call and put prices on another stock (DIV) that is the same in all respects except it pays a dividend. Which of the following statements is most accurate? Price of:A. DIV call will be less than price of NODIV call.B. NODIV call will equal price of NODIV put.C. NODIV put will be greater than price of DIV put.

2. I n a one-period binomial model, the hedge ratio is 0.35. To construct a riskless arbitrage involving 1,000 call options if the option is overpriced, what is theappropriate portfolio?

Calls StockA. Buy 1,000 options Short 350 sharesB. Buy 1,000 options Short 2,857 sharesC. Sell 1,000 options Buy 350 shares3. Which of the following statements is least accurate? The value of a:A. call option will decrease as the risk-free rate increases.B. put option will decrease as the exercise price decreases.C. call option will decrease as the underlying stock price decreases7. A synthetic European put option is created by:A. buying the discount bond, buying the call option, and short-selling the stock.B. buying the call option, short-selling the discount bond, and short-selling the stock.C. short-selling the stock, buying the discount bond, and selling the call option.SWAP1. T he current U.S. dollar ($) to Canadian dollar (C$) exchange rate is 0.7. In a $1 million plain vanilla currency swap, the party that is entering the swap to hedge existing exposure to a C$-denominated fixed-rate liability will:A. receive $1 million at the termination of the swap.B. pay a fixed rate based on the yield curve in the United States.C. receive a fixed rate based on the yield curve in Canada.2. A European-style payer swaption gives its holder the:A. right to enter a swap by paying the strike price at expiration of the option.B. right to enter a swap at expiration of the option as the fixed-rate payer.C. option to take either side of a swap at a certain date in the future, by making a current payment.3. At what point in a swaps life is the credit risk in an interest rate swap the greatest?A. In the middle.B. At the end, when the largest payments are due.C. At the beginning, because all the payments remain.4. C onsider a 6-year plain vanilla currency swap in which we will receive LIBOR semiannually and pay 9% fixed semiannually in British pounds (). The notional value of the swap is 50 million. The current spot rate is $1.50 per . Which of the following transactions would replicate the payoffs to the pay-fixed side of the swap? Bond to issue Bond to purchaseA. 9% fixed, 50 million LIBOR, $33 millionB. 9% fixed, 50 million LIBOR, $75 millionC. LIBOR, 112 million 9% fixed, $33 million11. The equity return receiver in an equity return-for-fixed-rate equity swap receives payments that are equivalent to which of the following strategies?A. Buy and hold the index stocks and issue a fixed-rate bond.B. I ssue a bond, short the index stocks, and adjust the short position value to the principal amount at each payment date.C. Buy the index stocks, adjust the portfolio value to the notional amount of the swap at each payment date, and issue a fixed-rate bond.