The common stock of Company XLT and its derivative securities currently trade in the market at the following prices and contract terms: Price ($)Exercise Price ($)Stock XLT $       21.50 $              -  Call Option on Stock XLT $         5.50 $       21.00Put Option on Stock XLT $         4.50 $       21.00Both of these options will expire 91 days from now, and the annualized yield for the 91-day Treasury bill is 3.0 percent.a. Briefly explain how to construct a synthetic Treasury bill position.b. Calculate the annualized yield for the synthetic Treasury bill in part (a) using the market price data provided. c. Describe the arbitrage strategy implied by the difference in yields for the actual and synthetic T-bill positions. Show the net, riskless cash flow you could generate assum­ing a transaction involving 21 actual T-bills and 100 synthetic T-bills.d. What is the net cash flow of this arbitrage strategy at the option expiration date, assuming that Stock XLT trades at $23 at expiration three months from now?

Question
Asked Apr 28, 2019
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The common stock of Company XLT and its derivative securities currently trade in the market at the following prices and contract terms:
  Price ($) Exercise Price ($)
Stock XLT  $       21.50  $              -  
Call Option on Stock XLT  $         5.50  $       21.00
Put Option on Stock XLT  $         4.50  $       21.00
Both of these options will expire 91 days from now, and the annualized yield for the 91-day Treasury bill is 3.0 percent.
a. Briefly explain how to construct a synthetic Treasury bill position.
b. Calculate the annualized yield for the synthetic Treasury bill in part (a) using the market price data provided. 
c. Describe the arbitrage strategy implied by the difference in yields for the actual and synthetic T-bill positions. Show the net, riskless cash flow you could generate assum­ing a transaction involving 21 actual T-bills and 100 synthetic T-bills.
d. What is the net cash flow of this arbitrage strategy at the option expiration date, assuming that Stock XLT trades at $23 at expiration three months from now?
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Expert Answer

Step 1

Part (a)

Let's denote current stock price as S0, call premium as C, put premium as P, strike price as K, present value of K as PV (K),  risk free rate as r and time to expiration as t then,

Call Put Parity Equation:

C + PV (K) = S0 + P

Or, C + K x (1 + r)-t = S0 + P

Hence, K x (1 + r)-t = S0 + P - C

Hence, a synthetic Treasury bill position is:

  • Buy (long) a stock
  • Buy (long) a put option
  • Short a call optionj

 

Step 2

Part (b)

K x (1 + r)-t = S0 + P - C

21 x (1 + r)-t = 21.50 + 4.50 - 5.50 = 20.50

t = 91 days = 91 / 365 year

Hence, (1 + r)-91/365 = 20.50 / 21

Hence, r = (20.50 / 21)-365/91 - 1 = 10.15%

Step 3

Part (c)

Actuall  bill rate, = 3%

Since, actual t bill rate is different from no arbitrage t bill rate, hence there exists an arbitrage opportunity. 

Assum­ing a transaction involving

  • Short 21 actual T-bills and
  • Long 100 synthetic T-bills i.e.
    • Buy 100 stocks
    • Buy 100 put options
    • Short 100 call options

Initial cash flows at t= 0 will be : 

  • Proceeds from sale of 21 actual T bil...

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