1. The European put option on Summer.com with an exercise price of $50 expiring
ID: 2732476 • Letter: 1
Question
1. The European put option on Summer.com with an exercise price of $50 expiring a year from now isselling for $10.60, and the risk free rate is 4%. Suppose that you would like to establish a long position in aone-year European call option written on the stock with a strike price of $50. You could buy the call in theoptions market, where it is currently trading for $7.90. The current stock price is $45. As you are about tosubmit your order, it just occurs to you that in F303 you learned that you could also create the positionsynthetically by using a replicating portfolio.
(a) Describe the replicating portfolio by demonstrating that the payoff from your portfolio isidentical to that of the call option no matter what the stock price is on the expiration day a year
from now. Please use a payoff diagram or payoff table to prove your claim.
(b) What is the price of the replicating portfolio? Is the call correctly priced relative to thereplicating portfolio?
(c) If the call is mispriced in the option market, describe the arbitrage trade you could engage in tocapitalize in on the mispricing.
Explanation / Answer
Replicating a portfolio is a method of determining a present value of a given set of cash flows by constructing combinations of investments that will give you that cash flow. To a certain extent , the concept of discounted cash flow is also a portfolio replication but with a hypothetical risk free investment or an estimate of the alternative opportunity investment returns 1 If you want to establish a long position in a one year Euro call, this can be replicated by buying the stock and going long a one year put. THis will have the same payoff structure as the long call. ie upside is theoetically unlimited, downside is limited when the stock price drops below $50. It's difficult to show this without a payoff diagram but it profit is on the y axis and time on the stock price on the x axis, the graph will start at a negative value on the y axis and proceed in a straight line to the right and start to increase in astraight line once $50 mark on the x axis is passed. Step 1- Buy Puts. A simple strategy to hedge your portfolio is to buy some option puts. A put is a contract that makes money when the stock or index that it represents falls in price Step 2- Sell Calls. Perhaps a better way to hedge your portfolio is for you to sell calls either on the stock This is called “Covered Call Writing”. This is when you own a stock and sell calls against that stock, thus generating a small amount of income Step 3- Sell calls on an index In regards to options, the problem is that unlike bonds or dividends, the cash flow is not defined nor obligated. When buying an at the money call option, the efficient market hypothesis puts the probability of it being in the money on expiration at 50% with the actual return dependent on the difference between the market price and the strike price on that date. The Black Scholes equation uses a log normal probability density function to estimate the future value and you could perhaps simplify the random walk to say that the most probable in the money price is the current price times e^( mu + sqrt( n ) * v ) where mu is the average of the logarithms of the growth factors (days closing price divided by previous days close), n is the number of trading days into the future and v is the volatility which would be the standard deviation of the logarithms of the growth factors. You could then use Fermat's Expected Value Theorem and multiple the profit from that future price by 0.5 to give you the expected value of the cash flow and then construct your replicated portfolio accordingly. 2 The concept of portfolio replication isn't really to build portfolios with the same risks and benefits as that would most easily be done by assuming such risks and benefits to begin with but to determine an equivalent present value. 3 However if you must construct a portfolio using put options to approximate the purchase of a call option, you could short a put option thats well in the money with a strike price above a price range that you deem likely effectively going long on the equity (you could just buy the stock too) and buy a put at the strike that you wished the strike of the call option that you are trying to synthesize at but even with the prices you've listed, it would be cheaper to just buy the call option. Attempting to replicate the portfolio with investments that are not derivatives of the stock in question would be probabilistic in nature and would require attentional information
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.