Homework Assignment 11 (Ch 16, 17) Question 6 (of 10) 1.00 points You are attemp
ID: 2801152 • Letter: H
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Homework Assignment 11 (Ch 16, 17) Question 6 (of 10) 1.00 points You are attempting to value a call option with an exercise price of $65 and one year to expiration. The underlying stock pays no dividends, its current price is $65, and you believe it has a 50% chance of increasing to $90 and a 50% chance of decreasing to $35. The risk-free rate of interest is 8%. Based upon your assumptions, calculate your estimate of the the call option's value using the two-state stock price model. (Do not round intermediate calculations. Round your answer to 2 decimal places.) Value of the call Check my workExplanation / Answer
The two assets on which the valuation depends are the call option and the underlying stock.
The underlying stock price can move from current $65 to either 50% chance of increasing to $90 or 50% chance of decreasing to $35 in one year’s time, and there are no other price moves possible.
In an arbitrage-free world, if we have to create a portfolio comprising of these two assets (call option and underlying stock) such that irrespective of where the underlying price goes ($90 or $65), the net return on portfolio always remains the same.
Suppose we buy ‘d’ shares of underlying and short one call option to create this portfolio.
If the price goes to $90, our shares will be worth $90*d and we’ll lose $25 on short call payoff.
The net value of our portfolio will be (90d – 25)
If the price goes down to $35, our shares will be worth $35*d, and option will expire worthless. The net value of our portfolio will be (35d).
If we want the value of our portfolio to remain the same, irrespective of wherever the underlying stock price goes, then our portfolio value should remain the same in either cases, i.e.:
=> (90d – 25) = 35d
=> d = 0.4545
i.e. if we buy 0.4545 a share (assuming fractional purchases are possible), we will manage to create a portfolio such that its value remains same in both possible states within the given timeframe of one year. (point 1)
This portfolio value, indicated by (35d) or (90d -25) = 15.905, is one year down the line.
To calculate its present value, it can be discounted by risk free rate of return (8%).
=> 35d * exp(8%*1 year) = 15.905* 0.9259 = 14.726 => Present value of the portfolio
Since at present, the portfolio comprises of 0.4545share of underlying stock (with market price $65) and 1 short call, it should be equal to the present value calculated above i.e.
=> 0.4545*65 – 1*call value = 14.726
=> Call value = $ 14.817 i.e. the call price as of today.
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