Suppose a stock now sells at S- $100, and the price will either increase to $130

Question

Suppose a stock now sells at S- $100, and the price will either increase to $130 or fall to $90 by year-end. However you don't know the probabilities. Suppose ,y-10%. Answer question 29-31 based on the information above. 29. What will be the year-end payoff to an investor who has purchased 1 share of this stock and written 2 calls at exercise price of S1102 a. $90 b. $130 c. $170 d. either $90 or $130 e. None of the above options is correct 30. What should be the value of the call option with exercise price of $110? a $4.04 b. $6.06 c. $9.09 d. $14.09 None of the above options is correct e. 31. An investor established a risk-free position by purchasing 1 share of the stock and 4 put options with exercise price of $100. What should be the value of the put option? a. $4.04 b. $4.55 c. $6.06 d. $18.18 e. None of the above options is correct

Explanation / Answer

We will use the one step binomial model and no arbitrgae assumption to solve this problem. Given that S0 can move to either $130 or $ 90 by the year end, a portfolio can be created where 'x' units of stock are purchased along with selling 1 call option. The net pay off for this portfolio after 1 year can be either (130x - 20) - the stock price moves to $ 130 and there will be loss of $ 20 on the short call option - or (90x) - the stock price is $ 90 and short call option value is zero - depending upon where the price of stock ends after 1 year. Under the no arbitrage assumption, the value of 'x' should be such that the pay off for this portfolio in both the cases should be same. Hence,

130x - 20 = 90x which when equated for 'x' will give x = 0.5 (assuming fractional purchase is possible). At x=0.5, the value of the portfolio in either price scenario shall $ 45. If we discount this portfolio value at risk free rate of 10%, the present value shall be $ 40.91

Given the present value after 1 year, the portfolio value today should be equal to this otherwise there will arbitrage possible. Hence 100*0.5 - 1 CE = 40.91; where CE is the call option. Solving for CE, we get the value of call option to be $ 9.09.

Now we can try to answer the questions above:

Ans. 29. The year end pay off for the investor purchasing 1 stock and selling 2 CE shall be as below:

If S1 = 130; then $130 - 20 - 20 = $ 90

If S1 = 90; then $ 90 (since the call options shall be zero in value)

Hence option (a) is correct since the pay off in both cases shall be $ 90.

Ans. 30. We calculated the CE price to $9.09 above; hence option (c)

Ans. 31. The pay off for purchasing 1 share and 4 PE at excercise price of $ 100 will have following pay off:

If S1 = 130; then $130 = $ 130 (since the put options shall be zero in value)

If S1 = 90; then $ 90 + 10 * 4 = $ 130 (pay off of $ 10 on each put option)

SInce the pay off is same at $ 130, its is a risk free portfolio. We can calculate the present value by discounting it by the risk free rate of 10%, and we get the present value = $ 118.18.

Given the present value after 1 year, the portfolio value today should be equal to this otherwise there will arbitrage possible. Hence 100 + 4 PE = 118.18. Solving for PE, we get the value of call option to be $ 4.55. Hence the option (b) is the correct answer.

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