Question

Question I Value Derivatives

An investor is very bullish on LewCo, a nondividend paying company. The current spot price of the company’s equity is $50 per share. The investor is confident the company will solve the social distancing issue and that the value of the stock will be at least $100 in 12 months.

The investor uses a 12-period binomial tree assuming

S0 = $50

T = 12 months

r = 30 basis points per month

u=1.1 per month

d = 1/u= 1.1-1 per month 1.

1. Apply the binomial tree model to value a 12-month European style call with a strike of $100. What is the call premium?

2. Use put-call parity to value the put on the same asset, for the same expiration, and with the same strike. What is the put premium?

3. The investor is considering two other calls. One with a strike of $200. Another with a strike of $0. The values of these structures should be evident to you. What is the premium on the call with a strike of $200? Why? What is the premium on the call with the strike of $0? Why?

4. Why do investors use options? To answer this, we consider an investor with $5,000 to invest. Assume the spot price in 12 months is $100, as predicted by the investor. Calculate the amount the investor will make spending the $5,000 to buy shares of stock, spending the $5,000 to buy puts with a strike of $100, or spending the $5,000 to buy calls with a strike of $100? Based on this analysis, what is the advantage of using options?

5. What would be the premium of a derivative structure paying the square of the call payoff in each node of the terminal distribution?

Answer 1

Call Premium is $0.18

Value of Node at T=12 = Max (stock price - strike price, 0)
Value of Node at T=11 = [(Prob of Up Move * Value of call option on up move) + (Prob of Down Move * Value of call option on the down node)] / (1 + Rf)

Probability of Up Move = [(1 + Rf) - d] / (u - d)
where u = up move
d = down move

  

2) Put-Call Parity

Stock + Put Option = Call Option + X/(1+Rf)n
50 + x = 0.18 + 100/(1+0.3%)12
50 + x = 0.18 + 96.47
x = 46.65

Put Option Premium is $46.65

3) Value of Call option if the strike price is $200 = $0. This is because the option is Out of the Money till expiry i.e. the price of the stock doesn't move more than the strike price of $200. The maximum the stock price can go at the end of 12 months is $156.92

Value of Call Option if the strike price is 0 = $50. This is because the option is In the Money throughout its life i.e. Price of stock at all the nodes is more than $0 and the least price of the stock at the end of 12 periods is $15.93 (which will still be in the money)

4) Assuming that 1 option contract has a market lot of 100 stocks

By investing $18 and $4665 in a call and put option respectively, I will get the same exposure instead of buying 100 stocks for $5000. The leftover amount in call and put option can be used for other purposes or invested at the risk-free rate. Thus with less capital deployed I can get the same amount of exposure by using option and the max loss in case of buying a call and put option is the premium amount paid i.e. $18 and $4665 for a call as well as the put option, respectively.