Question

A stock is currently priced at $35.00. The risk free rate is 3.2% per annum with continuous compounding. In 4 months, its price will be either $39.90 or $31.15.

Consider the portfolio with the following: long a European call with strike $39.00 expiring in 4 months; a short futures position on the stock with delivery date in 4 months and delivery price $40.00; a derivative which pays, in 4 months, three times the price of the stock at that time.

Using the binomial tree model, compute the price (or "value") of this portfolio.

Answer 1

Ans. :-

• First, we shall calculate the probability of stock price increasing to $39.90 and going down to $31.15 denoted as P & (1-P) respectively. Please find the detailed calculation in the image below:

• Here, Risk free rate = 3.12 % p.a. Hence, for the period of 4 months, we shall consider

(i). Value of the European call :-

• If price of the stock increases to $39.90, then only the call option will be exercised at $39 and it will give profit of $0.90
• If stock price goes to $31.15 in four months, then the call will not be made at exercise price of $39.
• To find the value of the call option, we shall calculate the present value of the future expected cash flows from the call after 4 months from now.
• Please find the detailed calculation in the image attached below :

(ii). Value of short futures position :-

• we have obligation to deliver a stock after 4 months at price=$40.
• For this deliver, we would have to buy the stock from the market at the spot rate after 4 months.

• Please find the detailed calculation in the image attached below :

(iii). Value of Derivative :-

• Derivative would pay 3 times the spot price of the stock after 4 months.
• if stock price increases to $39.90, it would pay $159.6
• if stock price goes down to $31.15 , it would pay $124.6
• Hence, to find the value of the derivative we shall calculate the present value of the cash flow that we might receive after 4 months.
• Please find the detailed calculation in the image attached below :

Price of the Portfolio :-

Particulars	Amount
Value of long call option	$0.43
Value of short future	$4.6
Value of Derivative	$139.87
Total value of portfolio	$144.9

(Note :- Calculations are rounded off upto 2 decimal points.)