Question

A stock price is currently $50. it is known that at the end of two months it will be either $53 or $48. the risk-free interest rate is 10% per annum with continuous compounding. what is the value of a two-month european call option with a strike price of $49? use no-arbitrage arguments.

The value of the option is therefore $2.23.
This can also be calculated directly from equations (12.2) and (12.3). u=1.06, d=0.96 that


p= (e⁰.¹⁰ˣ²/¹²) -0.96/1.06-0,96= 0.5681
and
f = e⁰.¹⁰ˣ²/¹² x 0.5681x4= 2.23

Answer 1

Bond valuation is subject to change based on market interest rates, with prices adjusting to maintain the no-arbitrage condition. An increase in interest rates will decrease the price of existing bonds, whereas a decrease in market rates will make existing bonds with higher rates more valuable. This concept ensures that the yield of a bond aligns with current market conditions.

Step-by-step explanation:

Assessing the value of a bond requires understanding the relationship between the bond's fixed payment streams and the prevailing market interest rates. When interest rates rise, newly issued bonds offer higher yields, making existing bonds with lower interest less attractive. To maintain the no-arbitrage condition, the price of these existing bonds must decrease to align the bond's yield with current rates.

Let's consider a $3,000 bond with an 8% coupon rate, maturing in two years. If the discount rate is also 8%, the bond's present value equals the sum of the discounted interest payments plus the discounted principal repayment. However, if the market interest rate rises to 11%, the bond's price will drop because the present value of its future payments becomes less when discounted at the higher rate.

Using the present value formula, we calculate:

• The bond's price with a discount rate of 8% (which will be equal to face value since the rates are identical).
• The bond's adjusted price with a discount rate of 11%, where it will be less than the face value to compensate investors for the now below-market coupon rate.

For a one-year bond originally valued at $1,000, if market rates go up to 12%, no investor would pay more than the price that would yield 12% over the course of the year. This price is calculated to be $964, as $964 invested at 12% interest would result in $1,080 after one year, matching the bond's expected payment.

Similarly, if an unattractive investment with a lower interest rate exists on the market, its price must decrease to offer a yield that matches the higher rates available elsewhere.