Consider an 8 -month European put option on a Treasury bond that currently has 14.25 years to ma

Consider an 8 -month European put option on a Treasury bond...

Key Concepts

Risk-Free Interest Rate

The risk-free interest rate is used in option pricing to discount future cash flows to their present value. It reflects the time value of money under a risk-neutral measure, ensuring that the valuation appropriately scales with the term to expiration of the option.

Strike Price Definition

The strike price in option pricing is the predetermined level at which the underlying asset can be bought or sold upon exercise. In bond options, the definition of the strike price—whether based on the cash price or the quoted price—directly affects the option's value, since it determines the basis of the payoff and how the underlying asset’s price is interpreted.

European Option

European options are financial derivatives that can only be exercised at expiration. This characteristic simplifies pricing because it removes the complexity associated with early exercise features, allowing the use of models that only consider the option's value at maturity.

Coupon Payments

Coupon payments represent periodic cash flows from a bond and affect option pricing by altering the underlying bond's value. In the context of bond options, these cash flows must be accounted for—similar to dividends in equity options—as they result in price adjustments on the ex-coupon dates.

Black’s Model

Black’s model is an adaptation of the Black-Scholes framework tailored for pricing options on instruments like bonds or interest rate derivatives. It assumes that the underlying asset follows a lognormal distribution and uses a forward or cash price specific to the instrument, incorporating volatility, time to expiration, and the risk?free rate to determine the option's value.

Transcript

00:01 This problem covers the concepts in coupon bonds and interest rates.

00:05 We're given a bond with a 6 % annual coupon rate and a face value of $1 ,000.

00:11 We're asked to find the current price of the bond given its ears to maturity and yield to maturity.

00:17 In other words, our task here is to fill in the rightmost column of the table below.

00:23 So to find the current price, in other words, the present value of the bond, let's first write out the formula for the present value.

00:31 So i'll write pv, short for present value.

00:34 Okay, it's computed like this.

00:38 It's computed as the coupon payment, okay, of the first year discounted, okay, by its, by 1 plus its yield to maturity.

00:48 I stands for its yield to maturity.

00:51 And that's the first year.

00:53 And we'll add the coupon payment in the second year, discounted two years ahead.

00:59 And coupon payment in the third year discounted three years ahead.

01:01 And all the way to the coupon payment of the nth year, discounted n years ahead, using the yield to maturity.

01:12 And lastly, we have to add in the future value discounted throughout the whole n years.

01:21 So this is how we compute the present value.

01:25 In a problem, we're given a 6 % annual coupon rate.

01:30 In other words, a coupon payment every year would be a 1 ,000.

01:35 Thousand dollars okay it's face value actually times six percent which is going to be sixty dollars and its face value like i mentioned before it's given as a thousand dollars okay and this is just a one -time payment at the very in the very last year that the bond matures so to find the current price okay so i'll write down you'll yield to maturity.

02:13 I'll copy the table down here.

02:16 The first column is yield to maturity.

02:18 The second column is, oh, sorry, the first column is years to maturity.

02:29 So i'll write years.

02:34 Tm stands to maturity.

02:36 And then the second column is yield to maturity.

02:43 And the third column is current price.

02:50 And our objective is to compute the third column.

02:53 The formula above, so first of all, let's populate these rows first.

03:06 And this is 4%, 6%, 6%, 4%, less than 8%.

03:20 Right.

03:21 So to find the current price or the present value of the bond for each row, we will follow this formula above and do the calculation of it...