European call option

Thus, we end up with the well known Black and Scholes -like formula for the price of a European call option on a zero-coupon bond... 

European call option For which the issuer can recall the bond only once on the call date ... 

In this section we consider the pricing of a European option on the money fund. (This is the same as a bank account when the initial value B(0) = 1.) Thus, the payoff of a European call option with exercise price K is max[B(T) - K,0], The continuous version of the Ho-Lee model is assumed for the short interest rate process. The risk-neutral valuation methodology provides the solution as ... 

Initially the first formulas on pricing options on pure discount bonds used the Vasicek model for the term structure of interest rates. Thus, given that r follows equation (18.6), the price of a European call option with maturity Tq with exercise price fC on a discount bond maturing at T(Tq < T) is... 

We will use the Vasicek model for pricing a 3-year European call option on a 10-year zero-coupon bond with face value 1 and exercise price K equal to 0.5. As in Jackson and Staunton, we use for the parameters of this model the values estimated by Chan, Karolyi, Longstaff, and Sanders for US 1-month Treasury bill yield from 1964 to 1989. Thus a = 0.0154, p = 0.1779, and o = 2%. In addition, the value of the short... 

EXHIBIT 18.6 Calculations of Elements for Pricing an European Call Option on a Zero-Coupon Bond when Short Rates are Following the Vasicek Model... 

Taking the same example as that developed to demonstrate the Vasicek model earlier, we now price the 3-year European call option on a 10-year pure discount bond using the CIR model for the short interest rates. Recall that face value is 1 and exercise price K is equal to 0.5. As in the example with the Vasicek model, we consider that o = 2% and tq = 3.75%. The CIR model overcomes the problem of negative interest rates (acknowledged as a problem for the Vasicek model) as long as 2a > o. This is true, for example, if we take a = 0.0189 and P = 0.24. Feeding this information into the above formulae is relatively tedious. A spreadsheet application is provided by Jackson and Staunton, After some work we get that the price of the call is... 

Hence, the value at time 0 of a European call option with maturity Tq and strike price K on the coupon bearing bond, under the one-factor HJM model described above, is given by... 

EXHBIT 18.7 Calculations Using Vasicek Model for Separate Zero-Coupon European Call Options the Bond Prices Shown are Calculated with the Estimated... 

We now revisit the earlier Vasicek example for short interest rates to consider the case where the underlying bond pays an annual coupon at a 5% rate (p = 0.05), all the other characteristics remain as before. In order to calculate the call price of the coupon-bond European option first we need to calculate the interest rate such that the present value at the maturity of the option of all later cash flows on the bond equals the strike price. This is done by trial and error using equation (18.48) and the value we get here is = 22.30%. Next, we map the strike price into a series of strike prices via equation (18.50) that are then associated with coupon payments considered as zero-coupon bonds and calculate the value of the European call options contingent on those zero-coupon bonds as in the above example. The calculations are described in Exhibit 18.7. 

It is clear now that this is the same as a European call option on a coupon-bearing bond when the exercise price is equal to 1. 

The Black-Scholes model is neat and intuitive. It describes a process for calculating the fair value of a European call option, but one of its many attractions is that it can easily be modified to handle other types, such as foreign-exchange or interest rate options. 

The B-S model is based on the resolution of partial differential equation (8.15), given the appropriate parameters. The parameters refer to the payoff conditions corresponding to a European call option. 

Equation (8.21) can be simplified as (8.22), the well-known Black-Scholes option pricing model for a European call option. It states that the fair value of a call option is the expected present value of the option on its expiry date, assuming that prices follow a lognormal distribution. 

Calculate the price of a European call option with a strike price of 100 and a maturity of one year, written on a bond with the following characteristics ... 

This model is based upon a treatment of the asset price as a random walk, the mathematics of which will be discussed in Chapter 7. While an analytical solution exists for this plain vanilla option, more realistic cases generally require numerical solution. Write a program that computes V(S, t) for a European call option. For more on this subject, consult Wilmott (2000). 

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