European options

Increasing competition for LNG supplies with North American and Pacific importers may constrain European options regarding additional large-scale gas deliveries dedicated to Europe. 

In the following, we derive a theoretical pricing framework for the computation of options on bond applying standard Fourier inversion techniques. Starting with a plain vanilla European option on a zero-coupon bond with the strike price K, maturity T of the underlying bond and exercise date To of the option, we have... 

Starting from the payoff function of a European option on a coupon bearing bond we can write the option price at the exercise date Tq as follows... 

European option B S is not very useful because it assumes the exercise of the option at maturity only, while it does not consider the possibility of exercising for several dates ... 

This is the same model as that proposed by Rendleman and Bartter. This model is the only log-normal, single-factor model that leads to closed formulae for pure discount bonds. Nonetheless there is no closed formula for a European option on a pure discount bond. 

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 ... 

Discount bond options are not very liquid, but they form an elementary component for pricing other options. For example, a floating-rate cap can be decomposed into a portfolio of European puts on discount bonds. Similarly with the European option contingent on the bank account we can price European options contingent to discount bonds. 

There is a similar put-call parity for European options contingent on a discount bond. If Pp(0, t) Tq-,k he price at r = 0 of a European put option on the discount bond with maturity T, then for B(0) = 1,... 

Under the one-factor HJM model corresponding to the Ho-Lee model, a European option on a coupon bond can be valued as a portfolio of options contingent on zero discount bonds with maturities Ti,T2,...,Tm- Let Tq be the maturity of such a European option. 

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. 

Since the possibility of early exercise, which represents the chief difference between American and European options, is rarely actualized, the two types of contracts generally have equivalent values. Pricing models, however, calculate the probabilities that different options will be exercised. For American options, this entails determining what circumstances make early exercise more likely and assigning the contracts higher prices in these situations. 

If the owner of an option elects to exercise it and enter into the underlying trade, the option writer is obliged to execute under the terms of the contract. The price at which an option specifies that the underlying asset may be bought or sold is the exercise, or strike, price. The expiry date of an option is the last day on which it may be exercised. Options that can be exercised anytime from the day they are struck up to and including the expiry date are called American options. Those that can be exercised only on the expiry date are known as European options. 

ABC has in effect embedded ten European options in the bond, each relating to 5 million nominal of the bonds and each expiring on December 1 of a different year, starting in 2009 and ending with 2018. The decision to exercise the options as they mature is made using the binomial-tree method discussed earlier. 

In a simple type of derivative, a European option, we purchase at time t the right to either buy (a call option) or sell (a put option) the underlying asset at some time r > r in the future at an exercise price E. Thus, at time T, if we purchase the option, we will have a payoff... 

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