The Black Model for Interest Rate Derivatives - Essay Example

? The Black Model for Interest Rate Derivatives Table of Contents Introduction 2 Overview and Development of Black Model2 Applications of Black Model in pricing European Options 4 Validity of Black Model 5 Other Applications of Black Model 6 Conclusion 7 Bibliography 8 Introduction The following is a discussion and evaluation of the Black Model for interest rate derivates. Despite existence of other models for interest rate derivatives, the Black Model has extensively been applied within financial markets. This paper attempts an evaluation of the model whilst developing a concluding remark on the model with reference to derivatives markets. Over the last two and half decades, finance has experienced tremendous and exciting developments especially with reference to derivatives markets. One of the reasons explaining the idea of tremendous and exciting developments within financial sector is the fact that both hedger and speculators within financial markets find it attractive to trade derivate specifically assets rather than trading on the assets themselves (Gupta and Subrahmanyam. 2005). Development of derivatives is considered as one of the most successful upcoming within capital markets (Brigo and Mercurio 2001). Within derivatives, there are three main traders; hedger, speculators, and arbitrageurs. Application of derivatives within financial markets helps in eliminating or reducing risk associated with the fluctuations in the prices of assets. Overview and Development of Black Model Financial markets have experienced an increase in the interest-rate contingent claims that include amongst others caps, swaptions, bond options, mortgage-backed securities, as well as captions. The main problem however that is currently experienced is the development of effective and efficient instruments for valuing such contingent claims. Different models have been developed and used in an attempt to find the best and most effective one. Nonetheless, there has been indifference amongst traders on the model effective and efficient enough to help in measuring, controlling, and supervision of interest-rate risks. Hull (234) identifies Black-Scholes Model as a major innovation is pricing of various stock options. During the early 1970s, Fischer Black, Myron Scholes, and Robert Merton developed a model that can be used effectively and efficiently in pricing stock options (Hull p234). In addition, Clewlow and Strickland (2000) confirm that Black Model has been frequently used in valuing bond options due to its effectiveness and efficiency. Black Model borrows extensively from the Black-Scholes Model (Black, 1976). Actually the former is an extension and modification of the latter. Black Model for pricing stock options assumes that the value of an interest rate, bond price, or other variables at a given time is future follows a lognormal distribution. One of the reasons that necessitated the extension and modification of the Black-Scholes Model to Black Model is the difficulty experienced in valuing interest rate derivatives as opposed to valuing foreign exchange derivative (Hull p508). The difficulty is experienced due to a number of reasons such as complications within the behavior of individual interest rate as compared to stock prices of exchange rates (Hull p508). In addition, there has been the need to develop a model that will help in evaluating the behavior of the entire derivate including the zero-coupon yield rate. Consequently, Black Model was developed, which derives most of its assumptions from the Black-Scholes-Merton differential equation that represents the model. For instance, the model assumes that there are no transactional costs of taxes involved in applying the model to value stock options (Black, 1976). What’s more, the model assumes that there are no dividends obtained during the derivatives’ life coupled with facts that arbitrate opportunities are termed as riskless. In this model, another important assumption is that the rate of risk-free interest is constant and equals for all the options’ maturities. Applications of Black Model in pricing European Options In pricing European options, Black Model uses the assumption that the value of a variable with V as the value at a given time (T) of maturity has a lognormal distribution having a standard deviation of In VT that is equated to a^/f where a is the volatility of F (Forward price of V for a contract with maturity time of T) (Hull 2010; p509). The other assumption that Black Model uses in pricing European options is that the expected value of VT is F­o (Value of Forward Price at a time that T = zero). After taking in the assumptions, the Black Model discounts expected payoffs at a given time when T is represented by year free risk rate. This is obtained by multiplying the T by P (0, T), which represents the price at time zero of a zero-coupon bond that pays approximately $ 1 at a given time T. When the model discounts the expected payoff at a particular time, then from the concept of lognormal assumptions it therefore follows that: E {VT) N (dx) - KN (d2) Since the assumption in this calculation is that the expected payoffs is equal to the value of the forward price (F) at time zero (T = 0) that is, E(VT) = F0; then the value of the European option under the Black Model will be given by the following formula: c = P (0, r) {[F0 N (4) – KN (d2)} The above formula may be simplified as follows: Where; X is the strike price; r is the zero-coupon yield for maturity T; T is the optional maturity; and F0 is the forward value of the variable in question; Where; E is the expected value of the payoff after or at a given time, T; V­T represents the value of the variable, V at a particular time, T; and K is the strike price of the option under valuation (Cakici and Zhu, 2001). Through this process, it has become possible to overcome most of the difficulties experienced earlier on with the classical versions of models that found it difficult in measuring or valuing the various interest rate derivates as opposed to the foreign exchange (Clewlow and Strickland, 2000). Therefore, through the development of the Black Model, it has been easier to overcome most of the problems and challenges experienced in pricing interest rate derivates resulting from the reasons identified earlier on in the discussion. Validity of Black Model From the above discussion, it is true concurring with Hull (2010, p510) perception and proposition that the Black Model is an appropriate and effective method for pricing stock options in cases where interest rates are assumed to be constant hence can easily be determined. The concept in applying Black Model to pricing different stock options is that the forward price is assumed to be the future price given the lognormal distribution assumptions of the model (Brigo and Mercurio 2001). What’s more, the validity of Black Model is attained when the two main assumptions offset each other during the pricing of stock options. It is noteworthy to point out that the validity obtained from applying Black Model makes it one of the most preferable models in pricing stock options especially the interest rate derivatives that have for a long time proven to be difficult in measuring due to specific reasons mentioned earlier on in this discussion (Gupta and Subrahmanyam. 2005). Other Applications of Black Model Black Model has been effective in pricing various stock options such as the caps and swaptions amongst others. A cap is usually a portfolio of “caplets” where as swaptions is a portfolio that provides the holder with a right to engage in swapping the future interest rate (Brigo and Mercurio 2001). Unlike the swaptions, every cap held by an investor is usually a call option on future LIBOR rate that normally has the payoff rate attained in arrears Clewlow and Strickland (2000). On a different perspective, swaptions are of two kinds; right of paying specified fixed interest rate in order to receive the LIBOR or the right of paying LIBOR through a specified fixed interest rate. Black Model has been extensively applied in pricing caps as well as swaptions. On the perspective of pricing caps, Black Model assumes that the underlying interest rate for each caplet follows a lognormal distribution (Gupta and Subrahmanyam. 2005). Therefore, in applying the Black Model to price caps, there is serious need to apply spot volatilities that belong to the different caplets under investigation or rather apply flat volatilities across all the caps under investigation. In order to find the value of a caplet under investigation for a period between tk and tk+1 (tk, tk+1); the following formula is used (Hull 2010, p621): Valuing swaption using Black Model on the other hand assumes that there is lognormal distribution on the swap rates. The payoff on each swap under investigation therefore follows the right to pay within a given period (Brigo and Mercurio 2001). For instance, if an investor has a right to pay a given amount of swap interest on a specified swap starting at a particular point, then the payoff of the swap under investigation can be calculated as follows: Conclusion There is no doubt that financial sector especially within capital markets have attained tremendous development and improvements with introduction of interest rate derivatives as opposed to trading on particular assets. Nonetheless, there have been serious difficulties in valuing various stock options. Consequently, this has led to development of many models. Development of many models has not been a solution to valuing interest rate derivatives given the various difficulties associated with the same. Through extension and modification of Black-Scholes Model into Black Model, financial analysts and investors have found the most effective and efficient way of pricing various stock options, interest rate derivatives inclusive. From the aforementioned discussion, it is obvious that Black Model is an effective and efficient tool in pricing various options of stock within the capital market. Bibliography Black, F., 1976, The Pricing of Commodity Contracts“, Journal of Financial Economics 3, March, pp. 167-179. Brigo D., and Mercurio F., 2001, Interest Rate Models: Theory and Practice. Springer-Verlag, Germany. Cakici, N. and J. Zhu, 2001, Pricing Eurodollar Futures Options with The Heath-Jarrow-Morton Model. Journal of Futures Markets, 21, 7, pp.655-680. Clewlow, L., and C. Strickland, 2000, Energy Derivatives: Pricing and Risk Management, Lacima Group. Gupta, A. and M. G. Subrahmanyam. 2005, Pricing and Hedging Interest Rate Options: Evidence from Cap-Floor Markets. Journal of Banking & Finance, 29, 701-733. Hull, C. J., 2010, Options, futures, and other derivatives, Prentice Hall, Upper Saddle River, NJ. Read More