Black-Scholes model

The introduction of the Black-Scholes model paved the way for the rapid development of options as liquid tradable products. B-S is widely used today to price options and other derivatives. Nevertheless, academics have pointed out several weaknesses related to the main assumptions on which it is based. The major criticisms involve the following  

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The flow of informaticHi to investors is described by the filtration process. The two sources of risk in the Black-Scholes model are the risk-canying underlying asset and the cash deposit which, though paying a riskless rate of interest, is at risk from the stochastic character of the interest rate itself. 

In the Black-Scholes model, the value of a 1 (or 1) deposit invested at the risk-free zero-coupon interest rate r and continuously compounded over a period t will have grown to the value given by the expression below, where M, is the value of the deposit at time t ... 

All valuation models must capture a process describing the dynamics of the asset price. This was discussed at the start of the chapter and is a central tenet of derivative valuation models. Under the Black-Scholes model for example, the price dynamics of a risk-bearing asset St under the risk-neutral probability function Q are given by... 

As explained in the introduction, the value of a convertible bond is the sum of two main components, the option-free bond and a call option on underlying security. The value of the option-free bond, or bond floor, is determined as the sum of future payments (coupon and principal at maturity). Therefore, the bond component is influenced by three main parameters, that is the maturity, the coupon percentage on par value and the yield to maturity (discount rate). Differently, the value of a call option can be found mainly through two option pricing models, Black Scholes model and binomial tree model. 

FIGURE 9.22 The comparison between the value of Black Scholes model and binomial tree. 

In order to find a fair value of the embedded option, the Black Scholes model is not suitable for the following reasons ... 

Cap prices can also be valued analytically using the Hull-White model. The cap prices calculated using the implied volatilities of interest rate caps and the Black-Scholes model serve as the calibrating instruments. After the Hull-White model has been calibrated, the parameters a and o that minimize a goodness-of-fit measure can be used to solve for the convexity bias. 

The pricing of a spread option is dependent on the underlying process. As an example we compare the pricing results for a spread option model, including mean reversion to the pricing results from a standard Black-Scholes model in Exhibit 21.14 and Exhibit 21.15. 

Appendix The Black-Scholes Model in Microsoft Excel.331... 

Most option pricing models use one of two methodologies, both of which are based on essentially identical assumptions. The first method, used in the Black-Scholes model, resolves the asset-price model s partial differential equation corresponding to the expected payoff of the option. The second is the martingale method, first introduced in Harrison and Kreps (1979) and Harrison and Pliska (1981). This derives the price of an asset at time 0 from its discounted expected future payoffs assuming risk-neutral probability. A third methodology assumes lognormal distribution of asset returns but follows the two-step binomial process described in chapter 11. 

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. 

Pricing Derivative Instruments Using the Black-Scholes Model... 

A number of option-pricing models exist. Market participants often use variations on these models that they developed themselves or that were developed by their firms. The best-known of the pricing models is probably the Black-Scholes, whose fundamental principle is that a synthetic option can be created and valued by taking a position in the underlying asset and borrowing or lending funds in the market at the risk-free rate of interest. Although Black-Scholes is the basis for many other option models and is still used widely in the market, it is not necessarily appropriate for some interest rate instruments. Fabozzi (1997), for instance, states that the Black-Scholes model s assumptions make it unsuitable for certain bond options. As a result a number of alternatives have been developed to analyze callable bonds. 

The figure on the following page shows the spreadsheet formulas required to build the Black-Scholes model in Microsoft Excel. The Analysis Tool-Pak add-in must be available, otherwise some of the function references may not work. Setting up the cells in the way shown enables the fair value of a vanilla call or put option to be calculated. The latter calculation employs the put-call parity theorem. 

For a detailed discussion of the mathematical basis of the Black-Scholes model, readers... 

To model these phenomena, different modification of the Black Scholes model have been proposed. The most two common models are the stochastic volatility model (see Hull White process (Hull White 1987)), CIR process (see (Cox Ross 1976)) and Ornstein-Uhmenbeck process (see (Heston 1993, Stein Stein 1991)) and the jump diffusion model. 

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