Black Scholes

The best modeling framework for R D options is, however, more contentious. The famous, or infamous, Black-Scholes formula [8], based on valuation of traded hnancial options, has in our view impeded the practical use of decision analysis methods by scientihc managers ... 

Let us consider then the key assumptions that form part of the economy of, for example, the Black-Scholes option pricing model. 

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 shown in previous sections, the credit spread on a corporate bond takes into account its expected default loss. Structural approaches are based on the option pricing theory of Black Scholes and the value of debt depends on the value of the underlying asset. The determination of yield spread is based on the firm value in which the default risk is found as an option to the shareholders. Other models proposed by Black and Cox (1976), Longstaff and Schwartz (1995) and others try to overcome the limitation of the Merton s model, like the default event at maturity only and the inclusion of a default threshold. This class of models is also known as first passage models . 

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 ... 

From oiu understanding of derivatives, we know that option pricing models such as Black-Scholes assume that asset price returns follow a lognormal distribution. The dynamics of interest rates and the term structure is the subject of... 

The key assumption in the derivation of the Black-Scholes option pricing model is that the asset price follows a lognormal distribution, so that if we assume the asset price is P we write... 

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. 

Structural models are characterized by modeling the firm s value in order to provide the probability of a firm default. The Black-Scholes-Merton option pricing framework is the foundation of the structural... 

Firm s assets evolve randomly. The probability of a firm default is determined using the Black Scholes Merton option pricing theory. 

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. 

Expiry in Six Months Risk-free rate = 10% Strike = 70 bps Credit spread = 60 bps Volatility = 20% Mean Reversion Model Price Standard Black Scholes Price Difference Between Standard Black Scholes and Mean Reversion Model Price... 

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

Swaptions are typically priced using the Black-Scholes or the Black pricing model. With a European swaption, the appropriate swap rate on the expiry date is assumed to be lognormal. The swaption payoff is given by equation (7.19). 

The price behavior of financial instruments. One of the key assumptions of option pricing models such as Black-Scholes (B-S), which is discussed below, is that asset prices follow a lognormal distribution— that is, the logarithms of the prices show a normal distribution. This characterization is not strictly accurate prices are not lognormally distributed. Asset returns, however, are. Returns are defined by formula (8.8). 

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... 

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. 

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 ... 

The N di) term in the Black-Scholes equation represents an option s delta. Delta indicates how much the contract s value, or premium, changes as the underlying asset s price changes. An option with a delta of zero does not move at all as the price of the underlying changes one with a delta of 1 behaves the same as the underlying. The value of an option with a delta of 0.6, or 60 percent, increases 60 for each 100 increase in the value of the underlying. The relationship is expressed formally in (9-1). 

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... 

Let S t) be the variable representing our concerned data at time t. S(t) is supposed to satisfy a Markov, continuous-time, geometric Stochastic Differential Equation (SDE). A classical model of stochastic process is the BS (Black Scholes) lognormal diffusion process. The BS model follows a basic stochastic differential equation given by ... 

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