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Financial calculus Introduction to Financial Option Valuation_6

Tham khảo tài liệu 'financial calculus introduction to financial option valuation_6', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14 Implied volatility OUTLINE the need for implied volatility properties of option value as a function of Ơ bisection and Newton for computing the implied volatility volatility smiles and frowns 14.1 Motivation We now put the bisection method and Newton s method to work on the problem of computing the implied volatility. 14.2 Implied volatility The Black-Scholes call and put values depend on S E r T t and Ơ2. Of these five quantities only the asset volatility Ơ cannot be observed directly. How do we find a suitable value for Ơ One approach is to extract the volatility from the observed market data - given a quoted option value and knowing S t E r and T find the Ơ that leads to this value. Having found Ơ we may use the Black-Scholes formula to value other options on the same asset. A Ơ computed this way is known as an implied volatility. The name indicates that Ơ is implied by option value data in the market. A completely different way to get hold of Ơ is described in Chapter 20. We focus here on the case of extracting Ơ from a European call option quote. An analogous treatment can be given for a put or alternatively the put quote could be converted into a call quote via put-call parity 8.23 . 14.3 Option value as a function of volatility We assume that the parameters E r and T and the asset price S and time t are known. In practice we will typically be interested in the time-zero case t 0 131 132 Implied volatility and s 50. We thus treat the option value as a function of ơ only and for the rest of this chapter denote it by C ơ . Given a quoted value C our task is to find the implied volatility ơ that solves C ơ C . Computing the implied volatility requires the solution of a nonlinear equation and hence from Chapter 13 we may use the bisection method or Newton s method. We will find that it is possible to exploit the special form of the nonlinear equation arising in this context. Since volatility is non-negative only values ơ G 0 TO are of interest. Let us look at