tailieunhanh - An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_7

Tham khảo tài liệu 'an introduction to financial option valuation: mathematics, stochastics and computation_7', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 109 Change of variables P 0 T S t Fig. . European put Black-Scholes surface with asset path superimposed. delta T 0 uni n Fig. . Black-Scholes surface for delta with three asset paths superimposed. 110 More on the Black-Scholes formulas We will introduce three new dimensionless quantities. First is the moneyness ratio Ser T t t m log-----E---- To interpret m we need to generalize into the formula Se l T t for the expected value of the asset at expiry given asset price S at time t Now we make the assumption that the asset growth rate equals the interest rate Ẳ. r. This assumption will be examined in detail in Chapter 12 for now we simply note that it leads to the following conclusions. If m 0 then the expected asset value at expiry is greater than the strike price. In a riskneutral expectation at expiry sense a call option is in-the-money and a put option is out-of-the-money. If m 0 then in the same sense call and put options are at-the-money. If m 0 then in the same sense a call option is out-of-the-money and a put option is in-the-money. Second we have the scaled volatility Ỹ Ơ T t Here the volatility is combined with the square root of the time to expiry. This is natural since for example volatility appears in the form Ơ2 ti 1 ti in the underlying asset model . The third step is to scale the option values by the asset price by letting c for a call option and P p S for a put optiom In these new variables d1 and d2 in and simplify to d1 m T 2 and d2 m x Ĩ 2 and from and the re-scaled call and put values become c m Ỹ N d1 e mN d2 and p m Ỹ e mN d2 N d1 see Exercise . Program of Chapter 11 and walkthrough Notes and references 111 Colour versions of Figures and can be downloaded from this book s website mentioned in the preface. EXERCISES . Consider the following explanation of why the Black-Scholes European call option value curve C S t lies above the payoff hockey stick max S t E 0