By Odegaard B.A.
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Extra info for Financial numerical recipes in C++
The Black Scholes price for a put option is: p = Ke−r(T −t) N (−d2 ) − SN (−d1 ) where d1 and d2 are as for the call option: + (r + 12 σ 2 )(T − t) √ σ T −t √ d2 = d1 − σ T − t d1 = ln S X and S is the price of the underlying secrutity, K the exercise price, r the (continously compounded) risk free interest rate, σ the standard deviation of the underlying asset, T − t the time to maturity for the option and N (·) the cumulative normal distribution. 1. Implement this formula. 2 Understanding the why’s of the formula To get some understanding of the Black Scholes formula and why it works will need to delve in some detail into the mathematics underlying its derivation.
References . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 63 63 63 66 69 70 70 70 70 74 75 76 Introduction We have shown binomial calculations given an up and down movement in chapter 5. However, binomial option pricing can also be viewed as an approximation to a continuous time distribution by judicious choice of the constants u and d. To do so one has to ask: Is it possible to find a parametrization (choice of u and d) of a binomial process uuS ✯ ✟ ✟ ✟✟ ✟✟ ✟✟ uS ✯ ❍❍ ✟ ✟ ✟ ❍ ❍❍ ✟✟ ✟ ❍ ❍ ❥ ❍ ✟✟ S ❍❍ ✯ ✟ ✟ ✟ ❍❍ ✟ ❍❍ ✟✟ ❍ ❥ ✟✟ ❍ ❍dS ❍❍ ❍ ❍❍ ❍ ❥ ❍ udS ddS which has the same time series properties as a (continous time) process with the same mean and volatility?
37 40 40 40 41 41 41 41 44 The pricing of options and related instruments has been a major breakthrough for the use of financial theory in practical application. Since the original papers of Black and Scholes  and Merton , there has been a wealth of practical and theoretical applications.
Financial numerical recipes in C++ by Odegaard B.A.