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This is effectively the Breeden-Litzenberger formula that expresses the risk neutral distribution (here for discrete states). You can see that the price of the butterfly is the finite difference estimate of the second derivative of the call price with respect to the price.

Let $Y$ be the state price density of the market such that $Y_0 =1$. We know that $$\mathbb E YS_1 = S_0, \quad \mathbb E Y ( S_1 - K)^ + = C(K), ext for K \in \ 1, \ldots, N-1 \ .$$

Consider a one-period market model with $N+1$ assets: a bond, a stock and $N-1$ call options. The prices of the bond are $B_0=1$ and $B_1 = 1+r$, where $r$ is a constant. The prices of the stock are given by a constant $S_0$ and a random variable $S_1$ taking values in $\ 0, 1, \ldots, N-1, N \$ for a given integer $N \geq 4$. Finally, let the time-0 price of the call option with strike $K \in \ 1, \ldots, N-1 \$ be denoted by $C(K)$. Now we introduce a contingent claim with time-1 payout $\xi_1 = g(S_1)$, where $g$ is the function $$g(M) = \mathbf 1 _ \ M = K_0 \ , \quad 0 \leq K_0 \leq N.$$ Assuming that the market has no arbitrage, we want to find the time-0 price $\xi_0$ in the following cases: $$2 \leq K_0 \leq N-2 \, ; \quad K_0 = N-1 \, ; \quad K_0 = 0 .$$

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For example, the binary numbers 1011 and 1010 are multiplied as follows:

The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:

In a fractional binary number such as 0.110101101012, the first digit is 1 2 \displaystyle \begin matrix \frac 1 2 \end matrix , the second ( 1 2 ) 2 = 1 4 \displaystyle \begin matrix ( \frac 1 2 )^ 2 = \frac 1 4 \end matrix , etc. So if there is a 1 in the first place after the decimal, then the number is at least 1 2 \displaystyle \begin matrix \frac 1 2 \end matrix , and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.

It is based on taoistic duality of yin and yang. 5 eight trigrams (Bagua) and a set of 64 hexagrams ("sixty-four" gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. 3

Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as borrowing. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value.

In the example below, the divisor is 1012, or 5 decimal, while the dividend is 110112, or 27 decimal. The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence: