Univariate Binomial Tree
Cox, Ross and Rubinstein (1979) developed the binomial option pricing model which converges to the Black-Scholes formula in the continuous limit and demonstrates the advantage in valuing American-style options. The model approximates the behavior of an asset price by the upward and downward changes in the asset price over a particular interval of time.
As shown in Figure 1, an asset with a current price of S follows a multiplicative binomial process in which the asset price can either go up to uS or down to dS, during the interval h = T/m, where m is the total number of moves over time. The figure 1 demonstrates a four moves binomial tree. Cox, Ross and Rubinstein (1979) has shown that the upward and downward …show more content…
The multiplicative returns over two moves will be (u2, A2), (u2, AB), (ud, AC) and (ud, AD), if the return of first move was (u,A).
The value of moves can be obtained as following:
where m is the number of moves over the period T, r is the continuous compounding risk free rate, di are continuous dividend yields, i are assets' volatilities, and is the correlation coefficient between two assets.
These parameters can be used to construct the forward moving structure of asset returns. At the end of the tree, values of the basket option can be evaluated with the asset pair at each node. Then, we work backwards by discounting four nodes into one node in each move, using the same probability of 1/4 for each nodes.
For example, consider an American put option on a two-asset basket, the payoff function at expiration is max(0, K - (n1S1 + n2S2)). For a two moves bivariate binomial tree, where m = 2, values of the option at expiration are:
C(u², A²) = max[ 0, K – (n1S1u² + n2S2A²) ]
C(u², AB) = max[ 0, K – (n1S1u² + n2S2AB) ]
C(u², B²) = max[ 0, K – (n1S1u² + n2S2B²) ]
C(ud, AC) = max[ 0, K – (n1S1ud + n2S2AC) ]
C(ud, BC) = C(du, AD) = max[ 0, K – (n1S1ud + n2S2BC)
Cox, Ross and Rubinstein (1979) developed the binomial option pricing model which converges to the Black-Scholes formula in the continuous limit and demonstrates the advantage in valuing American-style options. The model approximates the behavior of an asset price by the upward and downward changes in the asset price over a particular interval of time.
As shown in Figure 1, an asset with a current price of S follows a multiplicative binomial process in which the asset price can either go up to uS or down to dS, during the interval h = T/m, where m is the total number of moves over time. The figure 1 demonstrates a four moves binomial tree. Cox, Ross and Rubinstein (1979) has shown that the upward and downward …show more content…
The multiplicative returns over two moves will be (u2, A2), (u2, AB), (ud, AC) and (ud, AD), if the return of first move was (u,A).
The value of moves can be obtained as following:
where m is the number of moves over the period T, r is the continuous compounding risk free rate, di are continuous dividend yields, i are assets' volatilities, and is the correlation coefficient between two assets.
These parameters can be used to construct the forward moving structure of asset returns. At the end of the tree, values of the basket option can be evaluated with the asset pair at each node. Then, we work backwards by discounting four nodes into one node in each move, using the same probability of 1/4 for each nodes.
For example, consider an American put option on a two-asset basket, the payoff function at expiration is max(0, K - (n1S1 + n2S2)). For a two moves bivariate binomial tree, where m = 2, values of the option at expiration are:
C(u², A²) = max[ 0, K – (n1S1u² + n2S2A²) ]
C(u², AB) = max[ 0, K – (n1S1u² + n2S2AB) ]
C(u², B²) = max[ 0, K – (n1S1u² + n2S2B²) ]
C(ud, AC) = max[ 0, K – (n1S1ud + n2S2AC) ]
C(ud, BC) = C(du, AD) = max[ 0, K – (n1S1ud + n2S2BC)