$$c_{0}=\\frac{qc_{u}+(1-q)c_{d}}{1+r}$$<\/p>\r\n
Where:<\/p>\r\n
$$\\begin{align*}q&=\\frac{(1+r)-d}{u-d}\\\\&=\\frac{(1.04)-0.8}{1.20-0.80}\\\\&=0.6\\end{align*}$$<\/p>\r\n
$$c_{0}=\\frac{0.6\\times$10+(1-0.6)\\times0}{1.04}=$5.77$$<\/p>\r\n
Price of a European Put Option<\/h4>\r\n
$$p_{T}=max(K-S_{T},0)$$<\/p>\r\n
$$p_{u}=max($50-$60,0)=$0$$<\/p>\r\n
$$p_{d}=max($50-$40,0)=$10$$<\/p>\r\n
$$\\begin{align*}p_{0}&=\\frac{qp_{u}+(1-q)p_{d}}{1+r}\\\\&=\\frac{0.60\\times0+(1-0.60)\\times$10}{1.04}\\\\&=$3.85\\end{align*}$$<\/p>\r\n
Two-step binomial tree<\/h2>\r\n
The expectations approach can also be applied to the two-step binomial model to determine the value of options.<\/p>\r\n
Let q to be the risk-neutral probability of an up move, the price of a European call option can be determined using the two-step binomial model:<\/p>\r\n
$$c_{o}=\\frac{q^{2}c_{uu}+2q(1-q)c_{ud}+(1-q)^{2}c_{dd}}{(1+r)^2}$$<\/p>\r\n
The two-period European put value is given as:<\/p>\r\n
$$p_{0}=\\frac{q^{2}p_{uu}+2q(1-q)p_{ud}+(1-q)^{2}p_{dd}}{(1+r)^{2}}$$<\/p>\r\n
Example: Expectations Approach for two-Step Binomial Model<\/h3>\r\n
Suppose you have a stock that is currently trading at $60. A two-year European call option on the stock is available with a strike price of $60. The risk-free rate of 2% per annum. Given that the up-move factor is 1.10 and the down-move factor is 0.90, the value of the call option using a two-period binomial model is closest to<\/em>:<\/p>\r\n The risk-neutral probability of an up-move is given by:<\/p>\r\n $$\\begin{align*}q&=\\frac{(1+r)-d}{u-d}\\\\&=\\frac{1.02-0.90}{1.1-0.9}\\\\&=0.6\\end{align*}$$<\/p>\r\n The probability of down move \\((1-q) = 1-0.6 = 0.4\\)<\/p>\r\n The two-period binomial tree is shown below:<\/p>\r\n\r\n\r\n\r\n $$\\begin{align*}c_{o}&=\\frac{q^{2}c_{uu}+2q(1-q)c_{ud}+(1-q)^{2}c_{dd}}{(1+r)^{2}}\\\\&=\\frac{0.6^{2}\\times$12.60+2\\times0.6\\times0.4\\times0\\times0.4^{2}\\times0}{(1.02)^{2}}\\\\&=$4.36\\end{align*}$$<\/p>\r\n Nabi Gudka, CFA, applies the expectations approach to value a European call option on the common shares of Wipro Inc.<\/p>\r\n The expectation approach most likely<\/em> utilizes:<\/p>\r\n The correct answer is A:<\/strong><\/p>\r\n \u00a0Under the expectations approach, the expected future payoff is calculated using risk-neutral probabilities, and the expected payoff is discounted at the risk-free rate.<\/p>\r\n<\/blockquote>\r\n Reading 38: Valuation of Contingent Claims<\/em><\/p>\r\n LOS 38 (e):<\/em>\u00a0Describe how the value of a European option can be analyzed as the present value of the option\u2019s expected payoff at expiration;<\/em><\/p>\r\n","protected":false},"excerpt":{"rendered":" One-Step Binomial Tree The idea that a hedged portfolio returns the risk-free rate can determine the initial value of a call or put. The expectations approach calculates the values of the option by taking the present value of the expected…<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,302],"tags":[328,329,330],"class_list":["post-14769","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-expectations-valuation-approach","tag-one-step-binomial-tree","tag-two-step-binomial-tree","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"\nSolution<\/h3>\r\n
![\"Two-step](\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-step-Binomial-Model-Example.jpg\")
The two-period binomial value of the call option:<\/p>\r\n\r\n
Question<\/h2>\r\n
\r\n\t
Solution<\/h3>\r\n