![\"Two-period](\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Call-Option-scaled.jpg\")
Suppose that the notional principal of each option is $500,000, and the risk-neutral probability of an up jump is 0.5. The value of the European call option can be determined as follows:<\/p>\r\nInterest rate options are options with an interest rate as the underlying. A call option on interest rates has a positive payoff when the current spot rate is greater than the exercise rate.<\/p>\r\n
$$\\text{Call option payoff}=\\text{Notional Amount}\\times[\\text{Max}(\\text{Current spot rate}-\\text{Exercise rate, 0})]$$<\/p>\r\n
On the other hand, a put option on interest rates has a positive payoff when the current spot rate is less than the exercise rate.<\/p>\r\n
$$\\text{Put payoff}=\\text{Notional Amount}\\times[\\text{Max}(\\text{Exercise rate}-\\text{Current spot rate},0)]$$<\/p>\r\n
We can apply the binomial model to value such interest rate options.<\/p>\r\n
Consider a two-year European-style call option with a one-year spot rate compounded annually as the underlying. The exercise rate is 6%. The two-period interest rate tree is given below:<\/p>\r\n\r\n\r\n\r\n
Suppose that the notional principal of each option is $500,000, and the risk-neutral probability of an up jump is 0.5. The value of the European call option can be determined as follows:<\/p>\r\n
$$\\begin{align*}\\text{c}_{\\text{uu}}&=\\text{Max}(0,\\text{S}_{0}\\text{u}^{2}-\\text{K})\\\\&=\\text{Max}(0,0.11-0.06)\\\\&=0.05\\end{align*}$$<\/p>\r\n
$$\\begin{align*}\\text{c}_{\\text{ud}}&=\\text{Max}(0,\\text{S}_{0}\\text{ud}-\\text{K})\\\\&=\\text{Max}(0,0.07-0.06)\\\\&=0.01\\end{align*}$$<\/p>\r\n
$$\\begin{align*}\\text{c}_{\\text{dd}}&=\\text{Max}(0,\\text{S}_{0}\\text{d}^{2}-\\text{K})\\\\&=\\text{Max}(0,0.05-0.06)\\\\&=0\\end{align*}$$<\/p>\r\n
$$\\begin{align*}c_{u}&=PV_{1,2}[qc_{uu}+(1+q)c_{ud}]\\\\&=\\frac{1}{1.06}[0.5\\times0.05+(1-0.5)\\times0.01]\\\\&=0.028302\\end{align*}$$<\/p>\r\n
$$\\begin{align*}c_{d}&=PV_{1,2}[qc_{ud}+(1-q)c_{dd}]\\\\&=\\frac{1}{1.04}[0.5\\times0.01+(1-0.5)\\times0]\\\\&=0.004808\\end{align*}$$<\/p>\r\n
$$\\begin{align*}c_{0}&=PV_{0,1}[qc_{u}+(1-q)c_{d}]\\\\&=\\frac{1}{1.03}[0.5\\times0.028302+(1-0.5)\\times0.004808]\\\\&=0.01607\\end{align*}$$<\/p>\r\n
The call value at time 0 is then obtained by multiplying with the notional amount:<\/p>\r\n
$$\\text{Call value}=0.01607\\times$500,000=$8,035$$<\/p>\r\n
Consider a two-year European-style put option with the annually compounded one-year spot interest rate as the underlying. The exercise rate is 6%. The two-period interest rate tree is given below:<\/p>\r\n\r\n\r\n\r\n
The correct answer is A:<\/strong>\u00a0<\/strong><\/p>\r\n Similar to a call option, a put option will have three possible payoffs:<\/p>\r\n $$\\begin{align*}p_{uu}&=Max(0,K-S_{0}u^{2})\\\\&=Max(0,0.06-0.11)\\\\&=0\\end{align*}$$<\/p>\r\n $$\\begin{align*}p_{ud}&=Max(0,K-S_{0}ud)\\\\&=Max(0,0.06-0.07)\\\\&=0\\end{align*}$$<\/p>\r\n $$\\begin{align*}p_{dd}&=Max(0,K-S_{0}dd)\\\\&=max(0,0.06-0.05)\\\\&=0.01\\end{align*}$$<\/p>\r\n $$\\begin{align*}p_{u}&=PV_{1,2}[qp_{uu}+(1-q)p_{ud}]\\\\&=\\frac{1}{1.06}[0.5\\times0+(1-0.5)\\times0]\\\\&=0\\end{align*}$$<\/p>\r\n $$\\begin{align*}p_{d}&=PV_{1,2}[qp_{ud}+(1-q)c_{dd}]\\\\&=\\frac{1}{1.04}[0.5\\times0+(1-0.5)\\times0.01]\\\\&=0.004808\\end{align*}$$<\/p>\r\n $$\\begin{align*}p_{0}&=PV_{0,1}[qp_{u}+(1-q)p_{d}]\\\\&=\\frac{1}{1.03}[0.5\\times0+(1-0.5)\\times0.004808]\\\\&=0.00233398\\end{align*}$$<\/p>\r\n The put value at time 0 is then obtained by multiplying by the notional amount:<\/p>\r\n $$\\text{put value}=0.002334\\times$500,000=$1,167$$<\/p>\r\n Reading 38: Valuation of Contingent Claims<\/em><\/p>\r\n LOS 38 (d) calculate and interpret the value of an interest rate option using a two-period binomial model;<\/em><\/p>\r\n","protected":false},"excerpt":{"rendered":" Interest rate options are options with an interest rate as the underlying. A call option on interest rates has a positive payoff when the current spot rate is greater than the exercise rate. $$\\text{Call option payoff}=\\text{Notional Amount}\\times[\\text{Max}(\\text{Current spot rate}-\\text{Exercise rate,…<\/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":[216,304,327,324,326,325],"class_list":["post-14714","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-derivatives","tag-cfa-level-2","tag-derivatives","tag-payoffs-at-time-2","tag-valuation-of-an-interest-rate-option","tag-value-of-call-at-time-0","tag-value-of-call-at-time-1","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"\n![\"Two-period](\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/05\/Two-period-Interest-Rate-Tree-Put-Option-scaled.jpg\")
Suppose that the notional principal of each option is $500,000, and the risk-neutral (RN) probability of an up jump is 0.5. The value of the European put option is closest to:<\/em>\u00a0<\/em><\/p>\r\n\r\n\t
Solution<\/h3>\r\n
Put Payoffs at time 2<\/h4>\r\n
Value of put at time 1<\/h4>\r\n
Value of put at time 0<\/h4>\r\n