Unformatted text preview: The convexity C of a zero coupon bond with maturity T is equal to T(T+1)/(1+y)2, so C1 = 1.92 and C2 = 5.77. Thus, the weights should satisfy: w1 x 1 + w2 x 2 = 3.2 w1 x 1.92 + w2 x 5.77 = 6.7282 Hence w1 = 2.6, w2 = 0.3, and the cash position is 1‐w1‐w2 = ‐ 1.9. QUESTION 4. (15 points) Option pricing Part a. (2 points) Discuss the relationship between option prices and time to expiration, volatility of the underlying stocks, and the exercise price. The longer the time to expiration, the higher the premium because it is more likely that an option will become more valuable (more time for the stock price to change). The greater the volatility of the underlying stock, the greater the option premium; the more volatile the stock, the more likely it is that the option will become more valuable (e. g., move from an out of the money to an in the money option, or become more in the money). For call options, the lower the exercise price, the more valuable the option, as the option owner can buy the stock at a lower price. For a put option, the lower the exercise price, the less valuable the option, as the owner of the option may be required to sell the stock at a lower than market price. Part b. (2 points) Consider the following binomial tree for the evolution of the stock price over the period of 1 year, assuming 2 steps (t=0, t=1, t=2). The stock price can increase by 10% and decrease by 5% each period. The risk‐neutral probability of an upward movement is 43.33%. Calculate the annual interest rate. S2uu = 10.89 S1u = 9.9 S2ud = 9.405 S0 = 9 S2du = 9.405 S1d = 8.55 S2dd = 8.1225 The periodic interest rate rf is obtained using: Q = (S0(1+rf)‐Sd)/(Su‐Sd): 0.4333 = (9 x (1+rf) – 8.55)/(9.9 – 8.55), rf = 0.015 (periodic) or 0.03 (annual). 1 3 Part c. (7 points) A barrier optio...
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- Spring '14
- Interest Rates, risk‐free rate