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Lecture 19-21

# Lecture 19-21 - Options II 1.Binomial Option Pricing 1.1...

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Lily Qiu, Assistant Professor Economics Department, Brown University EC1710, Lecture 19-21, Spring 2010, page 1 Options, II 1.Binomial Option Pricing 1.1 Let’s start with a simple example: assume S 0 = \$100 at the beginning of the year, the price of the stock can go up to \$110 at the end of the year (by a factor of u=1.1), or drop to \$90 (by a factor of d=0.9). A call option on this stock has an exercise price of \$105 and a time to expiration of 1 year. The risk-free annual interest rate is 5%. Let’s use value “trees” to illustrate the payoffs: Stock Option / / 100 C 0 \ \ Now, we also form a leveraged portfolio. We buy one share at the beginning of the year at \$100, borrow \$85.71 at the 5% to finance the purchase. Its value tree at the end of the year: Leveraged portfolio / 14.29 \ If we buy 4 call options, we have the same cashflow pattern as the leveraged portfolio, then it must be that the cost of buying 4 call options is the same as the cost of constructing the leveraged portfolio: 4C 0 = 14.29 => C 0 = \$3.57 REMARK: This method of constructing a portfolio consisting of the risk-free and the underlying asset, which mimics the cashflow pattern of the option, is called replication => important concept behind most option pricing formulas. We can also create a perfect hedge by buying one share and write 4 calls. Portfolio value / \$110 \$20 = \$90 \$100 - \$14.29 = \$85.71 \ \$90 0 = \$90

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Lily Qiu, Assistant Professor Economics Department, Brown University EC1710, Lecture 19-21, Spring 2010, page 2 The portfolio is risk-less => a perfect hedge , it earns the risk-free return of 5%: \$85.71 * 1.05 = \$90. The hedge ratio = hedge perfect the in options of number the hedge perfect the in shares of number the _ _ _ _ _ _ _ _ _ _ _ _ _ _ => 1/4 in this example.
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