smile-lecture7 - E4718 Spring 2008 Derman Lecture 7:Local...

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Copyright Emanuel Derman 2008 E4718 Spring 2008: Derman: Lecture 7:Local Volatility Continued Page 1 of 18 3/27/08 smile-lecture7.fm Lecture 7: Local Volatility Continued
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Copyright Emanuel Derman 2008 E4718 Spring 2008: Derman: Lecture 7:Local Volatility Continued Page 2 of 18 3/27/08 smile-lecture7.fm 7.1 One More Remark About Static Hedging We showed in Section 5.4.1 that, in a Black-Scholes world with zero drift, the fair value for a down-and-out call with strike K and barrier B is given by Eq.7.1 You can write the payoff of the first term in Equation 7.1as Similarly, you can write the payoff of as This payoff represents that of K/B standard puts with strike . Thus, the payoff at expiration in Equation 7.1 is that of a long position in a call with strike K and a short position in K/B standard puts with strike . The payoff is displayed in the figure below. Roughly speaking the payoff of a down-and-out-call is that of an ordinary call and its price reflection (in log space) in the barrier. (In an arithmetic Brownian motion world, this would be accurate for price reflections rather than log price reflections.) You can see that the two payoff have positive and negative present discounted values that roughly cancel each other along the boundary B at all earlier times, and thus emulate the payoff of a down-and-out call with zero value on the barrier.This view is useful in thinking about the local volatilities that influence the option payoffs, which are sensitive to volatilities between and K . Similar notions of payoff cancellation are useful in arriving at approximately static replicating portfolios for other barrier options C DO S K , ( ) C BS S K , ( ) S B --- C BS B 2 S ----- K , = θ S K ( ) S K ( ) S B --- C BS B 2 S ----- K , S B --- B 2 S ----- K θ B 2 S ----- K B KS B ------ θ B 2 K ----- S K B --- θ B 2 K ----- S B 2 K ----- S = = B 2 K B 2 K S K B B 2 /K B 2 K
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Copyright Emanuel Derman 2008 E4718 Spring 2008: Derman: Lecture 7:Local Volatility Continued Page 3 of 18 3/27/08 smile-lecture7.fm 7.2 Difficulties with binomial trees The positions of the nodes of the local volatility tree and the transition proba- bilities we discussed are uniquely determined by forward rates and the local volatility function we specify. But if the local volatility varies too rapidly with stock price or time, then, for finite spacing between levels, you can obtain nodes with future stock prices that violate the no-arbitrage condition and result in binomial transition probabilities greater than 1 or less than zero. Here is an example with and . We have chosen the local volatility on level 3 at stock price 100 to be 0.21 rep- resenting a very rapid rise from the value of 0.1 at the start of the tree. The S' node, the up node in level 4 relative to the central nodes, is technically the next down connected to S u and should therefore lie below S u , but in fact lies above it, and so violates the no-arbitrage condition: the up and down nodes emanating from S u will both lie above the forward. These sorts of problems can be reme-
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  • Summer '10
  • john
  • Implied volatility, Mathematical finance, Volatility smile, EMANUEL DERMAN, COPYRIGHT EMANUEL DERMAN, local volatility

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