Microsoft PowerPoint - PoF.ppt

Example: American putƒ 221 Consider a traded asset (stock) with current price S = 100USD, an American put with E = 100USD, T = 1 ...

Example: Lookback optionƒ 222 Consider a traded asset (stock) with current price S = 4USD, u = 2, and d = 0.5, 3 periods and r = ...

Multi-period binomial asset pricing modelƒ 223 If we choose , and a continuous interest rate convention then as the number of pe ...

Completenessƒ 224 The binomial asset pricing in this section is called complete because every derivative can be replicated by tr ...

Exampleƒ 225 Example Take any q 1 such that the risk-neutral probabilities are between 0 and Two possible choices for the risk ...

The evolvement of r and S over timeƒ 226 In the Black-Scholes option pricing mo del (1973), there are two securities, a money ma ...

Wiener processƒ 227 Norbert Wiener, 1920: W is a t Wiener process, i.p. W t is a random (stochastic) real-valued continuous fu n ...

Geometric Brownian motionƒ ƒ 228 One realization of a geometric Brownian motion () [][][][][][][][][]() () dt S dt S N dS dt dt ...

Geometric Brownian motionƒ ƒ 229 Note: The probability of what S does next depend only on the current state, t. This is called M ...

Historyƒ 230 Robert Brown, 1828 : Experimental study of the motion of particles suspended in a liquid (Albert Einstein, 1905 and ...

Black-Scholes model: Main ideasƒ 231 The BS analysis is analogous to the anal ysis used to value options in the binomial asset p ...

232 Suppose, for example, that at a particular point in time we know that the delta of a call is given by 0.4 ( ∆ =dc/dS=0.4). c ...

ƒ 233 To remain riskless it must be frequently adjusted or “rebalanced”. In particular, we would have to rebalance continuously! ...

ƒ 234 The riskless-portfolio argument depends on basically the same assumptions as the replicating portfolio argument in the bin ...

ƒ 235 Black-Scholes formula where N(x) is cumulative standard no rmal distribution at x. In other words, it is the probability t ...

ƒ 236 The volatility is the only para meter which can not be directly observed in the market Ä We can either use ƒ historical es ...

237 ƒ The result is the following “volatility smile”:Note that we assumed a constant volatility which is obviously an incorrect ...

ƒ 238 Unfortunately no exact analytic formulae for the value of American calls and puts have been produced. There are however nu ...

ƒ 239 Black-Scholes price of a call and a put with E = 100, T = 1y, r = 10% p.a. and σ = 20% p.a.. Derivative securities: Option ...

Properties of the Black-Scholes pricesƒ 240 Call ƒ When S becomes very large , a call option is almost certain to be exercised . ...