現(xiàn)代金融中數(shù)學(xué)問題

1、Vanilla options,The payoff of a European (vanilla) option at expiry is ---call ---put where -- underlying asset price at expiry

2、 -- strike price The terminal payoff of a European vanilla option only depends on the underlying price at expiry.,Path-dependent options,Their payoffs depend on the historical underlying price during the options

3、’ life.Barrier options:Asian options：Lookback options：,Barrier options: an example,Barrier options are triggered by the action of the underlying hitting a prescribed value at some time before expiry.,For example, i

4、f the asset remains below a pre-determined barrier price during the whole life of the option, the contract will have a call payoff at expiry (red line). However, should the asset reach this level before expiry, then the

5、option becomes worthless because it has ``knocked out'‘ (blue line).,types of barrier options,The out option, that only pays off if a level is not reached. If the barrier is reached then the option is said to have kn

6、ocked out,The in option, that pays off as long as a level is reached before expiry. If the barrier is reached then the option is said to have knocked in.,continued,Up optionThe barrier is above the initial asset value,D

7、own optionThe barrier is below the initial asset value,continued,Call/Putup-out-call, down-out-call, up-in-call, down-in-call,up-out-put, down-out-put, up-in-put, down-in-put,,European/American,In-out parity,In + Out

8、= Vanilla,This parity holds for European, not for American,Pricing by Monte-Carlo simulation,For example: up-out-call,Pricing in PDE,V=V(S,t): the value of barrier option (before the barrier is reached),The details of th

9、e barrier feature come in through the specification of the boundary conditions and solution domains.,Boundary and final conditions,out,in,Solution domains,downup,Closed form solution,American out,American in,BTM,cont

10、inued,Hedging,Barrier options may have discontinuous delta at the barrier. So, it is very hard to hedge a barrier option. We refer interested students to Wilmott (1998) and references therein.,Other features,Time-depen

11、dent barrier level,continued,Double barrier optionsdouble outdouble inone in another out,Asian options,Payoff typesFixed strikeFloating strike,Type of Averaging,Payoff typesArithmeticGeometric,Monte-Carlo Simu

12、lation,V=exp(-r*T)E[final payoff],Extending Black-Scholes,Arithmetic Asian,Compared with the pricing model with an option on two assets,Geometric Asian,Early exercise,Formulas,Explicit formulas exist for all European-sty

13、le geometric Asian optionsNot for arithmetic Asian options,An example of reduction,Parity relation,Model independent results,Put-call parity for vanilla options, multi-asset optionsThe in-out parity for barrier optio

14、nsAmerican call options should never be early exercised if there is no dividend payment.,Model dependent results,Put-call symmetry relationPut-call parities for Asian options,Binomial tree method,Lookback options,Ty

15、pes of Payoff,Extending the Black-Scholes equations,Another variable,continued,continued,BTM,Equivalence of BTM and PDE,Similarity reduction,PDEEuropean/American floating strike options European fixed strike optionsB

16、TM,Exotic options,Barrier optionsAsian optionsLookback optionsForward start optionsShout optionsCompound options……,Forward Start Options,An option that starts in the future.An example: a forward start call,Prici

17、ng,Pricing Model,Shout Options,A shout call option: a vanilla option with an extra featureThe holder can at any time reset the strike priceThe strike price becomes the asset price when resetting The action of reset

18、ting is called “shouting”.,Two questions,How to value a shout optionWhat’s the optimal shouting policy?,Pricing Model,When shouting at time t, the option becomes an at-the-money option with Sf(t)Let V(S,t) be the val

19、ue of the shout option. Then,Continued,Compound Options,An option on another optioncall on callcall on putput on callput on putThe payoff two strike prices: X, X1 two expiration dates: T<T1,Future Options,An o

20、ption on a future,Pricing Model for Compound Options,Others,Parisian optionsEmployee reload optionsAsian barrier options……,Concluding remarks(1): closed-form solutions,Available (see Kwok (1998) or Hull (1998))Barrie

21、r optionsGeometric Asian optionsLookback optionsRussian optionsForward start optionsCompound optionsNot availableArithmetic Asian optionsShout optionsAmerican-style options,Concluding remarks(2):,Binomial tree m

22、ethod The BTM can be readily extended to all options discussed in this chapterBut it is not easy to implement for Asian options. (see Kwok (1998) for a modified BTM).Similarity reductionNot available for American-st