1. Lookbackoption Ticciati Marco Muhammad Naeem SantangeloGiusj Carmen

2. Definition Lookbackcall ( put) gives the optionholder the right tobuy (sell) asset at itslowest ( higher) price during the life of the option exercise based on the underlying asset's optimal value reduces uncertainties associated with the timing of market entry. Lookback option with floating strike The payoff is the maximum difference between the market asset's price at maturity and the floating strike. • call, the strike price is fixed at the asset's lowest price during the option's life • put, it is fixed at the asset's highest price, • LCfloat=max ( ST -Smin;0) • LPfloat=max( Smax-ST;0)

3. Lookback option with fixed strike The difference is that the option is not exercised at the price at maturity: the payoff is the maximum difference between the optimal underlying asset price and the strike. • call option, the holder chooses to exercise at the point when the underlying asset price is at its highest level. • put option, the holder chooses to exercise at the underlying asset's lowest price. • LCfix=max ( Smax-K;0) • LPfix= max(K-Smin;0)

5. Pricingmodels Thereexistclosedformulaeto price the instrumentbased on black and scholesframework Considering I(t) = min S(t) Being the lookbackcallanoption on the minimum price The samethingcouldbeappliedalsofor the put The onlydifferenceis the factthat the put is on the maximum Wehavetoinvert the formula

6. Princingmodels The formula depends on the PDE with the verysameboundaryconditionfor the plainvanillaeuropeanexceptforSmonwhichis the minimum price observed C(0,I,t)=max(S-Smin,0) and viceversa for the put I in the equationisonly a parameter dC/dI=0 since I the minimum value the stock couldreachinthetimeinterval

7. Pricingmodels Therecouldbealsousefultousemontecarlosimulation Purposeto monitor the min/maxvalueof the stock(somethingsimilartobarrieroption) Thing are simplerwhenyou price the option in the timewhenyoucouldexpiryit I=S Itissimilartoan american option We can also price the optionusingBinomialModel

8. Greeks • the Black-Scholes option valuation formulas depend upon S, t, and the parameters K, r and • we derive expressions for partial derivatives of the option values with respect to thesequantities • Useful: • traders like to know the sensitivity of the option value to changes in these quantities; the sensitivities can be measured bythesepartialderivatives • computing the partial derivatives allows us to conrm that the Black-Scholes PDE has been solved • examining the signs of the derivatives gives insights into the • underlyingformulas • the derivative V=S is needed in the delta hedging process • the derivatives V=sigma plays a role later when we discuss the impliedvolatility • we on the case of a call option

9. the delta of an at-the-money call option is close to 0,5. Delta moves to 1 as the call goes deep in the money. It moves to zero as the call goes deep out of the money. • the delta of an at-the-money put option is close to -0,5. Delta moves to -1 as the put goes deep in the money. It moves to zero as the call goes deep out ofthe money. • the parameter gamma measures the instability of delta with respect to S; note that gamma is identical for a call and put with identical characteristics • at-the-money options have the highest gamma, which indicates that changes very fast as S changes • both in-the money and out-of-the-money options have low gammas because their delta is constant, close to one or zero, respectively • as maturity nears the option gamma increases

10. Prospective and use Weuselookbackcalls in orderto take advantageofbullish trend of the stock sinceyouexercise the optionevenif the stock price isgreaterthan the minimum With the put wecould take advantageof the bearish trend of the stock after the maximum In graphbelowwecoulduselookbacks put after 2001 and lookbackcallafter 2003

11. Prospective and use • Lookbacks can begoodinstrumentsto take advantageof the possible slow down madebycrises • This type of option reduces uncertainties associated with the timing of market entry • While lookback options are appealing to investors, they can be expensive and are also considered to be quite speculative.