1. Ticciati Marco Muhammad Naeem Santangelo Giusj Carmen

2. Lookback call ( put) gives the option holder the right to buy (sell) asset at its lowest ( 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, LC float =max ( S T -S min ;0) LP float =max( S max -S T; 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. LC fix =max ( S max -K;0) LP fix = max(K-S min ;0)

5. Pricing models  There exist closed formulae to price the instrument based on black and scholes framework Considering I(t) = min S(t) Being the lookback call an option on the minimum price The same thing could be applied also for the put The only difference is the fact that the put is on the maximum We have to invert the formula

6. The formula depends on the PDE with the very same boundary condition for the plain vanilla european except for Smon which is the minimum price observed C(0,I,t)=max(S-Smin,0) and viceversa for the put I in the equation is only a parameter dC/dI=0 since I the minimum value the stock could reach inthe time interval

7. Pricing models  There could be also useful to use montecarlo simulation  Purpose to monitor the min/max value of the stock(something similar to barrier option)  Thing are simpler when you price the option in the time when you could expiry it  I=S  It is similar to an american option  We can also price the option using Binomial Model

8. 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 these quantities Useful: traders like to know the sensitivity of the option value to changes in these quantities; the sensitivities can be measured by these partial derivatives 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 underlying formulas the derivative V=S is needed in the delta hedging process the derivatives V=sigma plays a role later when we discuss the implied volatility 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  We use lookback calls in order to take advantage of bullish trend of the stock since you exercise the option even if the stock price is greater than the minimum  With the put we could take advantage of the bearish trend of the stock after the maximum  In graph below we could use lookbacks put after 2001 and lookback call after 2003

11.  Lookbacks can be good instruments to take advantage of the possible slow down made by crises  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.