These series of article aims to analyse **Exotic Options** and their implication in term of Greeks. Although the most of you already know the definition, a buyer of a call/put option has the right, but not the obligation, to buy/sell a security during a certain amount of time or on a specific date. Probably everyone knows the classical **Black & Scholes formula** for pricing European Options, but unfortunately, that formula is no longer available for pricing complex derivatives.

In order to get a fair price a simulation is needed. Our assumption is that **under risk neutral probabilities** the following stochastic process governs the movement of the underlying asset:

*dS = rS(t)dt + sigmaS(t)dWt , where W∼ D*

A simulation of this process looks like the following figure. These are 100 simulated paths for a stock with initial price of 10, risk free rate 0.01, implied volatility of 0.35 and stochastic shocks normally distributed.

In this first article about exotic options we are going to talk about the lookback floating strike option, analysing the behaviour in term of **Delta**, **Vega** and **Theta** compared to a classical **European call option**.

#### The Lookback call floating strike option

The lookback call option gives to the buyer the right to buy at maturity the underlying asset at the strike price, which has fixed at the lowest price reached during the life of the option. The** payoff of the lookback floating call option** is the following:

*Payoff = max [S(T) – mint (S(t)) , 0] , where t belongs to [0 , T]*

Looking at the Greeks the behaviour of this option is not straightforward. Starting from the Delta , it determines the magnitude of the change of the price of the option due to a unit change in the underlying asset. Being a call option our **delta** is **positive**.

*dOption*

*Delta => 0*

*dS*

Comparing the delta of a lookback call floating option with the delta of a classical European call options the analysis becomes tricky. The delta of our option is grater if, during the period of the contract, the minimum reached by the underlying asset is less than the strike price of the comparable call Option. While it is less if the minimum is greater than the strike price. In formula:

*DeltaLB call ≥ DeltaEu call iff min(S(t)) < K*

*DeltaLB call ≤ DeltaEu call iff min(S(t)) > K*

Although this tricky behaviour it is important to remember that the price of a lookback call floating strike option is always greater than the price of an European call option if the following equation holds.

*P(min(S(t)) < K) > 0*

Intuitively our option worth more than the European one if the characteristics of the underlying asset imply that it is possible that our call option will get a better strike price during the path.

The **Vega**, which describes how much a change in the implied volatility of the underlying asset affects the option price, is straightforward in our analysis. Since our option is highly speculative, a change in the volatility of the underlying asset has a positive impact in our price. Moreover, it is important to remember that the more the volatility of the underlying asset the more the probability of observing strange movements. Looking at the payoff of the lookback call floating strike option, we can deduce that this would be beneficial for our contract, in formula:

*dOption*

*V => 0*

*dsigma*

In the following chart, it is possible to compare the price of our option with the price of an **ATM call option** and their behaviour when the volatility increases.

We derived the price for the ATM option from the **B&S framework**. The price of the lookback options derives fromsimulation. In order to get a good approximation, we simulated 10000 possible path for each level of volatility.

Where can we see the Vega? The formula for the** Vega (V)** combined to a basic course inmath tell us that it is possible to observe the Vega of the option from the slope of the two lines in the chart:

*dy dOptionPrice*

*== Vega*

*dx dsigma*

As expected the lookback call option shows a greater sensitivity than the European option to the Vega. The **Theta** of our option is negative, since intuitively the less time to maturity the less the probability of getting better conditions to buy.

*dOption*

*theta =< 0*

*dt*

Comparing the Theta with the ATM call option the following graph says that the Lookback option has a greater sensitivity, in absolute value, to the passage of time than the European option does.

Why did we say that our theta is negative if the slope looks positive? It depends on how we look at the time. If we mean the time as on day nearer to maturity we should look at the graph from right to the left. Since it is the most common notation in the financial markets, our Theta is negative as expected.

We can conclude that the** lookback floating strike option** is a riskier asset than the** classical European Call** given its greater sensitivity for the most of the Greeks.

Everything we derived here can be easily applied to the lookback put option floating strike price symmetrically. Remembering that the **payoff** will be:

*Payoff = max[ maxS(t) – S(T) , 0] , where t belongs to [0 , T]*

Riccardo Aimone

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