# Financial options

## Course

An option is a right to buy (call) or to sell (put) an underlying asset, on the expiration date (European option) or during a given period (American option) at a price which is known in advance (Strike price).

It enables various trading strategies in order to speculate on

• the increase of the price of the underlying asset
• the decrease of the price of the underlying asset
• the volatility of the price of the underlying asset
• the stability of the price of the underlying asset.
Some combinations of options enable to make profits if the underlying asset's price is stable while having a floor for losses in case of volatilty: butterflies and condors.
The combination of positions on options and futures enables aribtrages, namely strategies the results of which do not depend on the price of the underlying asset and are therefore known in advance: reverse conversion and conversion.
The box spread is an arbitrage which relies only on options: it does not use any future contract.
Such derivative products can be valued in discrete time (Cox-Ross-Rubinstien) or in continuous time (Black & Scholes).
Such principles are detailed in the following handout:

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## Cox-Ross-Rubinstein's option pricing model

Cox, Ross and Rubinstein have proposed, in 1979, an option pricing model in discrete time. Their 2 main assumptions are the following ones:

• The time t between the option's valuation date and its expiration date can be divided into n periods
• During a given period, the underlying asset's spot price either goes up (and is mutiplied by a number equal to u), or goes down (and is multiplied by a number equal to d). Assuming k upward moves and n-k downward moves, the generic spot price of the underlying asset on expiration date is: uk.dn-k.S where k is a natural number taking its values in the [0,n] interval
On expiration date of the call, its premium (Cn) is equal to its intrinsic value (IV) which can be calculated for each possible spot price of the underlying asset: Cn=max(0;uk.dn-k.S -E). Then, the method consists in going backward till the valuation date to get the call's premium. This is based on a tree, in which each node is the value of a call that is defined by: C=1/(1+r)t/n[p.Cu+(1-p)Cd] where:
• r = risk free rate
• p = probability of an upward move during a period
• Cu = premium of the call at the end of the next period, assuming an upward move
• Cd = premium of the call at the end of the next period, assuming a downward move
• p= probability of an upward move during a period
The evidence of the formula and of the useful intermediary results is detailed in the first Handout of this page.

The following handout describes, step by step, the Cox-Ross-Rubinstein's valuation of the call's premium based on an illustrative example:

The calculations are based on the following underlying Excel file:

## Greek letters

Partial derivatives of the Black & Scholes formula enable to get sensivities of options premiums to 4 paramters. These sensitivites are often named by Greek letters

• Delta for the sensivitity to a change in the spot price of the underlying asset
• Vega for the sensivitity to a change in the volatility of the underlying asset
• Theta for the sensitiity to a change in time to expiration
• Rhô for the sensivitity to a change in the risk free rate.The delta also enables to calibrate a hedging strategy. But, as the delta depends on the spot price of the underlying asset, which is changing everyday, the delta is also changing and the hedging has to be adapted accordingly.

The Gamma provides a measure of the sensitivity of the delta to changes in the spot price of the underlying asset.
The following slides present the various Greek letters with an illustrative example

The following handout details the evidence of each formula for each Greek letter

The calculations are based on the following underlying Excel file: