MA3667 Computing assignmentBackground information for MA3667 Computing assignment.EUROPEAN CALL AND PUT OPTIONSLarge institutions will often trade securities known asderivatives. They are financial instru-ments whose value depends upon the value of more basic underlying stocks or assets. We willmeet a few types of derivative in this course, but among the most prominent will beoptions. Anoptionis a contract that gives the option holder the right to buy or sell a particular asset eitherup to or on a specified date in the future, for a price that is agreed upon now.Some terminology and symbols:Call Option. A call option gives the holder the right tobuythe underlying asset for aspecified price known as theexerciseorstrikeprice.Put Option. A put option gives the holder the right tosellthe underlying asset during acertain time window for a specified price known as theexerciseorstrikeprice.Exercising an option. If an option holder executes their right to buy or sell the underlyingasset then they are said toexercisethe option.Note that an option holder is not forcedto exercise their right to buy/ sell the asset.European Option. In European options the option holder is only allowed to exercise theoption at one specified time, known as theexpiration date.American Option. In American options the option holder can exercise the right to buy/sellatany time until expiration date.Maturity,Expiration date,Exercise date. These all mean the same thing – the date inthe future on which (European), or by which (American), the option to buy/sell the assetmust be exercised.Some symbols:T:= the expiration date.St:= the price of the asset at timet, so inparticularST:= the asset price (notthe option price!!) at expiration dateT.K:= theexercise price.2Option price. The buyer of the option gains protection against risk, whereas the seller isexposing themselves to it. The buyer of the option gains protection against unfavourablemovements in the underlying asset price: the holder of a put option knows that they canalways get at leastKfor it whatever happens to the price of the asset in the market, whereasthe holder of a call option knows that they will have to spend at mostKto buy the assetat timeT, however high the price of the asset may be in the market at timeT. In order togain this protection against risk the holder of the option has to pay the seller of the optiona fee – theoption price.More symbols: inthis documentwe will use:Pt(T; K) := the price at timetof a Europeanput option with expiration dateTand strike priceK. We will also use:Ct(T; K) := theprice at timetof a European call option with expiration dateTand strike priceK.Payoff. Suppose that you are the holder of a European call option to buy a stock for strikepriceKat timeT. Then you would only bother to exercise the option if it the market priceof the stock at timeTturned out to be more thanK, in which case the option would giveyou a payoff ofST