【正文】 ght, Martin Widdicks, 2021 Putcall parity: result ? So you are receiving $ now for no cost, hence this is an arbitrage opportunity. As such these portfolios must be equal in value at any point prior to maturity giving the following result (where X is the exercise price). European Put + Share = European Call + Present value of X ? This can be rearranged to give the arbitrage portfolio: Put + Share – Call – Present value of X = 0 ? Also it can allow you to determine the value of call and put options given the value of the other: Put = Call + Present value of X – Share Call = Put + Share – Present value of X Copyright, Martin Widdicks, 2021 Putcall parity: example ? Given our Microsoft data, does putcall parity hold? We assume that the current riskfree rate is % , there are 250 trading days in the year and given that the current share price is $ and exercise price is $25. The current value of the put option is $ giving Call = $ + $ – $*30/250 Call = $ Thus with these assumptions putcall parity holds. ? Note that there is a choice of how you calculate the present value, you can use discrete and continuous pounding but with these time scales then it is probably easier to use continuous pounding. Copyright, Martin Widdicks, 2021 Putcall parity:American options ? Note that put call parity also provides bounds for the value of European call and put options. As we know that both must be worth at least zero then this gives: European Call Share – Present value of X European Put Present value of X – Share . in the Microsoft example (assuming the same time and riskfree rate) then Call $ $ = $ ? This provides us with a very interesting result, as the American call option is worth at least as much as the European call then American call Share – Present value of X Copyright, Martin Widdicks, 2021 Putcall parity: American options ? However, with an American option it is always possible to exercise early and receive the difference between the current share price and the exercise price Share Price – X But, we know that the American call is always worth more than Share Price – Present value of X which is larger than Share price – X. Thus, If the share pays no dividends then it is never optimal to exercise an American call option. As such, it has the same price as a European Call option. Copyright, Martin Widdicks, 2021 Putcall parity: dividends ? When we consider shares which pay out dividends then the putcall parity changes to the following: Euro. Put + Share = Euro. Call + PV(X) + PV(Dividends) ? This again can be used to spot arbitrage opportunities or to value a call given the value of a put or vice versa. ? Note that the inclusion of a dividend may well mean that an American call option may be exercised before expiry thus making it more valuable than a European call. ? Note that we still have not arrived at a method for calculating the value of a European call or put option… Copyright, Martin Widdicks, 2021 Option pricing: BlackScholes ? It is actually a very plicated procedure to e up with the price of an option – even a simple European style option. ? In fact the value of an option is described by the BlackScholes equation: ? Where V is the value of the option, S is the value of the asset, t is the time from expiry, r is the riskfree rate and s the volatility of the underlying asset. ? The value of European Call and Put options are: Copyright, Martin Widdicks, 2021 Option pricing: BlackScholes ? Where, and, ? Unfortunately there do not exist ‘closed form’ or formulae expressions for the value of American options, thus one must use numerical techniques (. binomial trees, finitedifference methods, MonteCarlo methods or quadrature). Copyright, Martin Widdicks, 2021 Conclusions ? We have defined what an option is and seen the payoff from both writing and buying call and put options. We have also seen that unlike with futures contracts the buyer of the option must pay a premium to buy the option. ? We have been through some basics option strategies showing how flexible options are and how they can be used to both hedge and speculate on market movements. ? Although options are very difficult to price we have been able to derive some general results mainly by using the putcall parity which describes the relationship between call options, put options and the underlying asset. ? From this we were able to deduce that for underlying assets paying no dividends American call options are worth the same as European call options.