This relationship is known as the put-call parity principle between the price C of a European call option and the price P of a European put option, each with strike price K and underlying security value S. One can then calculate that the price of a call option with these assumptions is 1 1. At expiration this portfolio always has a value which is the strike price.
This example portfolio has total value 1 0 0. This is an illustration of the use of options for hedging an investment, in this case the extremely conservative purpose of hedging to preserve value.
Consider buying a put and selling a call, each with the same strike price K. But this payout is exactly what we would get from a futures contract to sell the stock at price K. This replicates the futures contract, so the future must have the same price as the initial outlay. Therefore we obtain the put-call parity principle:. Another way to view this formula is that it instructs us how to create synthetic portfolio. This same principle of linearity and the composition of more exotic options in terms of puts and calls allows us to create synthetic portfolios for exotic options such as straddles, strangles, and so on.
As noted above, we can easily write their values in closed form solutions. Knowing any two of S , C or P allows us to calculate the third. Of course, the immediate use of this formula will be to combine the security price and the value of the call option from the solution of the Black-Scholes equation to obtain the value of the put option:.
For the sake of mathematical completeness we can write the value of a European put option explicitly as:. Using the symmetry properties of the c. Value of the put option at maturity.
Now we use the Black-Scholes formula to compute the value of the option before expiration. Value of the call option at various times. Notice two trends in the value from this graph:. We can also plot the value of the put option as a function of security price and the time to expiration as a value surface.
Value surface from the Black-Scholes formula. This value surface shows both trends. This section is adapted from: Financial Derivatives by Robert W. Michael Steele, Springer, New York, , page The result can be plotted as functions of the security price as done in the text. In particular, the calculation of d 1 and d 2 uses broadcasting, also called binary singleton expansion, recycling, single-instruction multiple data, threading or replication. R script for Black-Scholes pricing formula for a put option.
Octave script for Black-Scholes pricing formula for a put option. Institute of Finance, Stochastic Calculus and Financial Applications. I check all the information on each page for correctness and typographical errors. Nevertheless, some errors may occur and I would be grateful if you would alert me to such errors.
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For a particular scripting language of your choice, modify the scripts to create a function within that language that will evaluate the Black-Scholes formula for a put option at a time and security value for given parameters.
For a particular scripting language of your choice, modify the scripts to create a script within that language that will plot the Black-Scholes solution for V P S , t as a surface over the two variables S and t.More...