Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
Mathematically Mature: may contain mathematics beyond calculus with proofs.
What does it mean to say that a diﬀerential equation is a linear diﬀerential equation?
between the price of a European call option and the price of a European put option, each with strike price and underlying security value .
The Black-Scholes equation is
With the additional terminal condition given, a solution exists and is unique. We observe that the Black-Scholes is a linear equation, so the linear combination of any two solutions is again a solution.
From the problems in the previous section (or by easy veriﬁcation right now) we know that is a solution of the Black-Scholes equation and is also a solution, so is a solution. At the expiration time , the solution has value .
Now if is the value of a call option at security value and time , then satisﬁes the Black-Scholes equation, and has terminal value . If is the value of a put option at security value and time , then also satisﬁes the Black-Scholes equation, and has terminal value . Therefore by linearity, is a solution and has terminal value . By uniqueness, the solutions must be the same, and so
This relationship is known as the put-call parity principle between the price of a European call option and the price of a European put option, each with strike price and underlying security value .
This same principle of linearity and the composition of more exotic options in terms of puts and calls allows us to write closed form formulas for the values of exotic options such as straps, strangles, and butterﬂy options.
Assume that an underlying security satisﬁes the assumptions of the previous sections. Assume further that:
One can then calculate that the price of a call option with these assumptions is .
Consider an investor with the following portfolio:
At expiration, the stock price could have many diﬀerent values, and those would determine the values of each of the derivatives. See Table 1 for some representative values.
At expiration this portfolio always has a value which is the strike price. Holding this portfolio will give a risk-free investment that will pay $100 in any circumstance. This example portfolio has total value . Therefore the value of the whole portfolio must equal the present value of a riskless investment that will pay oﬀ $100 in one year. 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.
The parameter values chosen above are not special and we can reason with general , and with parameters , , , and . Consider buying a put and selling a call, each with the same strike price . We will ﬁnd at expiration that:
But this payout is exactly what we would get from a futures contract to sell the stock at price . The price set by arbitrage of such a futures contract must be . Speciﬁcally, one could sell (short) the stock right now for , and lend dollars right now for a net cash outlay of , then at time collect the loan at dollars and actually deliver the stock. 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:
or more naturally
Another way to view this formula is that it instructs us how to create synthetic portfolio. A synthetic portfolio is a combination of securities, bonds, and options that has the same payout at expiration as another ﬁnancial instrument. Since
a portfolio “long in the underlying security, long in a put, short in bonds” replicates a call. 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 , or 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:
Usually one doesn’t see the solution as this full closed form solution. Instead, most versions of the solution write intermediate steps in small pieces, and then present the solution as an algorithm putting the pieces together to obtain the ﬁnal answer. Speciﬁcally, let
Using the symmetry properties of the c.d.f. , we obtain
For graphical illustration let be the value of a put option with strike price . The risk-free interest rate per year, continuously compounded is 12%, so , the time to expiration is measured in years, and the standard deviation per year on the return of the stock, or the volatility is . The value of the put option at maturity plotted over a range of stock prices surrounding the strike price is illustrated below:
Now we use the Black-Scholes formula to compute the value of the option before expiration. With the same parameters as above the value of the put option is plotted over a range of stock prices at time remaining to expiration (red), , (orange), (yellow), (green), (blue) and at expiration (black).
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.
This value surface shows both trends.
This section is adapted from: Financial Derivatives by Robert W. Kolb, New York Institute of Finance, Englewood Cliﬀs, NJ, 1993, page 107 and following. Parts are also adapted from Stochastic Calculus and Financial Applications by J. Michael Steele, Springer, New York, 2000, page 155.
For given parameter values for , , , and , the Black-Scholes-Merton solution formula for a put option is sampled at a speciﬁed array of times and at a speciﬁed array of security prices using vectorization and broadcasting. The result can be plotted as functions of the security price as done in the text. The calculation is vectorized for an array of values and an array of values, but it is not vectorized for arrays in the parameters , , , and . This approach is taken to illustrate the use of vectorization and broadcasting for eﬃcient evaluation of an array of solution values from a complicated formula.
In particular, the calculation of and uses broadcasting, also called binary singleton expansion, recycling, single-instruction multiple data, threading or replication.
The calculation relies on using the rules for calculation and handling of inﬁnity and NaN (Not a Number) which come from divisions by , taking logarithms of , and negative numbers and calculating the normal cdf at inﬁnity and negative inﬁnity. The plotting routines will not plot a NaN which accounts for the gap at in the graph line for .
R script for Black-Scholes pricing formula for a put option..
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Email to Steve Dunbar, sdunbar1 at unl dot edu
Last modiﬁed: Processed from LATEX source on August 12, 2016