Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010

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Put-Call Parity

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Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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QuestionofDay

Question of the Day

What does it mean to say that a differential equation is a linear differential equation?

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Key Concepts

The put-call parity principle links the price of a put option, a call option and the underlying security price.
The put-call parity principle can be used to price European put options without having to solve the Black-Scholes equation.
The put-call parity principle is a consequence of the linearity of the Black-Scholes equation.

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Vocabulary

The put-call parity principle is the relationship $C - P = S - K e^{- r (T - t)}$
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 $S$ .

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Mathematical Ideas

Put-Call Parity by Linearity of the Black-Scholes Equation

The Black-Scholes equation is

V_{t} + \frac{1}{2} σ^{2} S^{2} V_{S S} + r S V_{S} - r V = 0 .

With the additional terminal condition $V (S, T)$ 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 $S$ is a solution of the Black-Scholes equation and $K e^{- r (T - t)}$ is also a solution, so $S - K e^{- r (T - t)}$ is a solution. At the expiration time $T$ , the solution has value $S - K$ .

Now if $C (S, t)$ is the value of a call option at security value $S$ and time $t < T$ , then $C (S, t)$ satisfies the Black-Scholes equation, and has terminal value $max (S - K, 0)$ . If $P (S, t)$ is the value of a put option at security value $S$ and time $t < T$ , then $P (S, t)$ also satisfies the Black-Scholes equation, and has terminal value $max (K - S, 0)$ . Therefore by linearity, $C (S, t) - P (S, t)$ is a solution and has terminal value $C (S, T) - P (S, T) = S - K$ . By uniqueness, the solutions must be the same, and so

C - P = S - K e^{- r (T - t)} .

This relationship is known as the put-call parity principle

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.

Put-Call Parity by Reasoning about Arbitrage

Assume that an underlying security satisfies the assumptions of the previous sections. Assume further that:

The security price is currently $S = 100$ ,
The strike price is $K = 100$ ,
The expiration time is one year, $T = 1$ ,
The risk-free interest rate is $r = 0.12$ ,
The volatility is $σ = 0.10$ .

One can then calculate that the price of a call option with these assumptions is $11.84$ .

Consider an investor who buys the following portfolio:

Buy one share of stock at price $S = 100$ .
Sell one call option at $C = V (100, 0) = 11.84$ .
Buy one put option at unknown price.

Now at expiration, the stock price could have many different values, and those would determine the values of the derivatives, see the table for some representative values:

Security	Call	Put	Portfolio
80	0	20	100
90	0	10	100
100	0	0	100
110	-10	0	100
120	-20	0	100

This portfolio has total value which is the strike price (which happens to be the same as the current value of the security.) Holding this portfolio will give a risk-free investment that will pay $100 in any circumstance. 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 $S$ , $C$ and $P$ with parameters $K$ , $r$ , $σ$ , and $T$ . Consider buying a put and selling a call, each with the same strike price $K$ . We will find at expiration $T$ that

if the stock price $S$ is below $K$ we will realize a profit of $K - S$ from the put option that we own;
if the stock price is above $K$ , we will realize a loss of $S - K$ from fulfilling the call option that we sold.

But this payout is exactly what we would get from a futures contract to sell the stock at price $K$ . The price set by arbitrage of such a futures contract must be $K e^{- r (T - t)} - S$ . Specifically, one could sell (short) the stock right now for $S$ , and lend $K e^{- r (T - t)}$ dollars right now for a net cash outlay of $K e^{- r (T - t)} - S$ , then at time $T$ collect the loan at $K$ 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:

- C + P = K exp (- r (T - t)) - S

or more naturally

S - C + P = K exp (- r (T - t)) .

Synthetic Portfolios

Another way to view this formula is that it instructs us how to create synthetic portfolios: Since

S + P - K exp (- r (T - t)) = C

a portfolio “long in the underlying security, long in a put, short $K exp (- r (T - t))$ 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 the exotic options such as straddles, strangles, and so on. As noted above, we can easily write their values in closed form solutions.

Explicit Formulas for the Put Option

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:

P = C - S + K exp (- r (T - t))

For the sake of mathematical completeness write the value of a European put option explicitly as:

V_{P} (S, t) = S Φ (\frac{log (S ∕ K) + (r + σ^{2} ∕ 2) (T - t)}{σ \sqrt{T - t}}) - K e^{- r (T - t)} Φ (\frac{log (S ∕ K) + (r - σ^{2} ∕ 2) (T - t)}{σ \sqrt{T - t}}) - S + K exp (- r (T - t)) .

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

\begin{array}{rcl} d_{1} & = & \frac{log (S ∕ K) + (r + σ^{2} ∕ 2) (T - t)}{σ \sqrt{T - t}} \\ d_{2} & = & \frac{log (S ∕ K) + (r - σ^{2} ∕ 2) (T - t)}{σ \sqrt{T - t}} \end{array}

so that

\begin{array}{rcl} V_{P} (S, t) = S (Φ (d_{1}) - 1) - K e^{- r (T - t)} (Φ (d_{2}) - 1) . \end{array}

Using the symmetry properties of the c.d.f. $Φ$ , we obtain

V_{P} (S, t) = K e^{- r (T - t)} Φ (- d_{2}) - S Φ (- d_{1}) .

Graphical Views of the Put Option Value

For graphical illustration let $P$ be the value of a put option with strike price $K = 100$ . The risk-free interest rate per year, continuously compounded is 12%, so $r = 0.12$ , the time to expiration is $T = 1$ measured in years, and the standard deviation per year on the return of the stock, or the volatility is $σ = 0.10$ . The value of the put option at maturity plotted over a range of stock prices $0 \leq S \leq 150$ surrounding the strike price is illustrated below:

Figure 1:

Value of the put option at maturity

Now we use the Black-Scholes formula to compute the value of the option prior to expiration. With the same parameters as above the value of the put option is plotted over a range of stock prices $0 \leq S \leq 150$ at time remaining to expiration $t = 1$ (red), $t = 0.8$ , (orange), $t = 0.6$ (yellow), $t = 0.4$ (green), $t = 0.2$ (blue) and at expiration $t = 0$ (black).

Figure 2:

Value of the put option at various times

Notice a couple of trends in the value from this graph:

As the stock price increases, for a fixed time the option value decreases,
As the time to expiration decreases, for a fixed stock value price lass than the strike price the value of the option increases to the value at expiration.

We can also plot the value of the put option as a function of security price and the time to expiration as value surface:

Figure 3:

Value surface from the put-call parity formula

Sources

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.

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Problems to Work

Problems to Work for Understanding

Calculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum (compounded continuously) and the volatility is 30% per annum.
What is the price of a European put option on a non-dividend paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum (compounded continuously), the volatility is 35% per annum, and the time to maturity is 6 months?

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Books

Reading Suggestion:

References

[1] Robert W. Kolb. Financial Derivatives. Institute of Finance, 1993.

[2] J. Michael Steele. Stochastic Calculus and Financial Applications. Springer-Verlag, 2001. QA 274.2 S 74.

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Links

Outside Readings and Links:

Video with explanation of put-call parity..
Option Research and Technology Services. Provides important option trading terms and jargon, here is the link to definition of “Put-Call Parity”.

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Last modified: Processed from LATEX source on December 4, 2010