Put–call parity
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The validity of this relationship requires that certain assumptions be satisfied; these are specified and the relationship is derived below. In practice transaction costs and financing costs (leverage) mean this relationship will not exactly hold, but in liquid markets the relationship is close to exact.
Assumptions
Put–call parity is a
These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the
Replication assumes one can enter into derivative transactions, which requires leverage (and capital costs to back this), and buying and selling entails transaction costs, notably the bid–ask spread. The relationship thus only holds exactly in an ideal frictionless market with unlimited liquidity. However, real world markets may be sufficiently liquid that the relationship is close to exact, most significantly FX markets in major currencies or major stock indices, in the absence of market turbulence.
Statement
Put–call parity can be stated in a number of equivalent ways, most tersely as:
where is the (current) value of a call, is the (current) value of a put, is the
Now the spot price can be obtained by discounting the forward price by the factor . Using spot price instead of forward price gives us:
- .
Rearranging the terms gives a first interpretation:
- .
Here the left-hand side is a
That a long call with cash is equivalent to a long put with asset is one meaning of put-call parity.
Rearranging the terms another way gives us a second interpretation:
- .
Now the left-hand side is a cash-secured put, that is, a short put and enough cash to give the put owner should they exercise it. The right-hand side is a covered call, which is a short call paired with the asset, where the asset stands ready to be called away by the call owner should they exercise it. At expiry, the previous scenario is flipped. Both sides now have payoff equal to either the strike price or the value of the asset, whichever is lower.
So we see that put-call parity can also be understood as the equivalence of a cash-secured (short) put and a covered (short) call. This may be surprising as selling a cash-secured put is typically seen as riskier than selling a covered call.[1]
To make explicit the time-value of cash and the time-dependence of financial variables, the original put-call parity equation can be stated as:
where
- is the value of the call at time ,
- is the value of the put of the same expiration date,
- is the spot priceof the underlying asset,
- is the strike price, and
- is the present value of a zero-coupon bond that matures to $1 at time , that is, the discount factor for
Note that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price . Thus one way to read the equation is that a portfolio that is long a call and short a put is the same as being long a forward. In particular, if the underlying is not tradable but there exists forwards on it, we can replace the right-hand-side expression by the price of a forward.
If the bond interest rate, , is assumed to be constant then
Note: refers to the
When valuing European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes:
where represents the total value of the dividends from one stock share to be paid out over the remaining life of the options, discounted to present value.
We can rewrite the equation as:
and note that the right-hand side is the price of a forward contract on the stock with delivery price , as before.
Derivation
We will suppose that the put and call options are on traded stocks, but the
First, note that under the assumption that there are no
We will derive the put-call parity relation by creating two portfolios with the same payoffs (
Consider a call option and a put option with the same strike K for expiry at the same date T on some stock S, which pays no dividend. We assume the existence of a bond that pays 1 dollar at maturity time T. The bond price may be random (like the stock) but must equal 1 at maturity.
Let the price of S be S(t) at time t. Now assemble a portfolio by buying a call option C and selling a put option P of the same maturity T and strike K. The payoff for this portfolio is S(T) - K. Now assemble a second portfolio by buying one share and borrowing K bonds. Note the payoff of the latter portfolio is also S(T) - K at time T, since our share bought for S(t) will be worth S(T) and the borrowed bonds will be worth K.
By our preliminary observation that identical payoffs imply that both portfolios must have the same price at a general time , the following relationship exists between the value of the various instruments:
Thus given no arbitrage opportunities, the above relationship, which is known as put-call parity, holds, and for any three prices of the call, put, bond and stock one can compute the implied price of the fourth.
In the case of dividends, the modified formula can be derived in similar manner to above, but with the modification that one portfolio consists of going long a call, going short a put, and D(T) bonds that each pay 1 dollar at maturity T (the bonds will be worth D(t) at time t); the other portfolio is the same as before - long one share of stock, short K bonds that each pay 1 dollar at T. The difference is that at time T, the stock is not only worth S(T) but has paid out D(T) in dividends.
History
Forms of put-call parity appeared in practice as early as medieval ages, and was formally described by a number of authors in the early 20th century.
Michael Knoll, in The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage, describes the important role that put-call parity played in developing the equity of redemption, the defining characteristic of a modern mortgage, in Medieval England.
In the 19th century, financier Russell Sage used put-call parity to create synthetic loans, which had higher interest rates than the usury laws of the time would have normally allowed.[citation needed]
Nelson, an option arbitrage trader in New York, published a book: "The A.B.C. of Options and Arbitrage" in 1904 that describes the put-call parity in detail. His book was re-discovered by Espen Gaarder Haug in the early 2000s and many references from Nelson's book are given in Haug's book "Derivatives Models on Models".
Henry Deutsch describes the put-call parity in 1910 in his book "Arbitrage in Bullion, Coins, Bills, Stocks, Shares and Options, 2nd Edition". London: Engham Wilson but in less detail than Nelson (1904).
Mathematics professor
Its first description in the modern academic literature appears to be by
Implications
Put–call parity implies:
- Equivalence of calls and puts: Parity implies that a call and a put can be used interchangeably in any delta-neutral portfolio. If is the call's delta, then buying a call, and selling shares of stock, is the same as selling a put and selling shares of stock. Equivalence of calls and puts is very important when trading options.[citation needed]
- Parity of implied volatility: In the absence of dividends or other costs of carry (such as when a stock is difficult to borrow or sell short), the implied volatility of calls and puts must be identical.[4]
See also
- Spot-future parity
- Vinzenz Bronzin
References
- ^ Noël, Martin (17 May 2017). "Call Put Parity: How to Transform Your Positions". OptionMatters.ca. Bourse de Montréal Inc.
- JSTOR 2325677.
- S2CID 154820481.
- ISBN 0-13-009056-5.
External links
- Put-Call parity
- Put-call parity, tutorial by Salman Khan (educator)
- Put-Call Parity and Arbitrage Opportunity, investopedia.com
- The Ancient Roots of Modern Financial Innovation: The Early History of Regulatory Arbitrage, Michael Knoll's history of Put-Call Parity
- Put-call parity, tutorial by
- Other arbitrage relationships
- Arbitrage Relationships for Options, Prof. Thayer Watkins
- Rational Rules and Boundary Conditions for Option Pricing (PDFDi), Prof. Don M. Chance
- No-Arbitrage Bounds on Options, Prof. Robert Novy-Marx
- Tools
- Option Arbitrage Relations, Prof. Campbell R. Harvey