Put-call Parity For Binary Options

In fiscal mathematics,
put–call parity
defines a relationship betwixt the toll of a European call option and European put option, both with the identical strike toll and expiry, namely that a portfolio of a long call pick and a short put option is equivalent to (and hence has the same value every bit) a single forward contract at this strike price and death. This is considering if the cost at expiry is above the strike price, the telephone call volition be exercised, while if it is below, the put volition be exercised, and thus in either instance one unit of the nugget will be purchased for the strike price, exactly equally in a forward contract.

The validity of this relationship requires that sure assumptions exist satisfied; these are specified and the human relationship is derived below. In practise transaction costs and financing costs (leverage) mean this human relationship will not exactly agree, but in liquid markets the human relationship is close to exact.

Assumptions

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Put–telephone call parity is a static replication, and thus requires minimal assumptions, namely the beingness of a frontwards contract. In the absenteeism of traded forward contracts, the forward contract can be replaced (indeed, itself replicated) by the ability to buy the underlying asset and finance this by borrowing for fixed term (e.one thousand., borrowing bonds), or conversely to borrow and sell (brusque) the underlying asset and loan the received money for term, in both cases yielding a self-financing portfolio.

These assumptions do non crave whatever transactions between the initial date and expiry, and are thus significantly weaker than those of the Black–Scholes model, which requires dynamic replication and continual transaction in the underlying.

Replication assumes one can enter into derivative transactions, which requires leverage (and capital costs to back this), and ownership and selling entails transaction costs, notably the bid–inquire spread. The relationship thus just holds exactly in an ideal frictionless market with unlimited liquidity. Withal, real globe markets may be sufficiently liquid that the relationship is shut to verbal, almost significantly FX markets in major currencies or major stock indices, in the absence of market turbulence.

Argument

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Put–phone call parity
can be stated in a number of equivalent ways, most tersely as:

${\displaystyle C-P=D(F-G)}$

where


${\displaystyle C}$



is the (current) value of a telephone call,


${\displaystyle P}$



is the (current) value of a put,


${\displaystyle D}$



is the discount factor,


${\displaystyle F}$



is the forward price of the asset, and


${\displaystyle K}$



is the strike cost. Annotation that the spot price is given by


${\displaystyle D\cdot F=Southward}$



(spot price is nowadays value, forrard toll is future value, discount factor relates these). The left side corresponds to a portfolio of long a telephone call and brusque a put, while the right side corresponds to a forward contract. The assets


${\displaystyle C}$



and


${\displaystyle P}$



on the left side are given in current values, while the assets


${\displaystyle F}$



and


${\displaystyle Yard}$



are given in future values (frontwards price of asset, and strike price paid at expiry), which the disbelieve cistron


${\displaystyle D}$



converts to present values.

Using spot price


${\displaystyle Due south}$



instead of forward price


${\displaystyle F}$



yields:

${\displaystyle C-P=Southward-D\cdot K}$

Rearranging the terms yields a unlike interpretation:

${\displaystyle C+D\cdot K=P+Southward}$

In this case the left-hand side is a fiduciary call, which is long a call and enough greenbacks (or bonds) to pay the strike price if the phone call is exercised, while the right-hand side is a protective put, which is long a put and the asset, so the asset tin can be sold for the strike price if the spot is below strike at expiry. Both sides accept payoff
max(Due south(T),
G) at decease (i.east., at least the strike toll, or the value of the asset if more), which gives some other way of proving or interpreting put–call parity.

In more detail, this original equation can be stated every bit:

${\displaystyle C(t)-P(t)=S(t)-K\cdot B(t,T)}$

where

${\displaystyle C(t)}$



is the value of the call at time


${\displaystyle t}$


,

${\displaystyle P(t)}$



is the value of the put of the aforementioned expiration engagement,

${\displaystyle S(t)}$



is the spot toll of the underlying nugget,

${\displaystyle K}$



is the strike price, and

${\displaystyle B(t,T)}$



is the nowadays value of a zilch-coupon bond that matures to $1 at time


${\displaystyle T.}$



This is the present value factor for K.

Annotation that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price
Grand. Thus 1 way to read the equation is that a portfolio that is long a phone call and brusque a put is the same as being long a forward. In particular, if the underlying is not tradeable but there exists forward on it, we can supercede the correct-manus-side expression by the price of a forrard.

If the bail interest rate,


${\displaystyle r}$


, is assumed to be abiding and then

${\displaystyle B(t,T)=e^{-r(T-t)}}$

Note:


${\displaystyle r}$



refers to the force of interest, which is approximately equal to the effective almanac charge per unit for small interest rates. However, ane should take care with the approximation, especially with larger rates and larger fourth dimension periods. To find


${\displaystyle r}$



exactly, use


${\displaystyle r=\ln(ane+i)}$


, where


${\displaystyle i}$



is the constructive almanac interest charge per unit.

When valuing European options written on stocks with known dividends that volition be paid out during the life of the option, the formula becomes:

${\displaystyle C(t)-P(t)+D(t)=Southward(t)-K\cdot B(t,T)}$

where D(t) represents the total value of the dividends from 1 stock share to be paid out over the remaining life of the options, discounted to present value. We can rewrite the equation equally:

${\displaystyle C(t)-P(t)=S(t)-K\cdot B(t,T)\ -D(t)}$

and annotation that the right-hand side is the price of a forward contract on the stock with delivery price
K, every bit earlier.

Derivation

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Nosotros will suppose that the put and phone call options are on traded stocks, only the underlying can be any other tradeable nugget. The ability to buy and sell the underlying is crucial to the “no arbitrage” argument below.

Beginning, note that under the assumption that there are no arbitrage opportunities (the prices are arbitrage-free), ii portfolios that always have the aforementioned payoff at fourth dimension T must have the aforementioned value at any prior fourth dimension. To prove this suppose that, at some fourth dimension
t
before
T, one portfolio were cheaper than the other. So one could purchase (go long) the cheaper portfolio and sell (go short) the more than expensive. At time
T, our overall portfolio would, for any value of the share price, take nothing value (all the assets and liabilities have canceled out). The profit we fabricated at fourth dimension
t
is thus a riskless turn a profit, but this violates our assumption of no arbitrage.

Nosotros will derive the put-call parity relation by creating ii portfolios with the same payoffs (static replication) and invoking the above principle (rational pricing).

Consider a call choice and a put option with the same strike
M
for expiry at the aforementioned date
T
on some stock
S, which pays no dividend. We presume the existence of a bond that pays 1 dollar at maturity time
T. The bond price may be random (like the stock) simply must equal 1 at maturity.

Let the cost of
S
be S(t) at time t. At present assemble a portfolio by buying a call option
C
and selling a put option
P
of the same maturity
T
and strike
Chiliad. The payoff for this portfolio is
South(T) – K. At present assemble a 2d portfolio by buying one share and borrowing
K
bonds. Annotation the payoff of the latter portfolio is also
S(T) – K
at time
T, since our share bought for
Southward(t)
will exist worth
S(T)
and the borrowed bonds volition exist worth
K.

By our preliminary observation that identical payoffs imply that both portfolios must accept the aforementioned price at a general time


${\displaystyle t}$


, the post-obit relationship exists betwixt the value of the various instruments:

${\displaystyle C(t)-P(t)=S(t)-M\cdot B(t,T)\,}$

Thus given no arbitrage opportunities, the above relationship, which is known as
put-call parity, holds, and for any 3 prices of the phone call, put, bail and stock ane can compute the implied cost of the fourth.

In the case of dividends, the modified formula can be derived in like mode to above, but with the modification that 1 portfolio consists of going long a call, going short a put, and
D(T)
bonds that each pay i dollar at maturity
T
(the bonds will be worth
D(t)
at fourth dimension
t); the other portfolio is the same every bit before – long ane share of stock, short
Chiliad
bonds that each pay 1 dollar at
T. The difference is that at time
T, the stock is not only worth
Southward(T)
but has paid out
D(T)
in dividends.

History

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Forms of put-phone call parity appeared in do as early every bit medieval ages, and was formally described past a number of authors in the early 20th century.

Michael Knoll, in
The Ancient Roots of Modern Financial Innovation: The Early on History of Regulatory Arbitrage, describes the important role that put-call parity played in developing the disinterestedness of redemption, the defining characteristic of a modern mortgage, in Medieval England.

In the 19th century, financier Russell Sage used put-phone call parity to create synthetic loans, which had higher involvement rates than the usury laws of the time would have ordinarily allowed.^{[
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Nelson, an option arbitrage trader in New York, published a volume: “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’due south 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 Vinzenz Bronzin also derives the put-telephone call parity in 1908 and uses it as function of his arbitrage argument to develop a series of mathematical option models nether a series of unlike distributions. The piece of work of professor Bronzin was simply recently rediscovered by professor Wolfgang Hafner and professor Heinz Zimmermann. The original work of Bronzin is a volume written in German and is now translated and published in English language in an edited work by Hafner and Zimmermann (“Vinzenz Bronzin’s pick pricing models”, Springer Verlag).

Its first description in the modern academic literature appears to exist by Hans R. Stoll in the
Journal of Finance.
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Implications

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Put–call parity implies:

Encounter also

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• Spot-future parity
• Vinzenz Bronzin

References

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External links

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• Put-Call parity
  • Put-call parity, tutorial past Salman Khan (educator)
  • Put-Call Parity and Arbitrage Opportunity, investopedia.com
  • The Aboriginal Roots of Modern Fiscal Innovation: The Early History of Regulatory Arbitrage, Michael Knoll’s history of Put-Telephone call Parity
• Other arbitrage relationships
  • Arbitrage Relationships for Options, Prof. Thayer Watkins
  • Rational Rules and Boundary Conditions for Choice Pricing (PDFDi), Prof. Don K. Take chances
  • No-Arbitrage Bounds on Options, Prof. Robert Novy-Marx
• Tools
  • Choice Arbitrage Relations, Prof. Campbell R. Harvey