# Call option and put option formula

A call optionoften simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The seller or "writer" is obligated to sell the commodity or financial instrument to the buyer if the buyer so decides. The buyer pays a fee called a premium for this right. The term "call" comes from the fact that the owner has the right to "call the stock away" from the seller.

Option values vary with the value of the underlying instrument over time. The price of the call contract must reflect the "likelihood" or chance of the call finishing in-the-money.

The call contract price generally will be higher when the contract has more time to expire except in cases when a significant dividend is present and when the underlying financial instrument shows more volatility. Determining this value is one of the central call option and put option formula of financial mathematics.

The most common method used is the Black—Scholes formula. Importantly, the Black-Scholes formula provides an estimate of the price of European-style options. Adjustment to Call Option: When a call option is in-the-money i.

Some of them are as follows:. Similarly if the buyer is making loss on his position i. Trading options involves a constant monitoring of the option value, which call option and put option formula affected by the **call option and put option formula** factors:.

Moreover, the dependence of the option value to price, volatility and time is not linear — which makes the analysis even more complex. From Wikipedia, the free encyclopedia. This article is about financial options. For call options in general, see Option law. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.

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In financial mathematicsput—call parity defines a relationship between the price of a European call **call option and put option formula** and European put optionboth with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to and hence has the same value as a single forward contract at this strike price and expiry.

This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the call option and put option formula will be exercised, and thus in either case one unit of the asset will be purchased for the call option and put option formula price, exactly as in a forward contract.

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. Put—call parity is a static replicationand thus requires minimal assumptions, namely the existence of a forward contract.

In the absence of traded forward contracts, the forward contract can be replaced indeed, itself replicated by the ability to buy the underlying **call option and put option formula** and finance this by borrowing for fixed term e. These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than call option and put option formula of the Black—Scholes modelwhich 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 thisand buying and selling entails transaction costsnotably 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 call option and put option formula of market turbulence. The left side corresponds to a portfolio of long a call and short a put, while the right side corresponds to a forward contract.

The assets C and P on the left side call option and put option formula given in current values, while the assets F and K are given in future values forward price of asset, and strike price paid at expirywhich the discount factor D converts to call option and put option formula values. In this case the left-hand side is a fiduciary callwhich is long a call and enough cash or bonds to pay the strike price if the call is exercised, while the right-hand side is a protective putwhich is long a put and the asset, so the asset can be sold for the strike price if the spot is below strike at expiry.

Both sides have payoff max S TK at expiry i. Note that the right-hand side of the equation is also the price of buying a forward contract on the stock with delivery price K. Call option and put option formula 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 tradeable but there exists forwards on it, we can replace the right-hand-side expression by the price of a forward.

However, one should take care with the approximation, especially with larger rates and larger time periods. When call option and put option formula European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes:. We can rewrite the equation as:. We will suppose that the put and call options are on traded stocks, but the underlying can be any other tradeable asset. The ability to buy and sell the underlying is crucial to the "no arbitrage" argument below.

First, note that under the assumption that there are no arbitrage opportunities the prices are arbitrage-freetwo portfolios that always have the same payoff at time T must have the same value at any prior time. To prove this suppose that, at some time t before Tone portfolio were cheaper than the other.

Then one could purchase go long the cheaper portfolio and sell go short the more expensive. At time Tour overall portfolio would, for any value of the share price, have zero value all the assets and liabilities have canceled out.

The profit we made at time t is thus a riskless profit, but this violates our assumption of no arbitrage. We will derive the put-call parity relation by creating two portfolios with the same payoffs static replication and invoking the above principle rational pricing. Consider a call option and a put option with the same strike K for expiry at the same date Call option and put option formula on some stock Swhich 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 Tsince our share bought for S t will be worth S T and the borrowed bonds will be worth K. Thus given no arbitrage opportunities, the above relationship, which is known as put-call parityholds, 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 call option and put option formula 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 Tthe stock is not only worth S T but has paid out D T in dividends. 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.

The Early History of Regulatory Arbitragedescribes the important role that put-call parity played in developing the equity of redemptionthe 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.

Nelson, an option arbitrage trader in New York, published a book: His book was re-discovered by Espen Gaarder Haug in the early s and many references from Nelson's book are given in Haug's book "Derivatives Models on Models". Engham Wilson but in less detail than Nelson Mathematics professor Vinzenz Bronzin also derives the put-call parity in and uses it as part of his arbitrage argument to develop a series of mathematical option models under a series of different distributions.

The work of professor Bronzin was just recently rediscovered by professor Wolfgang Hafner and professor Heinz Zimmermann. The original work of Bronzin is a book written in German and is now translated and published in English in an edited work by Hafner and Zimmermann "Vinzenz Bronzin's option pricing models", Springer Verlag. Its first description in the modern academic literature appears to be by Hans R. Stoll in the Journal of Finance. From Wikipedia, the free encyclopedia.

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