# A vanilla call option can be replicated by. a vanilla option with the appropriate payoff. If the payoff is that of a vanilla call, the option is a down-and-in call. Up-and-in options are defined in an analogous way. Knock-out options can be further complicated in many ways. For example, the position of the knockout boundary may be a function of time; in particular it may.

## A vanilla call option can be replicated by. This post is based on problems and in, “Heard on the Street” by Timothy Falcon Crack. I was asked how to price a digital option in a job interview.

In financial mathematics , put—call parity defines a relationship between the price of a European call option and European put option , both 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 put will be exercised, and thus in either case one unit of the asset will be purchased for the strike 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 replication , and 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 asset 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 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 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.

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 are given in current values, while the assets F and K are given in future values forward price of asset, and strike price paid at expiry , which the discount factor D converts to present values. In this case the left-hand side is a fiduciary call , which 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 put , which 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 T , K 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. 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 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 valuing 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-free , two 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 T , one portfolio were cheaper than the other. Then one could purchase go long the cheaper portfolio and sell go short the more expensive. At time T , our 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 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. 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. 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 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. 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. Options, Futures and Other Derivatives 5th ed. Energy derivative Freight derivative Inflation derivative Property derivative Weather derivative. Retrieved from " https: Finance theories Mathematical finance Options finance. All articles with unsourced statements Articles with unsourced statements from June Articles with unsourced statements from July Views Read Edit View history.

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