CFA Level I Derivatives - Derivative Pricing and Replication

[Music] understanding how underlying assets are priced in the spot market is crucial to understanding how derivatives are priced to understand derivative pricing it's necessary to establish a linkage between the derivative market and the spot market that linkage occurs through arbitrage we've briefly introduced the concept of arbitrage in the previous lesson we know that based on the law of one price two different securities with identical payoffs in the future should have the same price otherwise an arbitrage opportunity exists looking deeper can you think of any two securities that would produce identical payoffs in the future you may be hard-pressed to think of any in markets for traditional securities we don't often encounter two assets that have identical future payoffs even for very similar companies in the same industry the picture changes however if we introduced derivatives for most derivatives the payoffs may not be identical but they come directly from the value of the underlying at the expiration of the derivative although no one can predict with certainty the value of the underlying at expiration as soon as that value is determined the value of the derivative at expiration becomes certain this property of derivatives allows investors to construct hedged portfolios a hedged portfolio can be constructed by taking a position in the underlying and taking the opposite position in the derivative at the same time this creates a portfolio with no uncertainty about its value at some future date to understand the concept of hedged portfolios let's go through an example let's say there's an asset that's traded at $100 at time zero at this time a forward contract that expires in exactly one year is priced at a hundred and three dollars so an example of a hedged portfolio is if an investor long as the underlying asset at $100 at time zero and simultaneously enters the short position of the forward at a contract price of a hundred and three dollars since there is no cash flow involved at the initiation of the forward the net cash outflow from the investor at time zero is $100 at expiration the investor is obligated to sell the asset at a hundred and three dollars to close the short position the investor simply delivers the asset which he owns and collects a hundred and three dollars regardless of the market price of the asset at expiration the investor earns three dollars in a year or a three percent return without taking any risk of the underlying asset other than the counterparty risk the investor will get a three percent return regardless of the price of the underlying at expiration now let's bring back the asset pricing equation that we've learned earlier let's assume that holding the asset Y has negligible costs and benefits the expected future price is therefore the price compounded by risk-free rate plus risk premium let's assume also that the risk-free rate at time zero is three percent if that's true this means that the risk premium in this case is zero now we know that for a risk averse investor zero risk premium would not hold so is there a pricing error of the forward at time zero let's say the forward is priced with a 2% risk premium at a hundred and five dollars instead if that's the case an arbitrageur can sense the opportunity and borrow $100 at the risk-free rate of 3% to buy the asset he will simultaneously enter a short position at the contract price of a hundred and five dollars at time zero the net cash flow from the arbitrator is exactly zero at expiration the arbitrator simply sells the asset for a hundred and five dollars to fulfill the contract and uses a hundred and three dollars from the proceeds to repay the loan as you can see the arbitrator earns two dollars without putting in any cash now we know that such arbitrage opportunities can't last for long for when many arbitrators realize the opportunity the buying pressure on asset why will push its price up and the increase in shorts will push the forward price down this continues until the forward price returns of this portfolio are equal to the risk-free rate of 3% where no arbitrage is possible and the risk premium is zero the important point to understand from this illustration is that while the risk aversion of investors is relevant to pricing assets it's not relevant to pricing derivatives as such derivatives pricing is sometimes called risk-neutral pricing so based on the principles of arbitrage and risk-neutral pricing only one price can exist for the derivative we call this the no arbitrage derivative price the no arbitrage derivative price has to satisfy this equation the purchase of the underlying asset at time zero plus the short position and the forward at time zero equal to the present value of the net payoff at time t discounted by the risk-free rate this equation can be generalized to a position in the underlying asset plus the opposite position in the derivative is equal to the present value of the net payoff at time T discounted by the risk-free rate now going back to our example do you realize that the cash flows and risk characteristics of this portfolio is exactly the same as a risk-free bond in both instances the investor places $100 at time zero and gets a return of the three percent risk-free rate after one year in other words an asset and a derivative on the asset can be combined to produce a physician equivalent to a risk-free asset it follows that the asset and the risk-free asset can be combined to produce the derivative alternatively the derivative and the risk-free asset can be combined to produce the asset this process of creating an asset or portfolio from another asset portfolio and all derivative is known as replication so for this example we show how a long forward position in an asset can be replicated by borrowing at the risk-free rate to buy the asset in taking a long forward position an investor agrees to sell why one year later at the initiation of the contract the investor does not fork out any cash so his net cash flow is zero on the other hand the investor can replicate the long forward position by borrowing $100 at the risk-free rate and using the money to purchase the asset similarly his net cash flow at time zero is zero at expiration the market price of the asset is now s $1 under the forward contract the investor has to buy the asset at a hundred and three dollars he can then sell the asset on the market for s $1 so his net cash flow at expiration is s1 minus a hundred and three dollars under the replicated portfolio the investor sells the asset at the market price of s1 dollars the investor also has to repay the loan by returning a hundred and three dollars so likewise his net cash flow at expiration is s1 minus a hundred and three dollars as you can see the net cash flows for both portfolios are identical this illustrates how a long forward can be replicated by buying an asset and borrowing at the risk-free rate you're watching an 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Keywords: CFA Level 1, CFA Level I, Derivatives, CFA Prep Course, CFA Prep Provider, CFA Lesson, CFA Tutorial, CFA Video

Id: UhQpjGJ3QDk

Length: 8min 42sec (522 seconds)

Published: Thu Jan 16 2020