ch04 by Salim Kasap

Finance and Derivatives: Theory and Practice SĂŠ bastien Bossu and Philippe Henrotte

Chapter 4 Derivatives

1. Introduction 



A derivative security is a financial security whose value depends on the value of other securities called underlying assets. For ease of analysis we only focus on derivatives with a single underlying asset. 



Let Dt be the value of the derivative at time t and St that of the underlying asset. There exists a function f of time and S such that: Dt = f(t, St)

Chapter 4 Derivatives

1. Introduction (2) 

There are securities: 



two

main

categories

derivative

Forward and futures contracts where two parties agree to exchange an asset at a pre-agreed price and date. Options, where one party has the right (but not the obligation) to exchange an asset at a pre-agreed price and date with the other party. This right is usually bought for a premium paid upfront to the option seller (comparable to a one-off insurance fee.)

Chapter 4 Derivatives

1. Introduction (3) 



There are also more complex derivative securities called exotic derivatives. Although it is financially conceivable to create perpetual derivatives, in practice all derivatives have a final date called the maturity date past which they cease to exist. The value of the derivative at maturity is called the payoff.

Chapter 4 Derivatives

2. Forward Contracts 

The financial characteristics of a forward contract are:  



The underlying asset to be exchanged in the future; A maturity date T: the date when the underlying will be exchanged (or ‘delivered’); A delivery price or strike K: the price to pay in exchange for the underlying asset.

Chapter 4 Derivatives

2. Forward Contracts (2) 

We will denote the value of a forward contract at time t from the buyer’s viewpoint as FCt



Clearly the seller lives in a symmgetric world where the contract value is –FCt



Note that the value FCt is for one unit of the underlying.

Chapter 4 Derivatives

2.1 Payoff 

 



The payoff of a forward contract at maturity is: FCT = ST – K For the seller the payoff is: –FCT = K – ST This formula corresponds to the profit or loss for the buyer at maturity T: the buyer receives one unit of the underlying whose value is ST and pays K to the seller. Note that such profit or loss is latent in the sense that to actually realize his gains or losses the buyer would need to immediately resell the underlying on the market.

Chapter 4 Derivatives

Figure p.49: Payoff for the Buyer FCT

ST – K

ST K

–K

Chapter 4 Derivatives

Figure p.49: Payoff for the Seller +K

–FCT

ST K K– ST

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2.2 Arbitrage Price 

Let us focus on the value of the forward contract at an arbitrary point in time t ≤ T and particularly on today’s value FC0.



Assume that: 



The underlying does not pay any cash flows (in particular no dividends or coupons); Investors can borrow and lend money over T years at interest rate r; There are no arbitrage opportunities, securities are infinitely liquid, and short selling is allowed.

Then the arbitrage price at t = 0 of a forward contract is

Chapter 4 Derivatives

2.2 Arbitrage Price (2) 

Then the arbitrage price at t = 0 of a forward contract is:

K FC0 = S0 − T (1 + r ) 

This formula is easily generalized for an arbitrary point in time t.

Chapter 4 Derivatives

2.2 Arbitrage Price (3) K



Proof: Suppose FC0 > S0 −



One may then follow the arbitrage strategy below:

(1+ r)

Chapter 4 Derivatives

2.2 Arbitrage Price (4) K



Suppose now FC0 < S0 −



Again one may follow the arbitrage strategy:

(1+ r)

Chapter 4 Derivatives

2.3 Forward Price 



In practice, when entering into a forward contract, the parties do not exchange any initial cash flows. This is made possible by choosing a delivery price K* such that the contract has zero initial value, i.e.:

K S0 − =0 T (1 + r )



Solving for K yields: K* = S0(1 + r)T.

Chapter 4 Derivatives

2.3 Forward Price (2) 



This price (K*) is called the forward price of the underlying at maturity T, while the current price of the underlying S0 is called the spot price. The forward price, commonly denoted F0 or F(0, T), is not to be confused with ST, the future spot price of the underlying at time t = T. F0 is indeed determined at t = 0 while ST is unknown until t = T and has no particular reason to be equal to F0 on that date.

Chapter 4 Derivatives

3. Plain vanilla options 3.1 Definitions & Notations 



A European call on an asset confers the right but not the obligation to buy this asset at a pre-agreed price and date. A European put on an asset confers the right but not the obligation to sell this asset at a pre-agreed price and date. An American call on an asset confers the right but not the obligation to buy this asset at a pre-agreed price until a certain date. An American put on an asset confers the right but not the obligation to sell this asset at a pre-agreed price until a certain date. These four categories are referred to by practitioners as plain vanilla options, in contrast to exotic options which are more sophisticated.

Chapter 4 Derivatives

3.1 Definitions & Notations (2) 



K: exercise price or strike: the price at which the underlying asset is exchanged; T: expiry or maturity: the date when or until when the underlying is exchanged; ct , CtUS : value at time t of a European and American call; pt , PtUS : value at time t of a European and American put. As with forward contracts, an option value is expressed per unit of underlying asset and from the option buyer’s viewpoint.

Chapter 4 Derivatives

3.2 Payoff 



Clearly a rational individual will only exercise his right to buy or sell the underlying asset conferred by a call or put option if it is profitable to do so 

For a call option this is the case when ST > K;



For a put option this is the case when ST < K.

Therefore, the respective payoffs of the European call and put with strike K and maturity T are given as: 

For the call: cT = max(0, ST – K);



For the put: pT = max(0, K – ST).

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Figure p.52: Call Payoff cT

max(ST – K, 0)

ST K

Chapter 4 Derivatives

Figure p.52: Put Payoff pT max(K – ST, 0) ST K

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3.3 Put-Call Parity 

European puts and calls with identical characteristics (underlying S, strike K, maturity T) satisfy the putcall parity equation:

K c0 − p0 = FC0 = S0 − (1 + r )T



In other words: ‘call minus put equals forward.’ Note that put-call parity does not hold for American options. Put-call parity shows that a European call is in fact exactly one half of a forward contract, the other half being a sold (‘short’).

Chapter 4 Derivatives

Figure p.53: Call – Put = Forward payoff

+cT

ST K