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4. Derivatives Notes

Finance Notes > Corporate Finance Notes

This is an extract of our 4. Derivatives document, which we sell as part of our Corporate Finance Notes collection written by the top tier of University Of London (examined By LSE) students.

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Derivatives
Derivative Properties and Pricing
Option Payoff and Profit
Option
A right but not an obligation to buy / sell a given asset at a specified price (X) during a specified period of time.
Holder
Call Option

Issuer

Payoff = max [(ST - X), 0]

X Payoff = min [-(ST - X), 0]

STT
S

X Profit (unlimited) = (ST - X) - C

Profit (max) = C

C

X X

X+C
X+C

STT

-C

Put Option

X Loss (max) = Call Premium (C)

Loss (unlimited) = (ST - X) - C

Payoff = max [(X - ST), 0]

Payoff = min [-(X - ST), 0]

X STT

X -X

Profit (max) = (X - ST) - P
P
-P

X-P
X -XP
X-PX

Profit (max) = P

STT
P-X

1 Loss (max) = Put Premium (P)

Loss (max) = (X - ST) - P

Types of Options
European
Features

American

Exercise on the expiration date Exercise on or before the expiration date
(T)

(T - ), where  = payoff at exercise date

Arbitrage
Attempt to make costless profits through exploiting price differences of essentially identical assets by buying at a lower price and immediately selling at a higher price in different markets.

Binomial Option Pricing
P(G) = Probability of good state

P(B) = Probability of bad state

S0 = Spot price

S1 = Price at period 1

SG1 = Price in good state at S1

SB1 = Price in bad state at S1

a = Number of units of stock

b = Number of units in the risk free asset

Call premium (Price) = C

Put premium (Price) = P

X = Strike price

R = Risk free rate

Portfolio Replication
Payoff
Portfolio Contents

Call Option Portfolio

Put Option Portfolio

Call option + Risk free asset

Put option + Risk free asset

when P(G) in S1

aSG1 + b(1 + R) = SG1 - X

aSG1 + b(1 + R) = 0

when P(B) in S1

aSB1 + b(1 + R) = 0

aSB1 + b(1 + R) = X - SB1

Cost of Portfolio in S0

aS0 + b = C

aS0 + b = P

Solve the equation to find C and P
Note: Continuous Compounding  beR x T

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