Financial Derivatives and Risk Management
A derivative is a financial instrument whose value depends upon the value of other,
more basic, underlying variables.
Some common types of derivatives: Options, Forwards, Futures, and Swaps
Participants in Derivatives Market
Hedger – when a firm uses derivatives to reduce the risk of its future cash flow.
- A hedger would normally have a risk exposure to the underlying asset.
Speculator- when derivatives are used with the sole aim of making a profit on how an
asset’s value will change in the future.
- A speculator may have no risk exposure to the underlying asset.
Arbitrageur- arbitrage involves locking in a riskless profit by simultaneously
entering into transactions in two or more markets.
- Transaction costs are very low for large investment banks.
Week 1: Financial Options
A financial option is a contract that gives its owner the right, but not the obligation, to
buy or sell an underlying asset at a fixed price, on or before a specified date in the
future.
Call option: A call option gives the holder the right to buy an asset.
Put option: A put option gives the holder the right to sell an asset.
An option is a contract between two parties, for every holder of a financial option,
there is also an option writer.
The seller (writer) of the option, however, has an obligation to honour the contract
if the holder decides to act; The holder of the option pays a premium to the writer
of the option.
Options can be OTC or exchange traded.
American options: the most common kind, options that allow their holder to exercise
the option on any date up to and including the expiration date.
European options: options that allow their holders to exercise the option only on the
expiration date.
Put-Call Parity
Price of underlying equity + Price of put = Price of call + Present value of
exercise price
S + P = C + PV(E)
This relationship between the value of the stock, the bond, and call can put options is
known as put-call parity.
It holds only if the put and the call have the same exercise price and the same
expiration date. The maturity date of the zero-coupon bond must be the same as
the expiration date of the options.
,Week 2: Option Valuation
1. The Binomial Option Pricing Model:
A technique for pricing options based on the assumption that each period, the stock’s
return can take on only two values.
Step 1: Draw Binomial Tree
A two-state single-period model:
Replicating portfolio: a portfolio consisting of a stock and a risk-free bond
that has the same value and payoffs in one period as an option written on the
same stock.
Example:
If share price rise, the value of option is 10.
If share price fall, the value of option is 0.
Binomial Tree:
Period 0 Period 1
Stock. Bond
Stock 50
Bond 1
Be the number of shares of stock, B be the initial investment in bonds.
Short position in bonds (borrowing money)
How much (costs) for you to replicate this portfolio?
50*0.5-18.8679= 6.13
Value of call option = the costs of replicating portfolio = 6.13
Good thing: do not require probabilities of the states in the binomial tree (the
probabilities of future states are part of investor beliefs and are very difficult to
, estimate)
2. Risk Neutral Probabilities
If all market participants are risk neutral, then all financial assets (including
options) have the same cost of capital, the risk-free rate of interest.
Calculate the probability of a rise and a fall under the assumption of risk
neutrality.
If ρ is the probability that the stock price will increase, then (1- ρ) is the
probability that it will go down.
The value of the stock today $50 must equal the present value of the expected price
next period discounted at the risk-free rate
Value of the call =
3. Making the Model Realistic: Black-Scholes Option Pricing Model
Shorten the period as much as possible (decrease the length of each period)
There are only two possibilities for the share price over the next infinitesimal instant.
A specific combination of equity and bond can indeed duplicate a call option over an
infinitesimal time horizon.
The only parameter in the Black-Scholes formula that we need to forecast is the
stock’s volatility.
A derivative is a financial instrument whose value depends upon the value of other,
more basic, underlying variables.
Some common types of derivatives: Options, Forwards, Futures, and Swaps
Participants in Derivatives Market
Hedger – when a firm uses derivatives to reduce the risk of its future cash flow.
- A hedger would normally have a risk exposure to the underlying asset.
Speculator- when derivatives are used with the sole aim of making a profit on how an
asset’s value will change in the future.
- A speculator may have no risk exposure to the underlying asset.
Arbitrageur- arbitrage involves locking in a riskless profit by simultaneously
entering into transactions in two or more markets.
- Transaction costs are very low for large investment banks.
Week 1: Financial Options
A financial option is a contract that gives its owner the right, but not the obligation, to
buy or sell an underlying asset at a fixed price, on or before a specified date in the
future.
Call option: A call option gives the holder the right to buy an asset.
Put option: A put option gives the holder the right to sell an asset.
An option is a contract between two parties, for every holder of a financial option,
there is also an option writer.
The seller (writer) of the option, however, has an obligation to honour the contract
if the holder decides to act; The holder of the option pays a premium to the writer
of the option.
Options can be OTC or exchange traded.
American options: the most common kind, options that allow their holder to exercise
the option on any date up to and including the expiration date.
European options: options that allow their holders to exercise the option only on the
expiration date.
Put-Call Parity
Price of underlying equity + Price of put = Price of call + Present value of
exercise price
S + P = C + PV(E)
This relationship between the value of the stock, the bond, and call can put options is
known as put-call parity.
It holds only if the put and the call have the same exercise price and the same
expiration date. The maturity date of the zero-coupon bond must be the same as
the expiration date of the options.
,Week 2: Option Valuation
1. The Binomial Option Pricing Model:
A technique for pricing options based on the assumption that each period, the stock’s
return can take on only two values.
Step 1: Draw Binomial Tree
A two-state single-period model:
Replicating portfolio: a portfolio consisting of a stock and a risk-free bond
that has the same value and payoffs in one period as an option written on the
same stock.
Example:
If share price rise, the value of option is 10.
If share price fall, the value of option is 0.
Binomial Tree:
Period 0 Period 1
Stock. Bond
Stock 50
Bond 1
Be the number of shares of stock, B be the initial investment in bonds.
Short position in bonds (borrowing money)
How much (costs) for you to replicate this portfolio?
50*0.5-18.8679= 6.13
Value of call option = the costs of replicating portfolio = 6.13
Good thing: do not require probabilities of the states in the binomial tree (the
probabilities of future states are part of investor beliefs and are very difficult to
, estimate)
2. Risk Neutral Probabilities
If all market participants are risk neutral, then all financial assets (including
options) have the same cost of capital, the risk-free rate of interest.
Calculate the probability of a rise and a fall under the assumption of risk
neutrality.
If ρ is the probability that the stock price will increase, then (1- ρ) is the
probability that it will go down.
The value of the stock today $50 must equal the present value of the expected price
next period discounted at the risk-free rate
Value of the call =
3. Making the Model Realistic: Black-Scholes Option Pricing Model
Shorten the period as much as possible (decrease the length of each period)
There are only two possibilities for the share price over the next infinitesimal instant.
A specific combination of equity and bond can indeed duplicate a call option over an
infinitesimal time horizon.
The only parameter in the Black-Scholes formula that we need to forecast is the
stock’s volatility.
