BASICS OF DERIVATIVE
PRICING AND VALUATION
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BASIC DERIVATIVE CONCEPTS
A derivative is a financial instrument that derives its performance
from the performance of an underlying asset. The two principal
types of derivatives are forward commitments and contingent
claims.
Forward Commitments
Contingent Claims
A forward commitment is an
obligation to engage in a
transaction in the spot market at a
future date at terms agreed upon
today. There are three types of
forward commitments:
A contingent claim is a derivative in
which the outcome or payoff is
determined by the outcome or payoff
of an underlying asset, conditional
on some event occurring. Contingent
claims include:
• Forwards
• Options
• Futures
• Credit Derivatives
• Swaps
• Asset-Backed Securities
PRICING THE UNDERLYING ASSET
The four main types of underlying assets on which
derivatives are based include:
• Equities
• Fixed-income securities
• Currencies
• Commodities
The price of a financial asset is often determined using a
present value of future cash flows approach. Determining a
rate at which to discount the expected future cash flows is
challenging.
PRICING EQUATIONS FOR SPOT ASSETS
The value of an asset today, S0, is the present value of the
expected future price of an asset with no interim cash flows,
E(ST), discounted at r (the risk-free rate) plus  (the risk
premium) over the period from 0 to T.
𝐸 𝑆𝑇
𝑆0 =
1+𝑟+λ 𝑇
If the asset has interim cash flows (benefits such as
dividends or convenience yield) or costs (such as storage),
we must adjust the pricing equation as follows:
𝐸 𝑆𝑇
𝑆0 =
−θ+γ
𝑇
1+𝑟+λ
Where γ: is the present value of any benefits and
θ: is the present value of any costs
EXECUTING AN ARBITRAGE
The “law of one price” states that assets that produce
identical results must sell for the same price. If the prices
differ, traders will engage in arbitrage until the prices
converge.
HEDGING THE UNDERLYING WITH A
DERIVATIVE
The “law of one price” states that assets that produce
identical results must sell for the same price. If the prices
differ, traders will engage in arbitrage until the prices
converge.
ARBITRAGE AND REPLICATION
Asset
Derivative
Risk-free asset
Asset
Risk-free asset
–Derivative
Derivative
Risk-free asset
–Asset
RISK-NEUTRAL PRICING
Risk aversion: The concept that investors do not accept
risk without the expectation of a return commensurate with
that risk.
Risk-neutral pricing: Arbitrage guarantees that a risk-free
portfolio consisting of the underlying and the derivative must
earn the risk-free rate.
Arbitrage-free pricing: The process of pricing a derivative
under the assumption of risk-neutral pricing and a market
free of arbitrage opportunities
PRICING AND VALUATION OF FORWARD
CONTRACTS
F0(T): The forward price established at the initiation date of
contract.
VT(T): The value at expiration of the forward contract.
At expiration (Time = T), the value of a forward contract is
VT(T) = ST – F0(T)
When a forward contract is initiated (Time = 0), it is
valueless to both the long and short positions.
V0(T) = 0
PRICING AND VALUATION OF FORWARD
CONTRACTS
F0(T): The forward price established at the initiation date of
contract.
S0(T): The initial price of the underlying asset.
T: The time to expiration of the forward contract.
r: The risk-free rate of interest.
Forward Pricing Equation:
F0(T) = S0(1 + r)T
FORWARD PRICING EXAMPLE
Assume an asset sells in the spot market for a price of $94,
the risk-free rate is 4%, and the forward contract expires in
six months. What is the initial value of the contract and the
correct forward price?
S0(T) = $94
The value of the contract at initiation is V0(T) = 0
T = 6 months = 0.5 years
Forward Pricing Equation
F0(T) = S0(1 + r)T = $94(1 + 0.04)0.5 = $95.86
The forward price established at the initiation date of the
contract is $95.86
VALUATION OF FORWARD CONTRACTS WITH
COSTS AND BENEFITS
Forward Pricing Equation
F0(T) = (S0 – γ + θ)(1 + r)T
Where:
F0(T): The forward price established at the initiation date of contract
S0(T): The initial price of the underlying asset
T: The time to expiration of the forward contract
r: The risk-free rate of interest
θ: Costs to hold the spot asset:
Monetary costs include storage, insurance
γ: Benefits to hold the spot asset:
Monetary benefits include dividends, interest
Nonmonetary benefits include convenience yield
FORWARD PRICING EXAMPLE WITH COSTS
AND BENEFITS
Assume an asset sells in the spot market for a price of $94.
The risk-free rate is 4%, and the forward contract on the asset
expires in six months. If the asset pays a dividend of $2.25 and
storage costs are $1 a year, what is the correct forward price?
S0(T) = $94
T = 6 months = 0.5 years
r = 4%
θ = $1 (storage costs)
γ = $2.25 (dividend)
Forward Price with carrying costs and benefits:
F0(T) = (S0 – γ + θ) (1 + r)T = ($94 – $2.25+$1)(1.04)0.5
= $94.59
FORWARD RATE AGREEMENTS
Forward Rate Agreements (FRAs): Forward contracts in
which the underlying is an interest rate. FRAs allow
participants to make a known interest payment at a later
date and receive in return an unknown interest payment.
Uses of an FRA:
• For borrowers: Someone borrowing at a floating rate
(such as Libor) can effectively lock in a fixed payment. By
entering into an FRA, they may receive a payment that
offsets the unknown Libor payment they will make on their
loan.
• For lenders: A lender can enter into an FRA to effectively
convert a floating rate loan exposure it will have at a future
date to a fixed rate.
FORWARD VS. FUTURES PRICES
Futures contracts are marked to market daily. Accumulated
futures gains and losses are built up in the margin account.
- Futures gains can be invested and collect interest.
- Futures losses must be borrowed and pay interest.
Forward contracts’ gains and losses are settled at expiration.
If interest rates are positively correlated with futures prices:
• futures prices > forward prices
If interest rates are uncorrelated with futures prices:
• futures and forwards prices will be the same
If interest rates are negatively correlated with futures prices:
• forward prices > futures prices
A SWAP AS A SERIES OF FORWARD
CONTRACTS
PRICING AND VALUATION OF SWAPS
Pricing a swap appeals to the principal of arbitrage.
• A swap is a series of forwards.
• To price a swap, find a single price FS0(T) such that the
value of all the forward contracts sum to zero.
• The initial value of the swap will be 0.
Valuation of the swap during its life again appeals to
replication and the principle of no arbitrage.
• While the value of the swap is 0 initially, it may change
during its lifetime.
• To value a swap during its life, we sum up the values of the
forward contracts.
VALUING A EUROPEAN CALL OPTION AT
EXPIRATION
The value of a European call at expiration is the exercise value, which is the
greater of zero or the value of the underlying minus the exercise price.
Let X be the exercise or strike price of the call.
If the underlying asset has a value at expiration of ST :
The value of the call option at expiration is:
cT = Max(0,ST – X)
Call Example 1: ST > X. If the strike price of a call option is X = $20 and the
asset price at expiration ST = $25.50, then the call value at expiration is:
cT = Max(0,25.50 – 20) = $5.50
Call Example 2: ST < X. If the strike price of a call option is X = $20 and the
asset price at expiration ST = $18.75, then the option will have no expiration
value:
cT = Max(0,18.75 – 20) = $0
VALUING A EUROPEAN PUT AT EXPIRATION
The value of a European put at expiration is the exercise value, which is the
greater of zero or the exercise price minus the value of the underlying.
Let X be the exercise or strike price of the put.
If the underlying asset has a value at expiration of ST:
The value of the put option at expiration is:
pT = Max(0,X – ST)
Put Example 1: ST < X. If the strike price of a put option is X = £100 and the
asset price at expiration ST = £90, then the put will have a value at
expiration of:
pT = Max(0,100 – 90) = £10
Put Example 2: ST > X. If the strike price of a put option is X = £35 and the
asset price at expiration ST = £40, then the put will have no value at
expiration:
pT = Max(0,35 – 40) = £0
MONEYNESS
Moneyness describes the relationship of the underlying
price to the strike price.
An option can be described as being:
• In-the-money: When the underlying is beyond the exercise
price in the appropriate direction (higher for a call, lower for
a put), the option is said to be in-the-money.
• At-the-money: When the underlying is precisely at the
exercise price, the option is said to be at-the-money.
• Out-of-the-money: When the underlying has not reached
the exercise price (lower for a call, higher for a put), the
option is said to be out-of-the-money.
FACTORS THAT DETERMINE THE VALUE OF A
EUROPEAN OPTION
•
The value of the underlying asset:
- The value of a European call option is directly related to
the value of the underlying.
- The value of a European put option is inversely related to
the value of the underlying.
• The exercise price:
- The value of a European call option is inversely related to
the exercise price.
- The value of a European put option is directly related to
the exercise price.
FACTORS THAT DETERMINE THE VALUE OF A
EUROPEAN OPTION CONTINUED
•
Time to expiration:
- The value of a European call option is directly related to the
time to expiration.
- The value of a European put option can be either directly or
inversely related to the time to expiration. The direct effect is
more common.
• The risk-free rate of interest:
- The value of a European call is directly related to the risk-free
interest rate.
- The value of a European put is inversely related to the risk-free
interest rate.
• The volatility of the underlying asset:
- The values of both the European call and the European put are
directly related to the volatility of the underlying.
PROTECTIVE PUT
FIDUCIARY CALL
PUT–CALL PARITY
PUT–CALL PARITY EQUATION
For European options on the same underlying asset with the
same strike price and expiration, the following equation must
hold:
𝑆0 + 𝑝0 = 𝑐0 + 𝑋Τ 1 + 𝑟 𝑇
Where
p0: European put price
c0: European call price
S0: current asset price
X: strike price
T: time to expiration
PUT–CALL PARITY EXAMPLE
Consider a non-dividend-paying stock with a current price of
$25/share. A 3-month European call with a strike price of
$24 is selling for $3. If the risk-free rate is 2%, what is the
correct price of the European put?
𝑆0 + 𝑝0 = 𝑐0 + 𝑋Τ 1 + 𝑟
𝑇
$25 + 𝑝0 = 3 + $24Τ 1 + 0.02
𝑝0 = $3 + $23.88 − $25
𝑝0 = $1.88
0.25
PUT–CALL FORWARD PARITY
PUT–CALL FORWARD PARITY
It follows that the cost of the fiduciary call must equal the
cost of the synthetic protective put, giving us what is referred
to as put–call–forward parity:
𝐹0 𝑇 Τ 1 + 𝑟 𝑇 + 𝑝0 = 𝑐0 + 𝑋Τ 1 + 𝑟 𝑇
BINOMIAL OPTION VALUATION
BINOMIAL OPTION VALUATION
A hedge portfolio can be created by taking a long position in h units
of the underlying asset and a short position in the call so that the
initial value of the portfolio is given by:
V0 = hS0 – c0
If the asset price jumps up at time 1, the hedge portfolio will be
worth
𝑉1+ = ℎ𝑆1+ − 𝑐1+
If the asset price jumps down at time 1, the hedge portfolio will be
worth
𝑉1− = ℎ𝑆1− − 𝑐1−
If we set the up and down up portfolios equal, we can solve for the
hedge ratio (h):
𝑐1+ − 𝑐1−
ℎ= +
𝑆1 − 𝑆1−
BINOMIAL OPTION VALUATION CONTINUED
Using the idea that a hedged portfolio will return the risk-free
rate, we can solve for the initial value of the option. The
result shows the value of a call today is a weighted average
of the next two possible call prices at expiration, as show in
the following equation.
π𝑐1+ + 1 − π 𝑐1−
𝑐0 =
1+𝑟
where the weights π and (1 – π) are given by
1+𝑟−𝑑
π=
𝑢−𝑑
The same approach can be used to value a put.
BINOMIAL OPTION VALUATION EXAMPLE
Let the initial stock price be $100 and the risk-free rate is 2%.
Assume the asset price can jump up 10% or down 5%, so the up
jump and down jump factors are u = 1.1 and d = 0.95. Price a call
option with X = $99 using a single period binomial.
𝑐1+ = Max(0,$110 – $99) = $11
𝑐1− = Max(0,$95 – $99) = $0
1 + 𝑟 − 𝑑 1 + 0.02 − 0.95 0.07
π=
=
=
= 0.467
𝑢−𝑑
1.1 − 0.95
0.15
The call value at time 0 is:
0.467 $11 + 1 − 0.467 $0
𝑐0 =
= $5.03
1.02
AMERICAN OPTION PRICING
American options cannot sell for less than European options.
Thus, we can state the following (using upper case letters for
American options and lower case letters for European
options):
𝐶0 ≥ 𝑐0
𝑃0 ≥ 𝑝0
The minimum value for an American call option is the same as
the minimum for the European call option:
𝐶0 ≥ max 0, 𝑆0 − 𝑋Τ 1 + 𝑟 𝑇
The minimum value for an American put option is:
𝑃0 ≥ max 0, 𝑋 − 𝑆0
AMERICAN OPTION PRICING EXAMPLES
Call Minimum Value Example:
Given that the risk-free rate is 5% and the current stock price
is $30, the minimum value of a 3-month American call option
with exercise price of $25 is:
𝐶0 ≥ max 0, 𝑆0 − 𝑋Τ 1 + 𝑟
𝑇
≥ max 0, $30 − $25Τ 1 + 0.05
0.25
≥ $5.30
Put Minimum Value Example:
Given that the current stock price is $30, the minimum value
of an American put option with exercise price of $40 is:
𝑃0 ≥ max 0, 𝑋 − 𝑆0 ≥ max 0, $40 − $30 ≥ $10
EARLY EXERCISE
• An American call on a non-dividend-paying stock
will never be exercised early.
• Under certain conditions, an American call on a
dividend-paying stock may exercised just prior to
the ex-dividend date.
• There is a critical point at which an American put
is so deep in-the-money that early exercise is
justified.
ARBITRAGE SUMMARY
• The price of the underlying asset is equal to the expected future
price discounted at the risk-free rate, plus a risk premium, plus
the present value of any benefits, minus the present value of
any costs associated with holding the asset.
• An arbitrage opportunity occurs when two identical assets or
combinations of assets (with the same payoffs) sell at different
prices, leading to the possibility of buying the cheaper asset and
selling the more expensive asset to produce a risk-free return
without investing any capital.
• In well-functioning markets, arbitrage opportunities are quickly
exploited, and the resulting increased buying of underpriced
assets and increased selling of overpriced assets returns prices
to equivalence.
DERIVATIVE PRICING SUMMARY
• Derivatives are priced by creating a risk-free combination
of the underlying and a derivative, leading to a unique
derivative price that eliminates any possibility of arbitrage.
• Derivative pricing through arbitrage precludes any need for
determining risk premiums or the risk aversion of the party
trading the option and is referred to as risk-neutral pricing.
FORWARD AND FUTURES PRICING SUMMARY
• The value of a forward contract at expiration is the value of the asset
minus the forward price.
• The value of a forward contract prior to expiration is the value of the
asset minus the present value of the forward price.
• The forward price, established when the contract is initiated, is the
price agreed to by the two parties that produces a zero value at the
start.
• Costs incurred and benefits received by holding the underlying affect
the forward price by raising and lowering it, respectively.
• Futures prices can differ from forward prices because of the effect of
interest rates on the interim cash flows from the daily settlement of
futures.
SWAPS PRICING SUMMARY
• Swaps can be priced as an implicit series of off-market
forward contracts.
• Each contract is priced the same, resulting in some
contracts being positively valued and some being
negatively valued.
• However, their combined values will equal zero.
AMERICAN VS. EUROPEAN OPTIONS
• At expiration, a European call or put is worth its exercise value, which
for calls is the greater of zero or the underlying price minus the
exercise price, and for puts is the greater of zero or the exercise price
minus the underlying price.
• European calls and puts are affected by the value of the underlying, the
exercise price, the risk-free rate, the time to expiration, the volatility of
the underlying, and any costs incurred or benefits received while
holding the underlying.
• Option values experience time value decay, which is the loss in value
due to the passage of time (approach of expiration), plus the
moneyness and the level of volatility.
• The minimum value of a European call is the maximum of zero or the
underlying price minus the present value of the exercise price.
• The minimum value of a European put is the maximum of zero or the
present value of the exercise price minus the price of the underlying.
AMERICAN AND EUROPEAN OPTION PRICING
• European put and call prices are related through put–call parity, which
specifies that the put price plus the price of the underlying equals the call price
plus the present value of the exercise price.
• European put and call prices are related through put–call–forward parity, which
shows that the put price plus the present value of a risk-free bond with face
value equal to the forward price equals the call price plus the present value of
the exercise price.
• The values of European options can be obtained using the binomial model,
which specifies two possible prices of the asset one period later and enables
the construction of a risk-free hedge consisting of the option and the
underlying.
• American call prices can differ from European call prices only if there are cash
flows on the underlying, such as dividends or interest; capturing these cash
flows are the only reason for early exercise of a call.
• American put prices can differ from European put prices, because the right to
exercise early always has value for a put, given that there is a lower limit on
the value of the underlying.