Derivative Securities
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Law of One Price
Payoff Diagrams for common derivatives
Valuation of Derivatives
Put-Call Parity
Derivative Securities
The Law of One Price
Two assets with identical cash flows in the future will have
the same current price in an arbitrage-free market
Sometimes the “assets” may be a portfolio, or combinations
of smaller assets in order to generate the similar cash flows
Payoff Diagram for
Forward Contracts
Suppose you know that on September 30th you will want to
sell a bushel of corn.
You have some alternatives:
1) Wait until 9/30 and take the then current price
2) Agree with another party on a price you both will honor
on 9/30 – this would take away some of the risk of the
price dropping drastically between now and 9/30
Payoff Diagram for
Forward Contracts
You enter into a forward contract with a cereal maker to sell
one bushel of corn to him at $15.30
What will be the gain or loss to you on 9/30?
Clearly dependent upon the price of a bushel of corn on
9/30…
Payoff Diagram for
Forward Contracts
Gain to you will be cash you receive for the bushel of corn
less the market value of the bushel of corn on 9/30
Example:
If corn is selling for $15.40 on 9/30, then you will have
received $15.30 from the cereal maker and given away an
asset worth $15.40
Your loss is $0.10 – Same as the cereal maker’s gain
Payoff Diagram for
Forward Contracts
Gain to you will be cash you receive for the bushel of corn
less the market value of the bushel of corn on 9/30
Example:
If corn is selling for $15.00 on 9/30, then you will have
received $15.30 from the cereal maker and given away an
asset worth $15.00
Your gain is $0.30 – Same as the cereal maker’s loss
Payoff Diagram for
Forward Contracts
Gain for agreeing to sell an asset through a forward contract
is (often called going “short” forward):
Cash Received – Current Value of Asset Sold
Gain for agreeing to buy an asset through a forward contract
is (often called going “long” forward):
Current Value of Asset Sold - Cash Paid
Notation
In notational terms, we can define:
St = Market Value of the underlying asset at time t
T = Time of maturity of futures
F(t, T) = Strike price of futures
Long Futures Payoff = ST - F(t, T)
Note that it is dependent on the Market Value at T
Forward Prices
We can use the Law of One Price to get at what the
theoretical strike price is for a forward contract…..
Consider Portfolio 1: Purchase the asset in the spot market
At time t, it is worth St
As we get closer to time T, there is a net cash benefit of
owning the asset, such as dividends received, less storage
costs, etc. Let these net benefits be D.
At time T, the portfolio is worth
ST + D
Forward Prices
We can use the Law of One Price to get at what the theoretical strike price
is for a forward contract…..
Consider Portfolio 2: Agree to pay F at time T, and also want to have a
total of D at time T
At time t, deposit (F+D) e-r(T-t) in an account to earn a risk free rate, r
As we get closer to time T, interest is earned on the account
At time T:
The account has grown to (F+D)
We pay F to acquire the asset
Our holdings are now worth ST + D
Forward Prices
We can use the Law of One Price to get at what the theoretical strike price
is for a forward contract…..
Portfolio 1 and Portfolio 2 are both have a payoff of ST + D at time T so
under the Law of One Price the two portfolios must be equal at time t
Implies
St = (F+D) e-r(T-t)
F = St er(T-t) – D
So the “correct theoretical” forward settlement price is the current spot
increased at risk free interest less the net benefit of owning the asset.
Market supply and demand, however, will assist in adjusting the
theoretical price to an actual price.
Sometime D is positive, sometimes negative, sometime negligible
For some assets, D is expressed as a rate of interest so that F = St e(r-d)(T-t)
Payoff Diagram for
Option Contracts
Call Option: The right to purchase an asset from another
party at a pre-determined price (K) on or before a point in
time (T)
Typically a price involved to purchase this right and is called
the call “premium” – denote this as c.
If market price of the asset at T is larger than K, then you will
exercise the call. You could buy the share for K,
immediately sell it for ST. Your payoff is ST – K - c
Payoff Diagram for
Option Contracts
Put Option: The right to sell an asset to another party at a
pre-determined price (K) on or before a point in time (T)
Typically a price involved to purchase this right and is called
the call “premium” – denote this as p.
If market price of the asset at T is less than K, then you will
exercise the put. You could buy the share for ST,
immediately sell it for K. Your payoff is K – ST - p
Payoff Diagram for
Option Contracts
Back to the Law of One Price…..
What is the payoff diagram for owning an asset?
Is there some way to combine puts, calls and asset to get
some similar pictures?
Payoff Diagram for
Option Contracts
We can use these ideas to create an equation called put-call
parity. If we know the prices of three of the following
four variables, we can solve for the remaining one.
1.
2.
3.
4.
p = Price of Put
c = Price of Call
S = Price of underlying asset
r = risk free rate of return
c – p = S – Ke-r(T-t)
Payoff Diagram for
Option Contracts
For put-call parity to hold, the following assumptions must
be made:
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No dividends payable on underlying asset
Borrowing and investing available at risk-free rate, r
No transaction fees or taxes
Short selling and borrowing allowed in any fractional
quantities
No arbitrage opportunities
Option Valuation
European versus American Options:
European options can be exercised only on the expiration date – no
interim right to exercise
American options can be exercised at any time up to and including the
expiration date
Most option valuation routines are derived assuming a European option as
an initial basis
- Easier to think about the defining event occurring at one point in
time
- Since American options have at least as many rights as European
options, the value of a European option can be thought to be a
minimum value for the value of an American option
Option Valuation
Mathematics of Option Valuation – probably one of the most
important areas of modern finance
Robert C. Merton and Myron S. Scholes won the 1997 Nobel
Prize in Economics for their work in this field
Unfortunately, Fischer Black died in 1995 before he could be
properly recognized
We’ll look a little bit of the math and then describe how you
can move from simple models to rigorous development
Option Valuation
Fundamental underlying concept is that the market value of a
European-style derivative contract based on an
underlying asset is equal to a combination of the market
values of the asset and a default-free bond
Essentially, values of derivatives can be replicated by forming
a portfolio with market value nS – B; the key is how
much n and how much B – that is what we’ll try to solve
for
Option Valuation
Perhaps a first easy example:
We’ll create a function defining the payoff of the derivative as f(s). So the
cash flow from the derivative would be f(ST) at maturity
Let f(ST) = ST
This is a derivative is the most simplest form.
To replicate the payoff of the derivatives, how much n and B do we need?
Clearly n = 1 and B = 0. Note that n and B can be negative if we need it
to be, implying selling the asset short or loaning risk-free amounts
instead of borrowing.
Option Valuation
In our simple model, let’s consider what would happen if one
of two ending states occur. Either…
The asset can appreciate at rate u and grow to Steu(T – t)
Or, the asset can depreciate at rate d and grow to Sted(T – t)
This is kind of a one-period Bernoulli trial
It has two possible outcomes and we could place probabilities
on the likelihood of each outcome
Option Valuation
Now let’s see what happens when we consider the two states in the
“derivative cash flow” function, f(s)
Let fu = f(Steu(T – t)) = cash flow in “up” state
Let fd = f(Sted(T – t)) = cash flow in “down” state
Then using our replicating portfolio, we can create two equations with two
unknowns:
fu = n(Steu(T – t)) – Ber(T-t)
fd = n(Sted(T – t)) – Ber(T-t)
asset will appreciate and loan will grow
asset will depreciate and loan will grow
Option Valuation
Solving for n and B:
nSt = (fu – fd) / (eu(T – t) - ed(T – t))
B = er(T – t) [(fu ed(T – t) - fd eu(T – t) ) / (eu(T – t) - ed(T – t))]
Option Valuation
Solving for n and B:
So….. If we use the most simple derivative payoff function,
f(ST) = ST
Then
nSt = (fu – fd) / (eu(T – t) - ed(T – t))
= [f(Steu(T – t)) – f(Sted(T – t))] / (eu(T – t) - ed(T – t))
= (Steu(T – t) – Sted(T – t)) / (eu(T – t) - ed(T – t))
= St (eu(T – t) – ed(T – t)) / (eu(T – t) - ed(T – t))
= St
This implies n = 1.
Option Valuation
Solving for n and B:
So….. If we use the most simple derivative payoff function,
f(ST) = ST
Then
B
= er(T – t) [(fu ed(T – t) - fd eu(T – t) ) / (eu(T – t) - ed(T – t))]
= er(T – t) [f(Steu(T – t)) ed(T – t) - f(Sted(T – t)) eu(T – t)] / (eu(T – t) - ed(T – t))
= er(T – t) [Steu(T – t) ed(T – t) - Sted(T – t) eu(T – t)] / (eu(T – t) - ed(T – t))
=0
Option Valuation
The General Result:
• When you know the payoff function of the derivative, you
can generate a replicating portfolio for the derivative in
the general form of nS – B
• If you write out “ns – B”, the value can generally be
thought of as a “discounted probability-weighted payoff”
nS – B = er(T – t) [ q fu + (1-q) fd ]
where q = (er(T – t) - ed(T – t) ) / (eu(T – t) - ed(T – t))
Option Valuation
So for a derivative contract we can imply that the value of the derivative is
er(T – t) [ q fu + (1-q) fd ]
For a specific derivative, say a call option, we can then factor in its
specific payoff function. Here, f(s) = (S – K) if S is high enough, or
f(s) = 0 if S is below the strike. We know that the strike, K, will have
to either fall below Sed(T – t), between Sed(T – t) and Seu(T – t), or above
Seu(T – t)
Situation
K > Seu(T – t)
Sed(T – t) < K < Seu(T – t)
K < Sed(T – t)
Payoff
0
er(T – t) [ q fu ]
er(T – t) [ q fu + (1-q) fd ]
Option Valuation
Rewriting the payoffs:
Situation
K > Seu(T – t)
Sed(T – t) < K < Seu(T – t)
K < Sed(T – t)
Payoff
0
er(T – t) q (Seu(T – t) – K)
er(T – t) [ q (Seu(T – t) – K)
+ (1-q) (Sed(T – t) – K) ]
The last payoff can also be written as:
er(T – t) [ q (Seu(T – t) – K) + (1-q) (Sed(T – t) – K) ]
er(T – t) [ q Seu(T – t) – qK + Sed(T – t) – K - q Sed(T – t) + qK]
er(T – t) [ q Seu(T – t) + Sed(T – t) – K - q Sed(T – t)]
er(T – t) [ q Seu(T – t) + (1-q) Sed(T – t) – K]
Option Valuation
Rewriting the payoffs:
Situation
K > Seu(T – t)
Sed(T – t) < K < Seu(T – t)
K < Sed(T – t)
Payoff
0
er(T – t) q (Seu(T – t) – K)
er(T – t) [ q Seu(T – t) + (1-q) Sed(T– t) – K]
From this, we can make some general conclusions for this “simplistic” value of a
call option:
The value is going to be dependent on the value of the underlying asset and some
sort of probability distribution of how that asset appreciates in value between
now and maturity
The value is going to be dependent on the present value of the strike price and
some sort of probability distribution of how the asset appreciates in value
between now and maturity
Option Valuation
Black and Scholes made some additional assumptions around this model and came
up with this valuation of the call option:
c = S N(d1) - K e-r(T-t) N(d2)
Where
d1 = [ ln (S/K) + (r + σ2 / 2) (T – t) ] / (σ
T t )
d2 = d1 - (σ T  t )
N(x) denotes the cumulative standard normal distribution function