Valuation of Contingent Claims
Pricing and Valuation Methodology
Our approaches to pricing and valuation are based on the
assumption that prices adjust to not allow arbitrage profits.
The arbitrageur abides by two fundamental rules:
• Rule #1: Do not use your own money.
• Rule #2: Do not take any price risk.
Law of one price: two investments with the same or equivalent
future cash flows, regardless of what will happen in the future, will
sell for the same current price.
One-Period Binomial Option Valuation Model
At time 1, there are only two possible outcomes and two resulting
values of the underlying, S+ (up occurs) and S− (down occurs).
The return implied by the up jump (the up factor) u = S+/S. The
down factor is d = S−/S.
One-Period Binomial Values
Recall, the value of the call option at expiration is the greater of
zero or the value of the underlying minus the exercise price (X). If
the underlying asset value jumps up at expiration to a value S+, the
call will have a value of:
c+
= Max(0, S+
− X)
The down jump value of the call at expiration will be:
c−
= Max(0, S−
− X)
The value of the put option at expiration is the greater of zero or
the value of the exercise price minus the underlying.
If the underlying asset value jumps up at expiration to a value S+,
the put will have a value of:
p+
= Max(0, X − S+
)
The down jump value of the put at expiration will be:
p−
= Max(0, X − S−
)
One-Period Binomial Option Valuation I
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:
V = hS − c
If the asset price jumps up at time 1, the hedge portfolio will be
worth:
V +
= hS+
− c+
If the asset price jumps down at time 1, the hedge portfolio will be
worth:
V −
= hS−
− c−
If we set the up and down portfolios equal, we can solve for the
hedge ratio:
h =
c+ − c−
S+ − S−
One-Period Binomial Option Valuation II
Using the idea that a hedged portfolio will return the risk-free rate,
we can solve for the initial value of the call or put. The expectations
approach computes the option values as the present value of the
expected terminal option payoffs:
c =
πc+ + (1 − π)c−
1 + r
p =
πp+ + (1 − π)p−
1 + r
where the probability of an up move, π, is given by:
π =
1 + r − d
u − d
One-Period Binomial Option Valuation Example
Assume an initial stock price of £60 and a risk-free rate of 5%. Assume
the asset price can move up 15% or down 10%, so the up jump and down
jump factors are u = 1.15 and d = 0.90. Price a European call and put
option with X = £60 using a single period binomial model.
For the call:
c+
= Max(0, £69 − £60) = £9 and c−
= Max(0, £54 − £60) = £0
For the put:
p+
= Max(0, £60 − £69) = £0 and p−
= Max(0, £60 − £54) = £6
The call and put values are:
π =
1 + r − d
u − d
=
1 + 0.05 − 0.90
1.15 − 0.90
=
0.15
0.25
= 0.60
c =
πc+
+ (1 − π)c−
1 + r
=
0.6(£9) + 0.4(£0)
1.05
= £5.143
p =
πp+
+ (1 − π)p−
1 + r
=
0.6(£0) + 0.4(£0)
6
= £2.286
Two-Period Binomial Option Valuation Model
At time 2, there are only three possible resulting values of the
underlying: S++ (up jump occurs twice), S+− (resulting from an up
move then a down move, or a down move then an up move), or
S−− (down move occurs twice).
Two-Period Binomial Values
At time 2, the call will have three possible payoffs depending on the
number of up and down moves in S, as follows:
c++
= Max(0, S++
− X) = Max(0, u2
S − X)
c+−
= Max(0, S+−
− X) = Max(0, udS − X)
c−−
= Max(0, S−−
− X) = Max(0, d2
S − X)
The put will have 3 possible payoffs:
p++
= Max(0, X − S++
= Max(0, X − u2
S)
p+−
= Max(0, X − S+−
= Max(0, X − udS)
p−−
= Max(0, X − S−−
= Max(0, X − d2
S)
Two-Period Binomial Option Valuation
The two-period binomial model can also be represented as the
present value of an expectation of future cash flows.
Again is the probability of an up move, so the price of a European
call option using the two-period binomial model is:
c =
π2c++ + 2π(1 − π)c+− + (1 − π)2c−−
(1 + r)2
And the two-period European put value is:
p =
π2p++ + 2π(1 − π)p+− + (1 − π)2p−−
(1 + r)2
Two-Period Binomial Valuation Example I
Assume an initial stock price of £100 and a risk-free rate of 2% per
year. The asset price can jump up 20% or down 20%, so the up
jump and down jump factors are u = 1.20 and d = 0.80. Price a
call and put option with two years to maturity and X = £95 using a
two-period binomial model.
First, we must solve for the probability of an up move.
Based on the RN probability equation, we have:
π =
FV (1) − d
u − d
=
1 + r − d
u − d
=
1 + 0.02 − 0.80
1.20 − 0.80
=
0.22
0.4
= 0.55
Two-Period Binomial Valuation Example II
Next, we must solve for the call and put payoffs at the terminal nodes:
c++
= Max(0, u2
S − X) = Max(0, 1.22
100 − 95) = £40
c+−
= Max(0, udS − X) = Max(0, 1.2(0.8)100 − 95) = £1
c−−
= Max(0, d2
S − X) = Max(0, 0.82
100 − 95) = £0
The put will also have 3 possible payoffs:
p++
= Max(0, X − u2
S) = Max(0, 0.95 − 1.22
100) = £0
p+−
= Max(0, X − udS) = Max(0, 0.95 − 1.2(0.8)100) = £0
p−−
= Max(0, X − d2
S) = Max(0, 0.95 − 0.82
100) = £31
The price of a European call option using the two-period binomial model is:
c =
0.552
(49) + 2(0.55)(0.45)(1) + 0.452
(0)
1.022
= £14.723
And the two-period European put value is:
p =
0.552
(0) + 2(0.55)(0.45)(0) + 0.452
(31)
1.022
= £6.034
Two-Period Valuation for an American Option
American-style options may be more valuable than similar European
options due to the early exercise feature.
• The difference in value is called the early exercise premium.
• Early exercise premium = American option value - European
option value
• American puts may have an early exercise premium and must
be checked for early exercise.
• American calls will not have an early exercise premium unless
the underlying asset pays a dividend.
Two-Period Binomial Valuation for American Options
American puts must be checked for early exercise. American calls may be
exercised early if the underlying asset pays a dividend.
American-style put options cannot be valued simply as the present value of
the expected future option payouts. We must work backward through the
binomial tree and address whether early exercise is optimal at each step.
At time 1, the unexercised values will be:
p+
1 =
πp++
+ (1 − π)p+−
(1 + r)
p−
1 =
πp+−
+ (1 − π)p−−
(1 + r)
Next, each node must be checked for early exercise. For example, the
value at the down jump node will be the maximum of the unexercised and
exercised values:
p−
= Max(p−
1 , X − dS)
Two-Period American Put Example I
Consider the previous example, with
S = £100, X = £95, r = 2%, u = 1.20, and d = 0.80.
p++
= Max(0, X − u2
S) = Max(0, 0.95 − 1.22
100) = £0
p+−
= Max(0, X − udS) = Max(0, 0.95 − 1.2(0.8)100) = £0
p−−
= Max(0, X − d2
S) = Max(0, 0.95 − 0.82
100) = £31
At time 1, the unexercised values will be:
p+
1 =
πp++ + (1 − π)p+−
(1 + r)
=
0 + 0
1.02
= 0
p−
1 =
πp+− + (1 − π)p−−
(1 + r)
=
0 + (0.45)31
1.02
= 13.677
Two-Period American Put Example II
Next, each node must be checked for early exercise.
p−
= Max(p−
1 , X − dS) = Max(13.68, 95 − 80) = 15
Recall, the European put value was £6.034. The American put
value is:
p =
πp+ + (1 − π)p−
(1 + r)
=
(0.55)0 + (0.45)15
1.02
= £6.618
In this example, the American put is worth more than its European
counterpart and has an early exercise premium of
£6.618 − £6.034 = £0.584
Black-Scholes-Merton Assumptions
The Black-Scholes-Merton (BSM) option valuation model presents a
formula to price European puts and calls. Assumptions of the BSM model
include:
• The underlying follows a geometric Brownian motion statistical
process.
• Geometric Brownian motion implies continuous prices (no jumps).
• The underlying instrument is liquid (it can be easily bought and sold).
• Continuous trading is available (one must be able to trade at every
instant).
• Short selling of the underlying instrument is permitted.
• There are no market frictions: no transaction costs, regulatory
constraints, or taxes.
• No arbitrage opportunities are available in the marketplace.
• The options are European-style (early exercise is not allowed).
• The continuously compounded risk-free interest rate is known and
constant.
• The volatility of the return on the underlying is known and constant.
• If the underlying instrument pays a yield, it is expressed as a
continuous, known, and constant yield at an annualized rate.
Black-Scholes-Merton Option Valuation Model
The BSM model for options on a stock is as follows:
European call: c = SN(d1) − e−rT XN(d2)
European put: p = e−rT XN(−d2) − SN(−d1), where
d1 =
ln S
X + r + σ2
2 T
σ
√
T
and d2 = d1 − σ
√
T
• N(x) denotes the standard normal cumulative distribution
function.
• r: annualized continuously compounded risk-free rate
• σ: annualized constant volatility
• S: stock price
• X: strike price
• e: constant that is the base of the natural logarithm (≈ 2.718)
• T: time to expiration for the option (in years)
Interpretation of the BSM Model I
The BSM model for the European call is:
c = SN(d1) − e−rT
XN(d2)
• The BSM model can be described as the present value of the
expected option payoff at expiration. Specifically, we can
express the BSM model for calls as:
c = PVr [E(cT )] where E(cT ) = SerT
N(d1) − XN(d2)
• The BSM model can be described as having two components:
a stock component and a bond component.
• For call options, the stock component is SN(d1), and the bond
component is e−rT XN(d2). The BSM model call value is the
stock component minus the bond component.
Interpretation of the BSM Model II
The BSM model can be interpreted as a dynamically managed
portfolio of the stock and zero-coupon bonds.
For both call and put options, we can represent the initial cost of
this replicating strategy as:
Replicating strategy cost = nSS + nBB
Here, the equivalent number of underlying shares is:
nS = N(d1) > 0 for calls and nS = −N(−d1) < 0 for puts
The equivalent number of bonds is:
nB = −N(d2) < 0 for calls and nB = N(−d2) > 0 for puts
BSM Model Component Interpretation for a Call
Suppose we are given the following information on call and put options on
a stock: S = 52, X = 50, r = 3%, T = 1.0, and σ = 21%.
Thus, based on the BSM model, it can be demonstrated that:
PV (X) = 48.522, d1 = 0.435, d2 = 0.225, N(d1) = 0.668, N(d2) = 0.589
These inputs result in a call price of c = 6.167.
The no-arbitrage approach to replicating the call option involves:
Purchasing nS = N(d1) = 0.668 shares of stock partially financed with
nB = −N(d2) = −0.589 shares of zero-coupon bonds priced at
B = Xe−r
T = 48.522 per bond.
Note that by definition the cost of this replicating strategy is the BSM call
model value.
Replicating Strategy cost is:
nSS + nBB = 0.668(52) + (−0.589)48.522 = 6.156
BSM Valuation With Carry Benefits
Carry benefits include dividends for stock options, foreign interest
rates for currency options, and coupon payments for bond options.
We assume the carry benefit pays a continuous yield, γ.
The BSM model adjusted for carry benefits is:
• European call: c = Se−γT N(d1) − e−rT XN(d2)
• European put: p = e−rT XN(−d2) − Se−γT N(−d1) where
d1 =
ln S
X + r − γ + σ2
2 T
σ
√
T
and d2 = d1 − σ
√
T
For an underlying equity paying a dividend, the BSM model
adjusts the stock price by e−γT to reflect the continuous
payout.
BSM Valuation for Currencies
The BSM model can be used to value currency options. The carry benefit
for a foreign exchange option (γ = rf
) is the continuously compounded
foreign risk-free interest rate.
• European call: c = Se−rf
T
N(d1) − e−rT
XN(d2)
• European put: p = e−rT
XN(−d2) − Se−rf
T
N(−d1) where
d1 =
ln S
X + r − rf
+ σ2
2 T
σ
√
T
and d2 = d1 − σ
√
T
Input example: A US auto exporter is planning to receive GBP for his cars.
To protect against a fall in the GBP exchange rate, the exporter purchases
a 3-month put struck at X = 1.20$/£. The current exchange rate,
1.25$/£, is the underlying (S, the value of the domestic currency per unit
of the foreign currency). The annualized US risk-free rate is (r), and the
British rate is (rf
). The time to expiration (T) is 0.25 years. The standard
deviation of the log return of the spot exchange rate (σ) is the last input
required to value the put.
Black Option Valuation Model
Black introduced a modified version of the BSM model approach
that is applicable to options on underlying instruments that are
costless to carry, such as options on futures contracts.
• European call: c = e−rT [F0(T)N(d1) − XN(d2)]
• European put: p = e−rT XN(−d2) − F0(T)N(−d1) where
d1 =
ln F0T
X + σ2
2 T
σ
√
T
and d2 = d1 − σ
√
T
Note that F0(T) denotes the futures price at time 0 that expires at
time T, and σ denotes the volatility related to the futures price.
The other terms are as previously defined.
Black Option Valuation Example
The S&P 500 Index is at 2300, and a futures contract on it trades at 2293.11.
The exercise price is 2250, the continuously compounded risk-free rate is 0.7%,
time to expiration is 0.2 years for the option and futures contract, volatility is
20%, and the S&P dividend yield is 2.2%. Based on this information, the
following results are obtained for options on the futures contract:
Calls Puts
N(d1) = 0.601 N(−d1) = 0.399
N(d2) = 0.566 N(−d2) = 0.434
c = $104.253 p = $61.203
The underlying asset is the current futures price of 2293.11. So, the value of the
European call option on the futures contract will be:
c = e−rT
[F0(T)N(d1) − XN(d2)]
c = e−0.007(0.2)
[2293.11(0.601) − 2250(0.566) = 104.45
The European put option value will be:
p = e−0.007(0.2)
[2250(0.434) − 2293.11(0.399)] = 61.46
Option Greeks
The option Greeks tell us how much the BSM option value will
change for a small change in a particular parameter, holding
everything else constant.
The Option Greeks (for stock options) include:
Delta: The change in an option value for a given small change in
the value of the underlying stock.
Gamma: The change in a given option delta for a given small
change in the underlying stock’s value.
Theta: The change in a option value for a given small change in
calendar time.
Vega: The change in an option value for a given small change in
the volatility of the underlying stock.
Rho: The change in a given option value for a given small change
in the risk-free interest rate.
Delta Hedging I
Delta hedging an option is the process of establishing a position in the
underlying stock of a quantity that is prescribed by the option delta, so as
to have no exposure to very small moves up or down in the stock price.
The delta of the hedging instrument is denoted as DeltaH.
The optimal number of hedging units, NH, is
NH = −(Portfolio delta/DeltaH)
A delta-neutral portfolio will not change in value for small changes in the
stock instrument. Delta neutral implies the portfolio delta plus NHDeltaH
is equal to 0.
• Stock Delta = +1.0
• Call Delta: Deltac = e−δT
N(d1). So, 0 ≤ delta of a call ≤ 1.
• Put Delta: Deltap = −eδT
N(−d1). So, -1 ≤ delta of a put ≤ 0.
Delta Hedging II
Example: If a portfolio consists of 1,000 shares of stock, then the
portfolio delta = +1,000.
If a call option with DeltaH = +0.40 is used as the hedging
instrument for the portfolio, then to achieve a delta neutrality, we
must sell 2,500 calls: NH = −(1, 000/0.40) = −2, 500
Delta Hedging Example
Suppose we know S = 50, X = 52, r = 3%, T = 1.0, σ = 30%, and the
underlying stock does not pay a dividend. We have a short position of
10,000 shares of stock.
Based on this information, we note Deltac = 0.668, and Deltap = -0.332.
The optimal number of hedging units, NH, is
NH = −(Portfolio delta/DeltaH)
We further note the portfolio delta = -10,000.
Now, if we hedge using call options with DeltaH = +0.668 as the hedging
instrument:
We must buy 14,970 calls, since NH = −(−10, 000/ + 0.668) = +14, 970
But, if we hedge using put options with DeltaH = −0.332 as the hedging
instrument:
We must sell 30,120 puts, since NH = −(−10, 000/ − 0.332) = −30, 120
Gamma Risk I
Gamma: The change in a given option delta for a given small change in
the underlying stock’s value, holding everything else constant.
Option gamma is a measure of the curvature in the option price in
relationship to the stock price.
Gamma of a long or short position in one share of stock is zero because
the delta of a share of stock never changes.
Gamma for a call and put option are the same and can be expressed as:
Gammac = Gammap =
e−δT
Sσ
√
T
n(d1)
where n(d1) is the standard normal probability density function.
Gamma is always non-negative and takes on its largest value when an
option is near at the money.
Gamma measures the non-linearity risk or the risk that remains once the
portfolio is delta neutral.
Gamma Risk II
For very small changes in the stock, the delta approximation works
well:
ˆc = c + Deltac(ˆS − S)
A portfolio that is delta neutral may be hedged for small changes in
the stock price.
For fairly large changes in the stock price, the delta-plus-gamma
approximation is more accurate:
ˆc = c + Deltac(ˆS − S) +
Gammac
2
(ˆS − S)2
Stock prices often jump rather than move continuously and
smoothly, which creates “gamma risk.”
Gamma risk is so-called because gamma measures the risk of stock
prices jumping when hedging an option position and thus leaving a
previously hedged option position suddenly unhedged.
Theta, Vega, and Rho
Theta: The change in an option value for a given small change in
calendar time. Theta measures time decay, the rate at which option time
value declines as time passes (and expiration approaches). Theta is
negative for options; as calendar time passes, time to expiration declines
and the option loses value. Note that stocks have no expiration date, so
stock theta is zero.
Vega: The change in an option value for a given small change in the
volatility of the underlying stock. An increase in volatility results in an
increase in option value. Vega is always positive, and vega of a call equals
vega of a similar put. Note that vega is based on an unobservable
parameter, future volatility. Of the five BSM variables, an option’s value is
most sensitive to volatility changes.
Rho: The change in a given option value for a given small change in the
risk-free interest rate. Rho is positive (negative) for a call (put). This is
because buying a call avoids the financing costs involved with purchasing
the stock, while buying a put delays the chance to earn interest on
proceeds of the stock sale.
Implied Volatility
For options, their value is particularly sensitive to volatility.
Unlike the price of the underlying, however, volatility is not an
observable value in the marketplace.
Volatility can be, and often is, estimated based on a sample of
historical data.
Volatility can also be inferred from option prices. This inferred
volatility is called the implied volatility.
The key advantage in using implied volatility is that it provides
information regarding the market’s perception of uncertainty going
forward.
Implied Volatility in Option Trading
Implied volatility has several uses in options trading.
• Implied volatility can be interpreted as the market’s view of option
value.
• In the option markets, participants use volatility as the medium in
which to quote options. Rather than quote a call price as $10,
traders may quote the implied volatility as 25%, which prices the
option at $10.
• Implied volatility can be used to assess the relative value of different
options, neutralizing the effects of moneyness and time to expiration.
• A trader can decide which options are cheap (lower implied volatility)
or expensive (higher implied volatility) using implied volatility rather
than price.
• Implied volatility is useful for revaluing existing positions over time.
• Regulators, banks, compliance officers, and most option traders use
implied volatilities to communicate information related to options
portfolios.
Arbitrage Summary
The arbitrageur would rather have more money than less and abides
by two fundamental rules: Do not use your own money and do not
take any price risk.
The no-arbitrage approach is used for option valuation and is built
on the key concept of the law of one price, which says that if two
investments have the same future cash flows regardless of what
happens in the future, then these two investments should have the
same current price.
The following key assumptions are made:
• Replicating instruments are identifiable and investable.
• There are no market frictions.
• Short selling is allowed with full use of proceeds.
• The underlying instrument price follows a known distribution.
• Borrowing and lending is available at a known risk-free rate.
Valuation Summary
• The two-period binomial model can be viewed as three oneperiod
binomial models, one positioned at time 0, and two
positioned at time 1.
• In general, European-style options can be valued based on the
expectations approach in which the option value is determined
as the present value of the expected future option payouts,
where the discount rate is the risk-free rate and the expectation
is taken based on the risk-neutral probability measure.
• Both American-style options and European-style options can be
valued based on the no-arbitrage approach, which provides
clear interpretations of the component terms; the option value
is determined by working backward through the binomial tree
to arrive at the correct current value.
• Interest rate option valuation requires the specification of an
entire term structure of interest rates, so valuation is often
estimated via a binomial tree.
Black-Scholes-Merton Model Summary
A key assumption of the Black-Scholes-Merton option valuation
model is that the return of the underlying instrument follows
geometric Brownian motion, implying a lognormal distribution of
the return.
The BSM model can be interpreted as a dynamically managed
portfolio of the underlying instrument and zero-coupon bonds.
BSM model interpretations related to N(d1) are that it is the basis
for the number of units of underlying instrument required to
replicate an option, that it is the primary determinant of delta, and
that it answers the question of how much the option value will
change for a small change in the underlying.
BSM model interpretations related to N(d2) are that it is the basis
for the number of zero-coupon bonds required to replicate an
option, and for estimating the risk-neutral probability of an option
expiring in the money.
Black Model Summary
The Black futures option model assumes the underlying is a futures or a
forward contract.
Interest rate options can be valued based on a modified Black futures
option model in which the underlying is an FRA, there is an accrual period
adjustment as well as an underlying notional amount, and that care must
be given to day-count conventions.
An interest rate cap is a portfolio of interest rate call options termed
caplets, each with the same exercise rate and with sequential maturities.
An interest rate cap can be used to hedge a set of floating rate loan
payments.
An interest rate floor is a portfolio of interest rate put options termed
floorlets, each with the same exercise rate and with sequential maturities.
An interest rate floor can be used to hedge a floating rate bond investment
or a floating rate loan.
A swaption is an option on a swap. A payer swaption is an option on a
swap to pay fixed and receive floating. A receiver swaption is an option on
a swap to receive fixed and pay floating.
Delta Summary
Delta is a static risk measure defined as the change in a given
portfolio for a given small change in the value of the underlying
instrument, holding everything else constant.
Delta hedging refers to managing the portfolio delta by entering
additional positions into the portfolio.
A delta-neutral portfolio is one in which the portfolio delta is set
and maintained at zero.
A change in an option price can be estimated with a delta
approximation.
Because delta is used to make a linear approximation of the nonlinear
relationship that exists between the option price and the
underlying price, there is an error that can be estimated by gamma.
Other Greeks
Gamma is a static risk measure defined as the change in a given portfolio
delta for a given small change in the value of the underlying instrument,
holding everything else constant.
Gamma captures the non-linearity risk or the risk—via exposure to the
underlying—that remains once the portfolio is delta neutral. A gamma
neutral portfolio is one in which the portfolio gamma is maintained at zero.
The change in an option price can be better estimated by a delta-plusgamma
approximation compared with just a delta approximation.
Theta is a static risk measure defined as the change in the value of a
portfolio given a small change in calendar time, holding everything else
constant.
Vega is a static risk measure defined as the change in a given portfolio for
a given small change in volatility, holding everything else constant.
Rho is a static risk measure defined as the change in a given portfolio for
a given small change in the risk-free interest rate, holding everything else
constant.
Implied Volatility Summary
Although historical volatility can be estimated, there is no objective
measure of future volatility.
Implied volatility is the BSM model volatility that yields the market option
price.
Implied volatility is a measure of future volatility, whereas historical
volatility is a measure of past volatility.
Option prices reflect the beliefs of option market participants about the
future volatility of the underlying.
The volatility smile is a two dimensional plot of the implied volatility with
respect to the exercise price.
The volatility surface is a three-dimensional plot of the implied volatility
with respect to both expiration time and exercise prices.
If the BSM model assumptions were true, then one would expect to find
the volatility surface flat, but in practice, the volatility surface is not flat.