Derivatives (Comm 4202)
Assignment #3 – Due March 6, 2020
1. It is March 5, 2020. A stock index is traded at a level of 3123, the risk-free rate
is 2.53% with continuous compounding, and the dividend yield is 3.08%. What
is the futures price for an index futures contract that expires on September 18,
2020?
2. The USD-CAD exchange rate is 1.3283, the 6-month Canadian risk-free rate is
1.73%, and the 6-month U.S. risk-free rate is 0.8%. Both risk-free rates are continuously compounded. From a U.S. investor’s perspective, what is the futures
price for a USD-CAD currency futures contract that expires in six months?
3. A stock is expected to pay a dividend of $1 per share in two months and in five
months. The stock price is currently traded at $50 per share, and the 2-month,
3-month, 5-month, and 6-month risk-free rates are 3%, 3.025%, 3.05%, and
3.10% per annum with continuous compounding, respectively.
(a) What is the futures price on the stock? What is the initial value of the
futures contract to a long position holder?
(b) If the price of the stock is $48 and the term structure of interest rates is
unchanged in three months, what will be the futures price and the value of
the short position in the futures contract?
4. A stock price is currently $40. Over each of the next two three-month periods it
is expected to go up by 10% or down by 10%. The risk-free interest rate is 5%
per annum with continuous compounding.
(a) What is the value of a six-month European call option with a strike price of
$42?
(b) What is the value of a six-month American put option with a strike price of
$42?
5. Consider an option on a stock when the stock price is $30, the exercise price is
$29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum,
and the time to maturity is four months.
1
(a) What is the price of the option if it is a European call or put based on the
Black-Scholes-Merton model?
(b) What is the price of the option if it is an American call or put based on a
two-period binomial model?
6. The price of a stock index futures contract is at 423.70, and the stock index level
is at 420.55. The three-month call option on the index with a strike price of 400
is traded at $26.25, and the three-month put option with the same expiration
is traded at $3.25. The three-month continuous compounding risk-free rate is
2.75%. Determine whether the futures and options are priced correctly in relation to each other. If they are not, construct a risk-free portfolio and show how
it will earn a rate better than the risk-free rate.
2
Option Valuation Models
Derivatives (Comm 4202)
Option Pricing Models
In the absence of arbitrage, the law of one price says any two financial securities having exactly the same payoffs in all possible future economic scenarios
must have the same price.
Especially, if a portfolio’s payoff (such as a risk-less bond) has no uncertainty,
its cost must be equal to the discounted value of the cash flows at the risk-free
rate.
1
A.
Binomial Models
A one-period binomial model
Suppose a stock is currently traded at S . It will either go up to Su or go down
to Sd with some probability law. The risk free rate is r with continuous compounding.
c Yonggan Zhao, Rowe School of Business, Dalhousie University
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Option Valuation Models
Derivatives (Comm 4202)
We want to evaluate a derivative security that will pay fu if the stock goes up
or fd if the stock goes down. Below is a diagram of the one-period prices (cash
flows) for the stock and the derivative security :
Our objective is to construct a riskless portfolio using the stock and the derivative security.
Suppose we hold ∆ shares of the stock and short the derivative security. The
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Option Valuation Models
Derivatives (Comm 4202)
portfolio’s payoff is
∆S − f ,
u
u
∆Sd − fd,
if the stock price goes up
(1)
if the stock price goes down
To make the portfolio risk-less, we just need set the payoffs equal to each other
across the possible states,
∆Su − fu = ∆Sd − fd.
That is,
∆=
fu − fd
.
(2)
Su − Sd
Hence, holding ∆ shares and shorting the derivative simultaneously yield a
riskless portfolio. The cost of this portfolio must be
∆S − f = (∆Su − fu)e−rT
(discounted at the risk free rate!)
where T is the time length in years to the expiration date. Hence,
f = ∆S − (∆Su − fu)e−rT
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(3)
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Option Valuation Models
Derivatives (Comm 4202)
Example 1.1. Suppose the current stock price is $20 and it will be either $22 or
$18 at the end of three months. Find the value of a call option to buy the stock
for $21 in three months. The risk free rate is 12%.
Solution:
S = 20, Su = 22, Sd = 18, r = 0.12, T = 3/12 = 0.25, K = 21. Option’s
payoff is
f
If the stock price goes up
u = max(22 − 21, 0) = 1,
fd = max(18 − 21, 0) = 0,
If the stock price goes down
Thus, by equation (2)
∆=
1−0
= 0.25.
22 − 18
and the option’s value (or price) is, by equation (3)
f = 0.25 ∗ 20 − (0.25 ∗ 22 − 1)e−0.12∗0.25 = 0.633
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Option Valuation Models
B.
Derivatives (Comm 4202)
The hedging portfolio
The replicating portfolio of the risk free asset is
∆ = 0.25 shares and a short position in the derivative security. It can be
deduced that the value of the derivative security can be replicated using the
stock and the risk-free security. To replicate the derivative security, one needs
to hold ∆ shares and a position in the risk free security:
B = −(∆S − f )
= −(∆Su − fu)e−rT
The portfolio composed of ∆ shares of the underlying asset and a loan of B
in the risk-free asset is termed the hedging portfolio of the derivative security.
The value of the derivative security can be represented as
f = ∆S + B.
∆ and B are called the hedging portfolio weights.
In Example 1.1, the hedging portfolio for the call option consists of ∆ = 0.25
shares and a position of B = −(0.25 ∗ 22 − 1)e−0.12∗0.25 = −4.3670 in the
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Option Valuation Models
Derivatives (Comm 4202)
risk-free asset. The value of the call option is then the cost of the hedging
portfolio:
f = ∆S + B = 0.25 ∗ 20 − 4.3670 = $0.6330.
C.
Risk-neutral valuation
It is surprising that the probabilities characterizing the likelihood of the market directional movement never appear in the pricing mechanism. Actually,
these important quantities are embedded in the dynamics of the underlying
stock prices.
We usually value a security on the bases of expectation of the cash flows.
Substitute the ∆ given by equation (2) into equation (3) and simplify,
f = e−rT [pfu + (1 − p)fd]
where
p=
erT − d
u−d
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(4)
.
6
Option Valuation Models
Derivatives (Comm 4202)
The quantity p must satisfy 0 ≤ p ≤ 1 in the absence of arbitrage, so p and
1 − p form a probability measure and are called risk neutral probabilities for
the two future market states, up and down.
With this definition, the valuation formula for the derivatives security given
by equation (4) can be interpreted as the expected payoff discounted at the
risk-free rate (Thrilling!!!).
Remark. The assumption of arbitrage free ensures
u ≥ erT ≥ d.
Verify!
Example 1.2. Suppose the current stock price is $100, and it is known that at
the end of three months it will be either up by 10% or down by 8%. The risk-free
rate is 5% per annum. Find the value of a put option to sell the stock for $105
in three months. Calculate the hedging portfolio weights.
Solution:
Su = 100 ∗ 1.1 = 110 and d = 100 ∗ 0.92 = 92. The risk neutral probability
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Option Valuation Models
Derivatives (Comm 4202)
for the upstate is
p=
100 ∗ e0.05∗0.25 − 92
110 − 92
So, the price of the put option is
= 0.5143
f = e−0.05∗0.25(0.5143∗max(105−110, 0)+0.4857∗max(105−92, 0)) = $6.24
The hedging portfolio for the put option consists of
∆ = max(105−110,0)−max(105−92,0) = −0.7222
110−92
B = −(∆Su − fu)e−rT = −(−0.7222 ∗ 110 − 0)e−0.05∗0.25 = $78.4552
which means shorting 0.7222 shares and investing $78.4552 at the risk free
rate.
We can now verify the valuation of the put option using the hedging portfolio
f = −0.7222 ∗ 100 + 78.46 = $6.24.
Note that, under the risk neutral probability measure, the expected returns on
both the underlying stock price and the option are equal to the risk free rate.
This is the fundamental principle of risk neutral valuation.
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Option Valuation Models
D.
Derivatives (Comm 4202)
A two-step binomial model
In the one-step model, the valuation approach works for both European and
American types of derivatives. However, the valuation process is very different
for multiple-period models.
Suppose a stock is currently traded at S . It will either go up to Su or go down to
Sd for the next two three-month periods. The risk free rate is r with continuous
compounding.
The diagram for the stock price dynamic:
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Option Valuation Models
Derivatives (Comm 4202)
The value of the derivative over time is calculated in the same way as for the
one-period model.
We first calculate the payoffs, fuu, fud, and fdd, of the derivative in period two,
using the contract specific parameters and terms.
We then find the values of the derivative, fu and fd, in period 1, using the risk
valuation approach:
f
−r12 (T −t)
[pfuu + (1 − p)fud]
u = e
fd = e−r12(T −t)[pfud + (1 − p)fdd]
where t stands for the time length between now and the end of period 1 and
r12 is the forward rate from period 1 to period 2.
Finally, we can calculate the value of the derivative:
f = e−r1t[pfu + (1 − p)fd],
where r1 is the spot rate for time t. A general formula for the two period pricing
model (only for European options not for American options) is
f = e−r2T [p2fuu + 2p(1 − p)fud + (1 − p)2fdd],
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Option Valuation Models
Derivatives (Comm 4202)
where r2 is the spot rate for time T . However, this formula cannot be used for
valuing American type of derivatives, because of early exercise opportunity.
Example 1.3. Suppose a stock is traded at $30 a share, and it will either increase by 15% or decrease by 10% for the next two three-month periods. The risk
free rate is 5% (flat term structure). Find the value of a European call option to
sell the stock for $31 in 6 months.
Solution:
S
u
Sd
= 30 ∗ 1.15 = 34.5
= 30 ∗ 0.9 = 27
Suu
Sud = Sdu
Sdd
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= 30 ∗ 1.152 = 39.675
= 30 ∗ 1.15 ∗ 0.9 = 31.05
= 30 ∗ 0.92 = 24.3
11
Option Valuation Models
Derivatives (Comm 4202)
The option’s payoffs in two periods are
fuu = max(39.675 − 31, 0) = 8.675
fud
fdd
= max(31.05 − 31, 0) = 0.05
= max(24.30 − 31, 0) = 0
The dynamics of the stock price and the option’s payoff are given below:
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12
Option Valuation Models
Derivatives (Comm 4202)
The risk neutral probability for the up move is
p=
e0.05∗0.25 − 0.9
= 0.4503.
1.15 − 0.9
Note that, if the term structure is NOT flat, the forward rate must be used for
valuation.
The values of the option for the up and down moves in period 1 are
f
−0.05∗0.25
[0.4503 ∗ 8.675 + 0.5497 ∗ 0.05] = 3.885
u = e
fd = e−0.05∗0.25[0.4503 ∗ 0.05 + 0.5497 ∗ 0] = 0.0222
The price of the option now is
f = e−0.05∗0.25[0.4503 ∗ 3.885 + 0.5497 ∗ 0.0222] = 1.7379
Using the general formula we also have
f = e−0.05∗0.5[0.45032∗8.675+2∗0.4503∗0.5497∗0.05+0.54972∗0] = 1.7379
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Option Valuation Models
2
Derivatives (Comm 4202)
Valuation of American Options
The valuation methodology introduced in the previous section works for European options, but it is not for American options due to early exercise opportunity.
However, in terms of valuation procedure, the only difference is that, at each
decision point, we need to see whether the value of exercising the option is
greater than that of holding it.
At a decision point, the value of the option is Vhold if it is not exercised, while
the cash flow received on exercise of the option is Vexercise. If Vexercise > Vhold,
we then exercise the option. Otherwise, we continue to hold the security. So,
the value of the option at this decision point is the greater of the two values.
As American calls have the same value as the European counterparts, we show
how to valuate an American put option.
Example 2.1. Suppose a stock is traded at $30 a share, and it will either increase by 15% or decrease by 10% for the next two three-month periods. The risk
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Option Valuation Models
Derivatives (Comm 4202)
free rate is 5%. Find the value of an American put option to sell the stock for
$31.5 in 6 months.
Solution: The model settings for the stock are the same as those in Example
1.3. The put option’s payoffs at expiration date are
fuu = max(31.5 − 39.675, 0) = 0
fud = max(31.5 − 31.05, 0) = 0.45
fdd = max(31.5 − 24.3, 0) = 7.2
The dynamics of the underlying stock price and the option’s payoff are as below
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15
Option Valuation Models
Derivatives (Comm 4202)
In period 1, if the put option is not exercised
f
−0.05∗0.25
[0.4503 ∗ 0 + 0.5497 ∗ 0.45] = 0.2443
u,hold = e
fd,hold = e−0.05∗0.25[0.4503 ∗ 0.45 + 0.5497 ∗ 7.2] = 4.1088
In period 1, if the put option is exercised
f
u,exercise = 31.5 − 34.5 = −3
fd,exercise = 31.5 − 27 = 4.5
Since fu,exercise < fu,hold, the put option is held at the up move in period 1.
Since fu,exercise > fu,hold, the option is exercised at the down move in period 1.
Hence
f
= max(fu,hold, fu,exercise) = 0.2443
fd
= max(fd,hold, fd,exercise) = 4.5
u
The price of the put option in the beginning is then, if not exercised,
fhold = e−0.05∗0.25 ∗ [0.4503 ∗ 0.2443 + 0.5497 ∗ 4.5] = 2.5516
If it is exercised in the beginning,
fexercise = max(31.5 − 30, 0) = 1.5
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16
Option Valuation Models
Derivatives (Comm 4202)
Thus, the option is not exercised now, and the price of the option is
f = max(fexercise, fhold) = 2.5516
Remark. An option is not exercised in the beginning, otherwise, there will be
an arbitrage opportunity.
The previous two-period valuation approach for European options does not give
the same solution:
e−0.05∗0.5[0.45032 ∗ 0 + 2 ∗ 0.4503 ∗ 0.5497 ∗ 0.45 + 0.54972 ∗ 7.2] = 2.3392
which is less than its American counterpart.
3
The Black-Scholes-Merton Model
The setting of the Black-Scholes-Model is quite different from the binomial
model. As observed, stock price changes very often with infinitely many possible outcomes – A continuous stochastic process is assumed. Within an arbitrarily small time intervalt to t + dt, stock returns are assumed to follow a
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17
Option Valuation Models
Derivatives (Comm 4202)
log-normal distribution
dSt
= µdt + σdZt
St
where St stands for price of the stock at time t and Zt being normally dis√
tributed with mean 0 and standard deviation σ t. µ and σ are called the
model drift and diffusion parameters. The stock price St can be written as
St =
1
(µ− σ 2 )t+σZt
S0 e 2
where S0 is the stock price at time now.
As a result, the expected stock return with continuous compounding, ln St/S0,
is normally distributed with mean
(µ − 12 σ 2)t
and standard deviation of
√
σ t.
σ is usually called the volatility. The expected price and the variance in time t
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18
Option Valuation Models
Derivatives (Comm 4202)
are, respectively,
E[St] = S0eµt
2
V [St] = S02e2µt(eσ t − 1)
Example 3.1. Suppose the current stock price is S0 = $10, µ = 0.25 and, σ =
0.4. What is expected return with continuous compounding and the volatility
of the stock price? What is the expected price and the variance of the stock in 6
months?
Solution:
The expected return (with continuous compounding) in 6 months
E[ln(ST /S0)] = (0.25 − 0.5 ∗ 0.42) ∗ 0.5 = 8.5%
and the standard deviation
√
V [ln(ST /S0)] = 0.4 0.5 = 11.31%
The expected price in 6 months is
E[ST ] = 10 ∗ e0.25∗0.5 = 11.3315
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Option Valuation Models
Derivatives (Comm 4202)
and the variance
2 2∗0.25∗0.5
V [ST ] = 10 e
A.
[e
0.42 ∗0.5
− 1] = 1.0694.
Basic concepts under the Black-Scholes-Merton model
• The option price and the stock price depend on the same underlying source
of uncertainty
• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty.
• The portfolio is instantaneously riskless and must instantaneously earn the
risk-free rate
B.
Risk-neutral valuation
As in the discrete model case, the probability is adjusted so that the expected
returns on the option and the underlying stock are equal to the risk free rate.
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Option Valuation Models
Derivatives (Comm 4202)
Thus, we need to adjust the parameter µ to the risk free rate, so that the stock
will earn the risk free rate with continuously compounding. The new price
dynamics under the risk neutral probability measure is
dSt
= r dt + σd Zt
St
and the price of a European option price with maturity T is
C = e−rT E[max(ST − K, 0)].
Let X = ln ST /S0, then X is normally distributed with mean (r − 12 σ 2)T and
variance σ 2T . Hence,
Z ∞
C = e−rT
max(S0ex − K, 0)φ(x)dx
−∞
where φ(x) is the normal density function:
φ(x) = √
1
2πT σ
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e−
1
x−(r− σ 2 )T
2
σ2 T
!2
.
21
Option Valuation Models
C.
Derivatives (Comm 4202)
The Black-Scholes-Merton formula
We just need to calculate the above integral to obtain the BSM pricing formula
for a European call option with a strike price K and T years to expire:
√
−rT
C = S0N (d) − Ke
N (d − σ t)
where
ln(S0/K) + (r + 21 σ 2)T
d=
√
σ T
and
Z x
2
1
− y2
e dy
N (x) = √
2π −∞
is the cumulative distribution function of the standard normal random variable.
Example 3.2. Suppose the risk free rate r = 0.08 and the stock price is traded
at $100 a share. The stock price follows
dSt
= 0.25 dt + 0.40 dZt.
St
Calculate a call option that can buy the stock for $100 in 6 months.
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Option Valuation Models
Derivatives (Comm 4202)
Solution:
ln(100/100) + (0.08 + 0.5 ∗ 0.42) ∗ 0.5
d=
= 0.282843
√
0.4 ∗ 0.5
We use Excel to find the cumulative distribution function for the standard normal random variable, normsdist(x),
√
N (0.282843) = 0.611351
and
N (0.282843 − 0.4 0.5) = 0.500000
Hence, the price of the call option is
c = 100 ∗ 0.611351 − 100 ∗ e−0.08∗0.5 ∗ 0.500000 = 13.095658.
D.
Hedging portfolios for European options
Similar to the binomial model, European options can be completely hedged
under BSM setting.
√
−rT
For calls, long N (d) shares of the stock and take a loan of Ke
N (d−σ T )
amount in the risk free asset.
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Option Valuation Models
Derivatives (Comm 4202)
−rT
For puts, short N (−d) shares and deposit Ke
free asset.
E.
√
N (−(d − σ t)) in the risk
Matching σ with u and d
The binomial model uses the up and down moves, u and d, while the BlackScholes-Merton model uses stock price volatility. How these quantities are
related?
The formulas are
u=e
√
σ ∆t
and
d = 1/u
where ∆t is the length of one time step on the tree.
With a dividend-style paying underlying asset at rate q , the risk-neutral probability is then represented as
p=
e
(r−q)∆t
√
σ
e ∆t
−e
−
√
−σ ∆t
√
−σ
∆t
e
√
The arbitrage-free condition guarantees −σ < (r − q) ∆t < σ .
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Option Valuation Models
Derivatives (Comm 4202)
In Example 3.2, r = 5%, q = 2%, ∆t = 0.25, then the stock volatility must
√
√
σ > |((r − q) ∆t| = |(0.05 − 0.02) 0.25| = 15%.
If σ = 30%,
u=e
√
0.30 0.25
and
= 1.1682
d = 1/1.1618 = 0.8607
The risk neutral probability for the up move is
p=
e
(0.05−0.02)∗0.25
√
0.3∗
0.25
e
−e
√
−0.3∗ 0.25
√
−0.3∗
0.25
e
= 0.4876
−
Example 3.3. Using a two-step binomial model, find the value of the call option
in Example 3.2.
Solution: As calculated above, the risk neutral probability for the up move is
p = 0.4876
The payoff of the call option in period 2 is
f uu = max(S0uu − 18, 0) = 9.2938
f ud = max(S0ud − 18, 0) = 2
f dd = max(S0dd − 18, 0) = 0
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Option Valuation Models
Derivatives (Comm 4202)
Thus
f
u
fd
= e−0.05∗0.25[0.4876 ∗ 9.2938 + 0.5124 ∗ 2] = 5.4874
= e−0.05∗0.25[0.4876 ∗ 2 + 0.5124 ∗ 0] = 0.9631
The price of the option is
c = e−0.05∗0.25[0.4876 ∗ 5.4874 + 0.5124 ∗ 0.9631] = 3.1298
Remark 1: Note the differences of the prices obtained using the binomial and
the BSM.
Remark 2: Sometimes matching with continuous compounding returns may
be more efficient.
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