1. science
2. mathematics
3. statistics

Option pricing/Leasing contract
The Binomial Option Pricing Model
(BOPM)
• option valuation
• We begin with a single period. Finding the risk neutral probability
• In a risk neutral world, all assets have risk free returns.
• Then, we combine single periods together to form the Multi-Period
Binomial Option Pricing Model.
• The Multi-Period Binomial Option Pricing Model is extremely flexible,
hence valuable; it can value American options (which can be exercised
early), and most, if not all, exotic options.
Assumptions of the BOPM
• There are two (and only two) possible prices for the underlying asset
on the next date. The underlying price will either:
– Increase by a factor of u% (an uptick)
– Decrease by a factor of d% (a downtick)
• The uncertainty is that we do not know which of the two prices will be
realized.
• No dividends.
• The one-period interest rate, r, is constant over the life of the option
(r% per period).
• Markets are perfect (no commissions, bid-ask spreads, taxes, price
pressure, etc.)
The Stock Pricing ‘Process’
Time T is the expiration day of a call option. Time T-1 is one period
prior to expiration.
ST,u
ST-1
ST,d
Suppose that ST-1 = 40, u = 25% and d = -10%.
ST,u = 50
40
ST,d = 36
The Option Pricing Process
CT,u = max(0, ST,u-K)
CT-1
CT,d = max(0, ST,d-K)
Suppose that K = 45. What are CT,u and CT,d?
CT,u = 5
CT-1
CT,d = 0
r d
p
ud
A Key Point
• If two assets offer the same payoffs at time T, then they must be priced
the same at time T-1.
• Here, we have set the problem up so that the equivalent portfolio offers
the same payoffs as the call.
• Hence the call’s value at time T-1 must equal the $ amount invested in
the equivalent portfolio.
CT-1 = ST-1 + B
So, in the Numerical Example….
ST-1 = 40, u = 25%, ST,u = 50, d = -10%, ST,d = 36, r = 5%, K = 45,
CT,u = 5 and CT,d = 0.
Finding the replicating portfolio of , B, and CT-1.
Att: Finding: THE RISK NEUTRALO PROBABILITY
p = (0.05 – (-0.1))/(0.25 – (-0.1)) = 0.15/0.35 = 0.428571429
(1-p) = 0.571428571
C = [(0.428571429)(5) + (0.571428571)(0)]/1.05 = 2.040816327
The two equations are
50  + 1.05B = 5
36  + 1.05B = 0
Solve, and
 = 0.357142857
B = -12.24489796
A Shortcut: discount method
C T 1
rd
ur
C T, u 
C T,d
u

d
u

d

(1  r)
or,
C T 1 
pC T, u  (1  p)C T,d
(1  r)
where,
rd
p
u d
In general:
C
(17 - 7)
and
pCu  (1  p)C d
(1  r)
(1  p) 
ur
u d
(17 - 8)
Interpreting p
r d
p
ud
• p is the probability of an uptick in a risk-neutral world.
• In a risk-neutral world, all assets (including the stock and the option)
will be priced to provide the same riskless rate of return, r.
• In our example, if p is the probability of an uptick then
ST-1 = [(0.428571429)(50) + (0.571428571)(36)]/1.05 = 40
• That is, the stock is priced to provide the same riskless rate of return as
the call option
The Equivalent Portfolio
Buy  shares of stock and borrow $B.
(1+u)ST-1 + (1+r)B =  ST,u + (1+r)B
 ST-1+B
(1+d)ST-1 + (1+r)B =  ST,d + (1+r)B
NB:  is not a
“change” in S…. It
defines the # of
shares to buy. For a
call, 0 <  < 1
Set the payoffs of the equivalent portfolio equal to CT,u and CT,d, respectively.
(1+u)ST-1 + (1+r)B = CT,u
(1+d)ST-1 + (1+r)B = CT,d
These are two equations with
two unknowns:  and B
What are the two equations in the numerical example with ST-1 = 40, u
= 25%, d = -10%, r = 5%, and K = 45?
 and B define the “Equivalent Portfolio” of a call
Δ
B
CT,u  CT, d
(u  d)S T 1

CT,u  CT, d
S T,u  S T, d
(1  u)CT, d  (1  d)CT,u
(u  d)(1  r)
; 0  Δ c  1 (17 - 1)
;
Bc  0 (17 - 2)
CT-1 = ST-1 + B
NB: a negative sign
now denotes borrowing!
(17-5)
Assume that the underlying asset can only rise by u% or decline by d%
in the next period. Then in general, at any time:
Δ
Cu  C d Cu  C d

(u  d)S Su  S d
(17 - 3)
B
(1  u)C d  (1  d)Cu
(u  d)(1  r)
(17 - 4)
C = S + B
(17-6)
Interpreting :
• Delta, , is the riskless hedge ratio; 0 < c < 1.
• Delta, , is the number of shares needed to hedge one call. I.e., if you
are long one call, you can hedge your risk by selling  shares of stock.
• Therefore, the number of calls to hedge one share is 1/. I.e., if you
own 100 shares of stock, then sell 1/ calls to hedge your position.
Equivalently, buy  shares of stock and write one call.
• Delta is the slope of the lines (where an option’s value is a function
of the price of the underlying asset).
• In continuous time,  = ∂C/∂S = the change in the value of a call
caused by a (small) change in the price of the underlying asset.
Two Period Binomial Model
ST,uu = (1+u)2ST-2
ST-1,u = (1+u)ST-2
ST,ud = (1+u)(1+d)ST-2
ST-2
ST-1,d = (1+d)ST-2
ST,dd = (1+d)2ST-2
CT,uu = max[0,(1+u)2ST-2 - K]
CT-1,u
CT-2
CT-1,d
CT,ud = max[0,(1+u)(1+d)ST-2 - K]
CT,dd = max[0,(1+d)2ST-2 - K]
Two Period Binomial Model: An
Example
ST,uu = 69.444
ST-1,u = 55.556
ST,ud = 50
ST-2 = 44.444
ST-1,d = 40.00
ST,dd = 36
CT,uu = _______
CT-1,u = ____
CT,ud = 5
CT-2
CT-1,d = 2.0408
CT,dd = 0
Two Period Binomial Model:
The Equivalent Portfolio
=1
B = -42.857143
 = 0.6851312
B = -24.1566014
T-2
 = 0.357142857
B = -12.24489796
T-1
Note that as S rises,  also rises. As S declines, so does .
Note that the equivalent portfolio is self financing. This means that the
cost of any purchase of shares (due to a rise in ) is accompanied by an
equivalent increase in required borrowing (B becomes more negative).
Any sale of shares (due to a decline in ) is accompanied by an
equivalent decrease in required borrowing (B becomes less negative).
The Multi-Period BOPM
• We can find binomial option prices for any number of
periods by using the following five steps:
(1) Build a price “tree” for the underlying.
(2) Calculate the possible option values in the last period
(time T = expiration date)
(3) Set up ALL possible riskless portfolios in the penultimate
period (next to last period).
(4) Calculate all possible option prices in the penultimate
period.
(5) Keep working back through the tree to “Today” (Time Tn in an n-period, (n+1)-date, model).
The ‘n’ Period Binomial Formula:
If n = 3:
C T 3 
p3CT,uuu  3p 2 (1  p)CT,uud  3p(1  p)2 CT,udd  (1  p)3 CT, ddd
(1  r)
3
(17 - 15)
The “binomial coefficient” computes the number of ways we can get j
upticks in n periods:
n
n!
  
 j  j! (n  j)!
Thus, the 3-period model can be written as:
C T 3
1

(1  r)3
3 j
 p (1  p)3 j max[0, (1  u) j (1  d)3 j ST 3  K].
j0  j 
3

The ‘n’ Period Binomial Formula:
In general, the n-period model is:
1
C
(1  r)n
n j
 p (1  p)n j [(1  u) j (1  d)n j ST n  K].
j a  j 
n

(17  17)
Where “a” in the summation is the minimum number of
up-ticks so that the call finishes in-the-money.
A Large Multi-period Lattice
Suppose that N = 100 days. Let u = 0.01 and d = -0.008. S0 = 50
135.241 = 50*(1.01^100)
132.830 = 50*(1.01^99)*(.992^1)
130.463 = 50*(1.01^98)*(.992^2)
51.51505
51.005
50.59696
50.50
50.00
50.096
49.69523
49.60
49.2032
48.80957
.
.
.
.
23.214 = 50*(1.01^2)*(.992^98)
22.801 = 50*(1.01^1)*(.992^99)
22.394 = 50*(.992^100)
T=0
T=1
T=2
T=3
T=100
Suppose the Number of Periods
Approachs Infinity
S
T
In the limit, that is, as N gets ‘large’, and if u and d are consistent
with generating a lognormal distribution for ST, then the BOPM
converges to the Black-Scholes Option Pricing Model (the
BSOPM is the subject of Chapter 18).
Stocks Paying a Dollar Dividend Amount
Figure 17.4: The stock trades ex-dividend
($1) at time T-2.
Figure 17.5: The stock trades ex-dividend
($1) at time T-1.
25.410
25.520
23.100
22 => 21
24.20 => 23.20
21.945
20.040
22.000
19.950
20.000
21.890
18.9525
21.780
20.000
20.90 => 19.90
18.905
19.800
19.000
18.755
19 => 18
18.810
18.05 => 17.05
17.100
16.1975
16.245
T-3
T-2
T-1
T
T-3
T-2
T-1
T
©David Dubofsky and 17-21
Thomas W. Miller, Jr.
American Calls on Dividend Paying
Stocks
• The key is that at each “node” of the lattice, the value of an American call
is:
 pCu  (1  p)C d

max 
, S  K .
(1  r)


(17  19)
If the first term in the brackets is less than the call’s intrinsic value,
then you must instead value it as equal to its intrinsic value. Moreover,
if the dividend amount paid in the next period exceeds K-PV(K), then
the American call should be exercised early at that node.
Binomial Put Pricing - I
PT,u = max(0,K-ST,u) = max(0,K-(1+u)ST-1)
ST,u = (1+u)ST-1
ST-1
PT-1
PT,d = max(0,K-ST,d) = max(0,K-(1+d)ST-1)
ST,d = (1+d)ST-1
(1+u)ST-1 + (1+r)B = ST,u + (1+r)B = PT,u
ST-1+B
(1+d)ST-1 + (1+r)B = ST,d + (1+r)B = PT,d
Binomial Put Pricing - II
• PT-1 = ST-1 + B
(17-24)
Where:
Δ
Pu  Pd
P  Pd
 u
(u  d)S Su  Sd
(17  22)
B
(1  u)Pd  (1  d)Pu
(u  d)(1  r)
(17  23)
-1 < p < 0
B>0
A put is can be replicated by selling  shares of stock short, and
lending $B.  and B change as time passes and as S changes.
Thus, the equivalent portfolio must be adjusted as time passes.
Binomial Put Pricing - III
pPu  (1  p)Pd
P 
(1  r)
(17  26)
Where:
p
r d
ud
and
(1  p) 
ur
ud
Binomial American Put Pricing

pP  (1  p)Pd 
P  max K  S, u

(1

r)


(17  27)
At any node, if the 2nd term in the brackets is less than the American
put’s intrinsic value, then value the put to equal its intrinsic value
instead. American puts cannot sell for less than their intrinsic value.
The American put will be exercised early at that node.
Binomial Put Pricing Example - I
79.86
The Stock
Pricing
Process:
u = 10%
d = -5%
r = 2%
K = 65
p = 0.466667
72.6
66
60
68.97
62.7
57
59.565
54.13
51.4425
T-3
T-2
T-1
T
Binomial Put Pricing Example - II
European Put Values:
0
0
1.485924
3.9776
0
2.84183
6.306976
5.435
9.57549
13.5575
T-3
T-2
T-1
T
Binomial Put Pricing Example - III
Composition of the
equivalent portfolio
to the European put:
Δ = 0.0
B = 0.0
Δ = -0.2870535
B = 20.431458
Δ = -0.5356724
Δ = -0.5778841
B = 36.117946
B = 39.075163
Δ = -0.7875626
B = 51.198042
Δ = -1.0
B = 63.72549
T-3
T-2
T-1
Binomial Put Pricing Example - IV
American put pricing: If
eqn. 17.25 yields an
amount less than the
put’s intrinsic value, then
the American’s put value
is K – S (shown in bold),
and it should be
exercised early.
0
0
1.485924
4.86284
0
2.84183
5
6.97339
5.435
8
9.57549
10
13.5575
T-3
T-2
T-1
T
1 Types of Leases
• The Basics
– A lease is a contractual agreement between a
lessee and lessor.
– The agreement establishes that the lessee has the
right to use an asset and in return must make
periodic payments to the lessor.
– The lessor is either the asset’s manufacturer or an
independent leasing company.
Operating Leases
• Usually not fully amortized. This means that the
payments required under the terms of the lease are
not enough to recover the full cost of the asset for
the lessor.
• Usually require the lessor to maintain and insure the
asset.
• Lessee enjoys a cancellation option. This option gives
the lessee the right to cancel the lease contract
before the expiration date.
Financial Leases
The exact opposite of an operating lease.
1. Do not provide for maintenance or service by
the lessor.
2. Financial leases are fully amortized.
3. The lessee usually has a right to renew the lease
at expiry.
4. Generally, financial leases cannot be cancelled,
i.e., the lessee must make all payments or face
the risk of bankruptcy.
Sale and Lease-Back
• A particular type of financial lease.
• Occurs when a company sells an asset it already
owns to another firm and immediately leases it from
them.
• Two sets of cash flows occur:
– The lessee receives cash today from the sale.
– The lessee agrees to make periodic lease
payments, thereby retaining the use of the asset.
Leveraged Leases
• A leveraged lease is another type of financial lease.
• A three-sided arrangement between the lessee, the
lessor, and lenders.
– The lessor owns the asset and for a fee allows the
lessee to use the asset.
– The lessor borrows to partially finance the asset.
– The lenders typically use a nonrecourse loan. This
means that the lessor is not obligated to the
lender in case of a default by the lessee.
2 Accounting and Leasing
• In the old days, leases led to off-balance-sheet
financing.
• In 1979, the Canadian Institute of Chartered
Accountants implemented new rules for lease
accounting according to which financial leases must
be “capitalized.”
• Capital leases appear on the balance sheet—the
present value of the lease payments appears on both
sides.
Accounting and Leasing
Balance Sheet
Truck is purchased with debt
Truck
$100,000
Land
$100,000
Total Assets
$200,000
Debt
Equity
Total Debt & Equity
$100,000
$100,000
$200,000
Operating Lease
Truck
Land
Total Assets
$100,000
$100,000
Debt
Equity
Total Debt & Equity
$100,000
$100,000
Capital Lease
Assets leased
Land
Total Assets
$100,000
$100,000
$200,000
Obligations under capital lease
Equity
Total Debt & Equity
$100,000
$100,000
$200,000
Lease form
38
Financial lease
The essential point of financial lease agreement is
that it contains a condition whereby the lessor
agrees to transfer the title for the asset at the end
of the lease period at a nominal cost.
At lease it must give an option to the lessee to
purchase the asset he has used at the expiry of the
lease. Under this lease the lessor recovers 90% of
the fair value of the asset as lease rentals and the
lease period is 75% of the economic life of
the asset.
39
Sale and lease back
40
Leveraged lease
41
Capital Lease
• A lease must be capitalized if any one of the
following is met:
– The present value of the lease payments is at least 90percent of the fair market value of the asset at the
start of the lease.
– The lease transfers ownership of the property to the
lessee by the end of the term of the lease.
– The lease term is 75-percent or more of the estimated
economic life of the asset.
– The lessee can buy the asset at a bargain price at
expiry.
3 Taxes and Leases
•
•
The principal benefit of long-term leasing is tax reduction.
Leasing allows the transfer of tax benefits from those who
need equipment but cannot take full advantage of the tax
benefits of ownership to a party who can.
• If the CCRA (Canada Customs and Revenue Agency)
detects one or more of the following, the lease will be
disallowed.
1. The lessee automatically acquires title to the property
after payment of a specified amount in the form of rentals.
2. The lessee is required to buy the property from the lessor.
3. The lessee has the right during the lease to acquire the
property at a price less than fair market value.
4 The Cash Flows of Leasing
Consider a firm, ClumZee Movers, that wishes to
acquire a delivery truck.
The truck is expected to reduce costs by $4,500 per
year.
The truck costs $25,000 and has a useful life of five
years.
If the firm buys the truck, they will depreciate it
straight-line to zero.
They can lease it for five years from Tiger Leasing with
an annual lease payment of $6,250.
4 The Cash Flows of Leasing
• Cash Flows: Buy
Cost of truck
After-tax savings
Depreciation Tax Shield
Year 0
–$25,000
Years 1-5
4,500×(1-.34) =
5,000×(.34) =
–$25,000
$2,970
$1,700
$4,670
• Cash Flows: Lease
Year 0
Lease Payments
After-tax savings
Years 1-5
–6,250×(1-.34) =
4,500×(1-.34) =
• Cash Flows: Leasing Instead of Buying
Year 0
$25,000
Years 1-5
–$1,155 – $4,670 = –$5,825
–$4,125
$2,970
–$1,155
4 The Cash Flows of Leasing
• Cash Flows: Leasing Instead of Buying
Year 0
$25,000
Years 1-5
–$1,155 – $4,670 = –$5,825
• Cash Flows: Buying Instead of Leasing
Year 0
–$25,000
Years 1-5
$4,670 –$1,155 = $5,825
• However we wish to conceptualize this, we need to
have an interest rate at which to discount the
future cash flows.
• That rate is the after-tax rate on the firm’s secured
debt.
5 Discounting and Debt Capacity with
Corporate Taxes
• Present Value of Riskless Cash Flows
– In a world with corporate taxes, firms should
discount riskless cash flows at the after-tax riskless
rate of interest.
• Optimal Debt Level and Riskless Cash Flows
– In a world with corporate taxes, one determines
the increase in the firm’s optimal debt level by
discounting a future guaranteed after-tax inflow at
the after-tax riskless interest rate.
6 NPV Analysis of the Lease-vs.-Buy
Decision
• A lease payment is like the debt service on a secured
bond issued by the lessee.
• In the real world, many companies discount both the
depreciation tax shields and the lease payments at the
after-tax interest rate on secured debt issued by the
lessee.
• The various tax shields could be riskier than lease
payments for two reasons:
1. The value of the CCA tax benefits depends on the
firm’s ability to generate enough taxable income.
2. The corporate tax rate may change.
NPV Analysis of the Lease-vs.-Buy Decision
•
There is a simple method for evaluating leases: discount all cash flows at the
after-tax interest rate on secured debt issued by the lessee. Suppose that
rate is 5-percent.
NPV Leasing Instead of Buying
Year 0
$25,000
Years 1-5
–$1,155 – $4,670 = -$5,825
5
$5,825
NPV  $25,000  
 $219.20
t
t 1 (1.05)
NPV Buying Instead of Leasing
Year 0
Years 1-5
-$25,000
$4,670 – $1,155 = $5,825
5
NPV  $25,000  
t 1
$5,825
 $219.20
t
(1.05)