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’

 ST,d

ST,u

ST‑1

Suppose that ST-1 = 40, u = 25% and d = -10%.

40ST,u = 50

ST,d = 36

Time T is the expiration day of a call option. Time T-1 is one period prior to expiration.

The Option Pricing Process

CT,d = max(0, ST,d‑K)

CT,u = max(0, ST,u‑K)

CT‑1

Suppose that K = 45. What are CT,u and CT,d?

CT‑1

CT,u = 5

CT,d = 0

dudrp

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 = 2.040816327The two equations are

50 + 1.05B = 536 + 1.05B = 0

Solve, and = 0.357142857B = -12.24489796

A Shortcut: discount method

durup)(1and

dudrp

where,

7)-(17 r)(1

p)C(1pCC

or,r)(1

CduruC

dudr

C

dT,uT,1T

dT,uT,

1T

8)-(17 r)(1

p)C(1pCC du

In general:

Interpreting p

• 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

dudrp

The Equivalent Portfolio

(1+d)ST‑1 + (1+r)B = ST,d + (1+r)B

(1+u)ST‑1 + (1+r)B = ST,u + (1+r)B

ST‑1+B

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?

Buy shares of stock and borrow $B.

NB: is not a “change” in S…. It defines the # of shares to buy. For a call, 0 < < 1

and B define the “Equivalent Portfolio” of a call

2)-(170 B ;r)d)(1(ud)C(1u)C(1

B

1)-(171 Δ 0 ;SSCC

d)S(uCC

Δ

cuT,dT,

cdT,uT,

dT,uT,

1T

dT,uT,

Assume that the underlying asset can only rise by u% or decline by d% in the next period. Then in general, at any time:

4)-(17 r)d)(1(ud)C(1u)C(1

B

3)-(17 SSCC

d)S(uCC

Δ

ud

du

dudu

CT‑1 = ST‑1 + B (17-5)

C = S + B (17-6)

NB: a negative sign now denotes borrowing!

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,dd = (1+d)2ST-2

ST,uu = (1+u)2ST-2

ST-1,u = (1+u)ST-2

ST,ud = (1+u)(1+d)ST-2

ST-1,d = (1+d)ST-2

ST-2

CT,dd = max[0,(1+d)2ST-2 - K]

CT,uu = max[0,(1+u)2ST-2 - K]

CT-1,u

CT,ud = max[0,(1+u)(1+d)ST-2 - K]

CT-1,d

CT-2

Two Period Binomial Model: An Example

ST,dd = 36

ST,uu = 69.444

ST-1,u = 55.556

ST,ud = 50

ST-1,d = 40.00ST-2 = 44.444

CT,dd = 0

CT,uu = _______

CT-1,u = ____

CT,ud = 5CT-1,d = 2.0408

CT-2

Two Period Binomial Model: The Equivalent Portfolio

= 1B = -42.857143

= 0.357142857B = -12.24489796

= 0.6851312B = -24.1566014

T-2 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 T-n in

an n-period, (n+1)-date, model).

The ‘n’ Period Binomial Formula:

15)-(17

r)(1Cp)(1Cp)3p(1p)C(13pCp

C 3dddT,

3uddT,

2uudT,

2uuuT,

3

3T

If n = 3:

j)!(nj!n!

jn

The “binomial coefficient” computes the number of ways we can get j upticks in n periods:

.K]Sd)(1u)(1max[0,p)(1pj3

r)(11C

3

0j3T

j3jj3j33T

Thus, the 3-period model can be written as:

The ‘n’ Period Binomial Formula:

In general, the n-period model is:

17)(17.K]Sd)(1u)[(1p)(1pjn

r)(11C

n

ajnT

jnjjnjn

Where “a” in the summation is the minimum number of up-ticks so that the call finishes in-the-money.

A Large Multi-period LatticeSuppose 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)

50.0050.50

51.00551.51505

49.6049.2032

48.80957

50.09650.59696

49.69523

T=0 T=1 T=2 T=3T=100

23.214 = 50*(1.01^2)*(.992^98)22.801 = 50*(1.01^1)*(.992^99)22.394 = 50*(.992^100)

.

.

.

.

Suppose the Number of Periods Approachs Infinity

S

TIn 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).

©David Dubofsky and 17-21 Thomas W. Miller, Jr.

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

23.100

22 => 21 21.945

19.950

20.000 18.952521.780

19.800

19 => 18 18.810

17.100

16.245

T-3 T-2 T-1 T

25.520

24.20 => 23.20

20.04022.000

21.890

20.000 20.90 => 19.90

18.90519.000

18.755

18.05 => 17.05

16.1975

T-3 T-2 T-1 T

American Calls on Dividend Paying Stocks

• The key is that at each “node” of the lattice, the value of an American call is:

19)(17 .KS,r)(1

p)C(1pCmax du

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

ST,u = (1+u)ST‑1

ST‑1 ST,d = (1+d)ST‑1

PT,u = max(0,K-ST,u) = max(0,K-(1+u)ST‑1)

PT‑1

PT,d = max(0,K-ST,d) = max(0,K-(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)

22)(17SSPP

d)S(uPP

Δdu

dudu

23)(17r)d)(1(ud)P(1u)P(1

B ud

Where:

-1 < p < 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.

B > 0

Binomial Put Pricing - III

26)(17r)(1

p)P(1pPP du

durup)(1and

dudrp

Where:

Binomial American Put Pricing

27)(17

r)(1p)P(1pP

S,KmaxP du

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

79.8672.6

66 68.9760 62.7

57 59.56554.13

51.4425

T-3 T-2 T-1 T

The Stock Pricing Process:

u = 10%d = -5%r = 2%K = 65p = 0.466667

Binomial Put Pricing Example - II

00

1.485924 03.9776 2.84183

6.306976 5.4359.57549

13.5575

T-3 T-2 T-1 T

European Put Values:

Binomial Put Pricing Example - IIIΔ = 0.0B = 0.0

Δ = -0.2870535B = 20.431458

Δ = -0.5356724 Δ = -0.5778841B = 36.117946 B = 39.075163

Δ = -0.7875626B = 51.198042

Δ = -1.0B = 63.72549

T-3 T-2 T-1

Composition of the equivalent portfolio to the European put:

Binomial Put Pricing Example - IV0

01.485924 0

4.86284 2.841835

6.97339 5.4358

9.5754910

13.5575

T-3 T-2 T-1 T

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.

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 LeasesThe 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 SheetTruck is purchased with debtTruck $100,000 Debt $100,000Land $100,000 Equity $100,000Total Assets $200,000 Total Debt & Equity $200,000

Operating LeaseTruck DebtLand $100,000 Equity $100,000Total Assets $100,000 Total Debt & Equity $100,000

Capital LeaseAssets leased $100,000 Obligations under capital lease $100,000Land $100,000 Equity $100,000Total Assets $200,000 Total Debt & Equity $200,000

38

Lease form

39

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 ofthe asset.

40

Sale and lease back

41

Leveraged lease

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 90-

percent 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 LeasingConsider 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

Year 0 Years 1-5Cost of truck –$25,000After-tax savings 4,500×(1-.34) = $2,970Depreciation Tax Shield 5,000×(.34) = $1,700

–$25,000 $4,670

• Cash Flows: LeaseYear 0 Years 1-5

Lease Payments –6,250×(1-.34) = –$4,125After-tax savings 4,500×(1-.34) = $2,970

–$1,155• Cash Flows: Leasing Instead of Buying

Year 0 Years 1-5$25,000 –$1,155 – $4,670 = –

$5,825

4 The Cash Flows of Leasing• Cash Flows: Leasing Instead of Buying

Year 0 Years 1-5$25,000 –$1,155 – $4,670 = –$5,825

• Cash Flows: Buying Instead of Leasing Year 0 Years 1-5–$25,000 $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

20.219$)05.1(

825,5$000,25$5

1

t

tNPV

20.219$)05.1(

825,5$000,25$5

1

t

tNPV

NPV Buying Instead of Leasing

Year 0 Years 1-5-$25,000 $4,670 – $1,155 = $5,825

• 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 BuyingYear 0 Years 1-5$25,000 –$1,155 – $4,670 = -$5,825