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’ S T,d S T,u ST‑1ST‑1 Suppose that S T-1 = 40, u = 25% and d = -10%. 40 S T,u = 50 S T,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 C T,d = max(0, S T,d ‑ K) C T,u = max(0, S T,u ‑ K) CT‑1CT‑1 Suppose that K = 45. What are C T,u and C T,d ? CT‑1CT‑1 C T,u = 5 C T,d = 0

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. C T-1 =  S T-1 + B

So, in the Numerical Example…. S T-1 = 40, u = 25%, S T,u = 50, d = -10%, S T,d = 36, r = 5%, K = 45, C T,u = 5 and C T,d = 0. Finding the replicating portfolio of , B, and C T-1. Att: Finding: THE RISK NEUTRALO PROBABILITY p = (0.05 – (-0.1))/(0.25 – (-0.1)) = 0.15/0.35 = (1-p) = C = [( )(5) + ( )(0)]/1.05 = The two equations are 50  B = 5 36  B = 0 Solve, and  = B =

A Shortcut: discount method 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 S T-1 = [( )(50) + ( )(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  (1+d)S T ‑ 1 + (1+r)B =  S T,d + (1+r)B  (1+u)S T ‑ 1 + (1+r)B =  S T,u + (1+r)B  S T ‑ 1 +B Set the payoffs of the equivalent portfolio equal to C T,u and C T,d, respectively.  (1+u)S T ‑ 1 + (1+r)B = C T,u  (1+d)S T ‑ 1 + (1+r)B = C T,d These are two equations with two unknowns:  and B What are the two equations in the numerical example with S T-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 Assume that the underlying asset can only rise by u% or decline by d% in the next period. Then in general, at any time: C T ‑ 1 =  S T ‑ 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 S T,dd = (1+d) 2 S T-2 S T,uu = (1+u) 2 S T-2 S T-1,u = (1+u)S T-2 S T,ud = (1+u)(1+d)S T-2 S T-1,d = (1+d)S T-2 S T-2 C T,dd = max[0,(1+d) 2 S T-2 - K] C T,uu = max[0,(1+u) 2 S T-2 - K] C T-1,u C T,ud = max[0,(1+u)(1+d)S T-2 - K] C T-1,d C T-2

Two Period Binomial Model: An Example S T,dd = 36 S T,uu = S T-1,u = S T,ud = 50 S T-1,d = S T-2 = C T,dd = 0 C T,uu = _______ C T-1,u = ____ C T,ud = 5 C T-1,d = C T-2

Two Period Binomial Model: The Equivalent Portfolio  = 1 B =  = B =  = B = T-2T-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: If n = 3: The “binomial coefficient” computes the number of ways we can get j upticks in n periods: Thus, the 3-period model can be written as:

The ‘n’ Period Binomial Formula : In general, the n-period model is: 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 = S 0 = = 50*(1.01^100) = 50*(1.01^99)*(.992^1) = 50*(1.01^98)*(.992^2) T=0T=1T=2T=3 T= = 50*(1.01^2)*(.992^98) = 50*(1.01^1)*(.992^99) = 50*(.992^100)

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 S T, then the BOPM converges to the Black-Scholes Option Pricing Model (the BSOPM is the subject of Chapter 18).

©David Dubofsky and 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.

American Calls on Dividend Paying Stocks The key is that at each “node” of the lattice, the value of an American call is: 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 S T,u = (1+u)S T ‑ 1 S T ‑ 1 S T,d = (1+d)S T ‑ 1 P T,u = max(0,K-S T,u ) = max(0,K-(1+u)S T ‑ 1 ) P T ‑ 1 P T,d = max(0,K-S T,d ) = max(0,K-(1+d)S T ‑ 1 )  (1+u)S T ‑ 1 + (1+r)B =  S T,u + (1+r)B = P T,u  S T ‑ 1 +B  (1+d)S T ‑ 1 + (1+r)B =  S T,d + (1+r)B = P T,d

Binomial Put Pricing - II P T ‑ 1 =  S T ‑ 1 + B (17-24) 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 Where:

Binomial American Put Pricing At any node, if the 2 nd 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 T-3T-2T-1T The Stock Pricing Process: u = 10% d = -5% r = 2% K = 65 p =

Binomial Put Pricing Example - II T-3T-2T-1T European Put Values:

Binomial Put Pricing Example - III Δ = 0.0 B = 0.0 Δ = B = Δ = Δ = B = B = Δ = B = Δ = -1.0 B = T-3T-2T-1 Composition of the equivalent portfolio to the European put:

Binomial Put Pricing Example - IV T-3T-2T-1T American put pricing: If eqn 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 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$100,000Debt$100,000 Land$100,000Equity$100,000 Total Assets$200,000Total Debt & Equity $200,000 Operating Lease TruckDebt Land$100,000Equity$100,000 Total Assets$100,000Total Debt & Equity $100,000 Capital Lease Assets leased$100,000Obligations under capital lease$100,000 Land$100,000Equity$100,000 Total Assets$200,000Total Debt & Equity$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 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 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 Year 0Years 1-5 Cost of truck – $25,000 After-tax savings4,500×(1-.34) = $2,970 Depreciation Tax Shield5,000×(.34) = $1,700 – $25,000$4,670 Cash Flows: Lease Year 0Years 1-5 Lease Payments–6,250×(1-.34) = –$4,125 After-tax savings4,500×(1-.34) = $2,970 –$1,155 Cash Flows: Leasing Instead of Buying Year 0Years 1-5 $25,000 – $1,155 – $4,670 = – $5,825

4 The Cash Flows of Leasing Cash Flows: Leasing Instead of Buying Year 0Years 1-5 $25,000 –$1,155 – $4,670 = –$5,825 Cash Flows: Buying Instead of Leasing Year 0Years 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 NPV Buying Instead of Leasing Year 0Years 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 Buying Year 0Years 1-5 $25,000 –$1,155 – $4,670 = -$5,825