Car Lease Option Valuation Case
Binomial Option Pricing and Lease Decisions

Table of Contents

1. Instructions

In this case, you will use a two-period binomial option pricing model to value the option to buy a car at the end of a lease. Then you will compare the value of this option to the extra lease payments you would make for having that option.

Show your work clearly in a spreadsheet.

For background, here is a paper I wrote on valuing the option in an auto lease.

2. Case Description

You are about to graduate from college and have just accepted a full-time job in a nearby city. The job will require a daily commute, so you need a reliable car, but you don’t want to commit to owning one long term until you see how the new job and city work out.

At a local dealership, you find a compact SUV with the following details:

  • Current market value (what the dealer says the car is worth today): $25,000
  • Model: a popular 3-year-old compact SUV with good resale demand

The salesperson offers you a 2-year closed-end lease with two options:

2.1. Lease A – with purchase option

  • 24 monthly payments (exact amount not important for this analysis)
  • At the end of 2 years, you may:
    • Return the car with no further obligation (subject to mileage/wear rules), or
    • Buy the car for a fixed “residual” price of $23,000

2.2. Lease B – no purchase option

  • Identical to Lease A in every respect except:
    • The monthly payment is $180 lower than under Lease A
    • There is no option to buy the car at the end; you must return it

The salesperson claims that Lease A is “basically the same deal” as Lease B, just with more flexibility because of the purchase option. You aren’t sure that’s true and you want to know:

How much is the purchase option in Lease A actually worth today?

Is the extra $180 per month a fair price for that flexibility?

From your finance coursework, you recognize that the right to buy the car at a fixed price in the future is similar to a call option: at the end of the lease, you can choose to buy the car for $23,000, but only if it is worth more than that in the used-car market.

To value this option, you will use a two-period binomial option pricing model.

3. Numerical Data for the Binomial Model

Use the following assumptions throughout the assignment:

  • Current value of the car:
    • \( S_0 = \$25{,}000 \)
  • Time to lease maturity:
    • \( T = 2 \) years
  • Number of binomial steps:
    • \( N = 2 \) (each step is 1 year)
  • End-of-lease purchase (exercise) price:
    • \( K = \$23{,}000 \)

  • Uncertainty in future car value (per year, binomial model): Over each 1-year period, the car’s value can:

    • Increase by 10% (an “up” move), or
    • Decrease by 10% (a “down” move)

    Therefore:

    • Up factor: \( u = 1.10 \)
    • Down factor: \( d = 0.90 \)
  • Risk-free interest rate (per year):
    • Annual risk-free rate: \( r = 4\% \)
    • Per-period gross return (per year): \( R = 1 + r = 1.04 \)
  • Lease payment difference:
    • Lease A (with purchase option) is $180 more per month than Lease B
    • Number of months: 24
    • Assume a monthly discount rate of \( 0.04 / 12 \) when computing the present value of the extra $180 payments.

4. Questions

4.1. Car Value Tree

4.1.1. Using the binomial model, compute the possible car values at the end of 2 years (after two 1-year steps). Compute the car value in each state:

  • Up-Up (UU)
  • Up-Down (UD)
  • Down-Up (DU)
  • Down-Down (DD)

Note: Under this model, UD and DU will have the same value.

Write your answers:

  • \( S_{UU} = \) __
  • \( S_{UD} = S_{DU} = \) __
  • \( S_{DD} = \) __

4.1.2. In a spreadsheet, construct a two-period binomial price tree showing the possible car values at each node (time 0, time 1, and time 2). Label each node clearly with the car value.

4.2. Option Payoffs at Maturity

At the end of the lease (after 2 years), you have the right (but not the obligation) to buy the car for $23,000.

4.2.1. For each final node (UU, UD/DU, DD), compute the payoff of the purchase option:

\[ \text{Option payoff} = \max(S_T - K,\ 0), \]

where:

  • \( S_T \) is the car’s value at the end of year 2
  • \( K = \$23{,}000 \)

Compute:

  • Payoff at UU: \( \max(S_{UU} - 23{,}000,\ 0) = \) __
  • Payoff at UD/DU: \( \max(S_{UD} - 23{,}000,\ 0) = \) __
  • Payoff at DD: \( \max(S_{DD} - 23{,}000,\ 0) = \) __

4.3. Risk-Neutral Valuation and Option Value Today

4.3.1. Compute the risk-neutral probability \( p^* \) for an up move using:

\[ p^* = \frac{R - d}{u - d}, \]

where:

  • \( R = 1.04 \)
  • \( u = 1.10 \)
  • \( d = 0.90 \)

Show your calculation and result:

  • \( p^* = \) __

4.3.2. Starting from the option payoffs at the final nodes, work backward through the tree using risk-neutral valuation to find the value of the option at each time-1 node, then at time 0.

Hint: At each node, the option value is:

\[ C = \frac{1}{R}\left(p^* C_{\text{up}} + (1 - p^*) C_{\text{down}}\right), \]

where \( C_{\text{up}} \) and \( C_{\text{down}} \) are the option values in the next period (up and down branches).

a) Option value at the time-1 up node: \( C_{U} = \) __

b) Option value at the time-1 down node: \( C_{D} = \) __

c) Current value (time 0) of the purchase option: \( C_0 = \) __

4.3.3. Interpretation:

In 1–2 sentences, explain in words what the value \( C_0 \) represents.

4.4. Comparing to the Extra Lease Payments

4.4.1. Compute the present value (PV) of the extra lease cost in Lease A

relative to Lease B.

  • Extra payment per month: $180
  • Number of months: 24
  • Monthly discount rate: \( i = 0.04 / 12 \)

Use the present value of an annuity formula:

\[ \text{PV} = \text{PMT} \times \frac{1 - (1 + i)^{-n}}{i}, \]

where:

  • \( \text{PMT} = 180 \)
  • \( i = 0.04 / 12 \)
  • \( n = 24 \)

Show your calculation and write the result:

  • PV of extra payments for Lease A (relative to Lease B):
    • PV = __

4.4.2. Compare:

  • Binomial model value of the purchase option: \( C_0 = \) __
  • PV of extra monthly payments: PV = __

Are you overpaying, underpaying, or paying about a fair price for the purchase option embedded in Lease A?

4.5. 5. Conceptual and Discussion Questions

4.5.1. In which future scenarios (states of the world) would you choose to exercise the purchase option (i.e., buy the car at the end of the lease)? When would you walk away and simply return the car?

4.5.2. How would the value of the purchase option change if:

  • The car’s price were more volatile (larger up and down moves each year)?
  • The purchase price \( K \) were higher or lower?
  • The risk-free rate were higher?

For each factor, state whether the option value would tend to increase, decrease, or be ambiguous, and give a short explanation.

4.5.3. From the dealership’s perspective, why might they be willing to offer a purchase option that appears underpriced (if that’s what you find)? What other business considerations or customer behaviors might affect how they set the lease terms?

Created: 2025-11-24 Mon 22:16

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