9.3 Contingent Claims: Binomial Option Pricing Model & Arbitrage Replication

9.3 Contingent Claims: Binomial Option Pricing Model & Arbitrage Replication

Core Insight: Contingent claims (options) possess asymmetric payoffs that depend on the price path of an underlying asset. The Binomial Option Pricing Model provides a discrete-time, arbitrage-free methodology for valuing European and American style options across any asset class. Rather than relying on investor risk preferences, the model uses Risk-Neutral Valuation—weighting payoffs by pseudo-probabilities $\pi$ and discounting at the risk-free rate $r$—or Replication, constructing a synthetic portfolio of stock shares (delta $\Delta$) and risk-free borrowing ($B$) that perfectly mirrors option payoffs across all states of nature.

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1. The One-Period Binomial Option Model

Consider an underlying asset with current spot price $S_0$. Over a time step $\Delta t$, the price can move up to $S_u = S_0 u$ with payoff $c_u$, or down to $S_d = S_0 d$ with payoff $c_d$, where $u > 1 + r > d > 0$.

                             ONE-PERIOD BINOMIAL LATTICE
                                         ┌──────────────────────────┐
                                         │ State Up: S_u = S_0 * u  │
                                    ┌───►│ Option Payoff: c_u       │
       ┌────────────────────────┐   │    └──────────────────────────┘
       │ Root Node: S_0         │───┤    Risk-neutral Prob = π
       │ Option Price: c_0 = ?  │   │    
       └────────────────────────┘   │    ┌──────────────────────────┐
                                    └───►│ State Down: S_d = S_0 * d│
                                         │ Option Payoff: c_d       │
                                         └──────────────────────────┘
                                         Risk-neutral Prob = 1 - π

Model Parameters

Using the Cox-Ross-Rubinstein (CRR) parameterization matching asset volatility $\sigma$:

$$\nu = e^{\sigma \sqrt{\Delta t}}, \qquad d = \frac{1}{u} = e^{-\sigma \sqrt{\Delta t}}$$

Risk-Neutral Probability ($\pi$)

In a risk-neutral world, the expected return on the underlying asset must equal the risk-free rate $r$ (adjusted for continuous dividend yield $q$):

$$S_0 e^{(r - q)\Delta t} = \pi S_u + (1 - \pi) S_d = \pi S_0 u + (1 - \pi) S_0 d$$ $$\mathbf{\pi = \frac{e^{(r - q)\Delta t} - d}{u - d}} \qquad \text{(or } \mathbf{\pi = \frac{(1 + r) - d}{u - d}} \text{ in discrete compounding)}$$

Option Valuation Formula

$$\mathbf{c_0 = e^{-r\Delta t} [\pi c_u + (1 - \pi) c_d]}, \qquad \mathbf{p_0 = e^{-r\Delta t} [\pi p_u + (1 - \pi) p_d]}$$

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2. No-Arbitrage Replication Portfolio & Delta Hedging

An option can be replicated perfectly by purchasing $\Delta$ shares of stock and borrowing cash amount $B$ at the risk-free rate:

$$\text{Portfolio Value at } t = \Delta S_0 - B$$

To match the option payoffs in both the up state and down state:

1. Up state: $\Delta S_u - B(1 + r) = c_u$
2. Down state: $\Delta S_d - B(1 + r) = c_d$

Subtracting equation 2 from equation 1 yields the Hedge Ratio (Option Delta, $\Delta$):

$$\mathbf{\Delta = h = \frac{c_u - c_d}{S_u - S_d}}$$

Solving for the Borrowing Amount ($B$):

$$\mathbf{B = \frac{\Delta S_d - c_d}{1 + r} = e^{-r\Delta t} (\Delta S_d - c_d)}$$ $$\mathbf{c_0 = \Delta S_0 - B}$$

Interpretation: The call option is economically identical to a leveraged long stock position. The put option is economically identical to a short stock position with cash lending: $p_0 = B - \Delta_{put} S_0$, where $\Delta_{put} = \frac{p_u - p_d}{S_u - S_d} < 0$.

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3. Two-Period Binomial Tree & Backward Induction

In a two-period tree ($T = 2\Delta t$), the price progresses through three stages:

• $t = 0$: Root node $S_0$
• $t = 1$: Up node $S_u = S_0 u$, Down node $S_d = S_0 d$
• $t = 2$: $S_{uu} = S_0 u^2$, $S_{ud} = S_0 ud = S_0$, $S_{dd} = S_0 d^2$

                               TWO-PERIOD BINOMIAL TREE
                                                          ┌──────────────────┐
                                                          │ Node uu: S_uu    │
                                                     ┌───►│ c_uu = max(0,S-K)│
                                ┌──────────────────┐ │    └──────────────────┘
                                │ Node u: S_u      │─┤
                           ┌───►│ c_u = (π*c_uu... │ │    ┌──────────────────┐
      ┌──────────────────┐ │    └──────────────────┘ └───►│ Node ud: S_ud    │
      │ Root: S_0        │─┤                              │ c_ud = max(0,S-K)│
      │ c_0 = (π*c_u...  │ │    ┌──────────────────┐ ┌───►└──────────────────┘
      └──────────────────┘ └───►│ Node d: S_d      │─┤
                                │ c_d = (π*c_ud... │ │    ┌──────────────────┐
                                └──────────────────┘ └───►│ Node dd: S_dd    │
                                                          │ c_dd = max(0,S-K)│
                                                          └──────────────────┘

Backward Induction Algorithm

1. Step 1: Calculate terminal payoffs at $t = 2$ for all three nodes ($c_{uu}, c_{ud}, c_{dd}$).
2. Step 2: Roll back to $t = 1$ to calculate continuation values: $$c_u = e^{-r\Delta t} [\pi c_{uu} + (1 - \pi) c_{ud}], \qquad c_d = e^{-r\Delta t} [\pi c_{ud} + (1 - \pi) c_{dd}]$$
3. Step 3: Roll back to $t = 0$: $$c_0 = e^{-r\Delta t} [\pi c_u + (1 - \pi) c_d]$$

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4. American Option Valuation on Binomial Trees

American options grant the holder the right to exercise early at any node prior to expiration. At every individual node in the tree, the American option price equals the maximum of the Continuation Value (holding the option) and the Intrinsic / Early Exercise Value:

$$\mathbf{\text{American Node Value} = \max\left( \text{Continuation Value}, \; \text{Early Exercise Value} \right)}$$ $$\text{Continuation Value} = e^{-r\Delta t} [\pi V_{up} + (1 - \pi) V_{down}]$$ $$\text{Early Exercise Value (Call)} = \max(0, S_{node} - K)$$ $$\text{Early Exercise Value (Put)} = \max(0, K - S_{node})$$

Early Exercise Decision Boundaries

• American Calls on Non-Dividend Paying Stocks: It is never optimal to exercise early. The live option always has positive time value ($C_{American} = C_{European}$). Selling the option in the market always generates more cash than exercising.
• American Calls with Dividends: Early exercise can be optimal immediately before an ex-dividend date if the dividend received exceeds the remaining time value of the option plus interest earned on the strike price ($D > K [1 - e^{-r(T-t_D)}]$).
• American Puts: Early exercise is frequently optimal when the put is deep in-the-money and interest rates are positive ($r > 0$). Exercising early accelerates receipt of strike price $K$, allowing the investor to earn risk-free interest on cash immediately ($P_{American} > P_{European}$).

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5. Comprehensive Worked Numerical Calculation: 2-Period American Put

Data:

• Spot stock price $S_0 = \text{\textdollar}50.00$
• Strike price $K = \text{\textdollar}52.00$
• Up-factor $u = 1.20$, Down-factor $d = 0.80$ ($d = 1/u$)
• Risk-free rate per period $r = 5.00\%$ ($1 + r = 1.05$)
• Number of periods = 2

Step 1: Calculate Risk-Neutral Probability ($\pi$)

$$\pi = \frac{(1 + r) - d}{u - d} = \frac{1.05 - 0.80}{1.20 - 0.80} = \frac{0.25}{0.40} = \mathbf{0.625}, \qquad 1 - \pi = \mathbf{0.375}$$

Step 2: Construct Stock Price Tree

• $t = 0$: $S_0 = \text{\textdollar}50.00$
• $t = 1$: $S_u = 50 \times 1.20 = \text{\textdollar}60.00$, $S_d = 50 \times 0.80 = \text{\textdollar}40.00$
• $t = 2$:
  • $S_{uu} = 60 \times 1.20 = \text{\textdollar}72.00$
  • $S_{ud} = 60 \times 0.80 = \text{\textdollar}48.00$
  • $S_{dd} = 40 \times 0.80 = \text{\textdollar}32.00$

Step 3: Calculate Terminal Put Payoffs at $t = 2$

• $p_{uu} = \max(0, 52 - 72) = \mathbf{\text{\textdollar}0.00}$
• $p_{ud} = \max(0, 52 - 48) = \mathbf{\text{\textdollar}4.00}$
• $p_{dd} = \max(0, 52 - 32) = \mathbf{\text{\textdollar}20.00}$

Step 4: Backward Induction to $t = 1$ with Early Exercise Check

Node $u$ ($S_u = \text{\textdollar}60.00$):

$$\text{Continuation Value} = \frac{\pi p_{uu} + (1 - \pi) p_{ud}}{1 + r} = \frac{(0.625 \times 0) + (0.375 \times 4)}{1.05} = \frac{1.50}{1.05} = \$1.4286$$ $$\text{Early Exercise Value} = \max(0, 52 - 60) = \$0.00$$ $$\mathbf{p_u = \max(\$1.4286, \$0.00) = \$1.4286} \quad \text{(Do not exercise)}$$

Node $d$ ($S_d = \text{\textdollar}40.00$):

$$\text{Continuation Value} = \frac{\pi p_{ud} + (1 - \pi) p_{dd}}{1 + r} = \frac{(0.625 \times 4) + (0.375 \times 20)}{1.05} = \frac{2.50 + 7.50}{1.05} = \frac{10.00}{1.05} = \$9.5238$$ $$\text{Early Exercise Value} = \max(0, 52 - 40) = \mathbf{\$12.00}$$ $$\mathbf{p_d = \max(\$9.5238, \$12.00) = \$12.00} \quad \mathbf{\text{(EARLY EXERCISE OPTIMAL!)}}$$

Crucial Step: Because Early Exercise ($12.00) exceeds Continuation Value ($9.5238), the node value is set to $12.00.

Step 5: Roll Back to Root Node $t = 0$

$$\text{Continuation Value at } t = 0 = \frac{\pi p_u + (1 - \pi) p_d}{1 + r} = \frac{(0.625 \times 1.4286) + (0.375 \times 12.00)}{1.05} = \frac{0.8929 + 4.5000}{1.05} = \frac{5.3929}{1.05} = \mathbf{\$5.1361}$$ $$\text{Early Exercise Value at } t = 0 = \max(0, 52 - 50) = \$2.00$$ $$\mathbf{P_{American, 0} = \max(\$5.1361, \$2.00) = \$5.14}$$

(Note: A European put with no early exercise at Node $d$ would be worth $\frac{0.8929 + (0.375 \times 9.5238)}{1.05} = \frac{4.4643}{1.05} = \text{\textdollar}4.25$. Early exercise adds $\text{\textdollar}5.14 - \text{\textdollar}4.25 = \text{\textdollar}0.89$ of value).

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6. Extending the Binomial Model to Interest Rates

The same backward-induction logic applies when the underlying is an interest rate rather than a stock price, with two differences: the payoff is defined on a rate rather than a price, and each node's own one-period rate is used for discounting. Section 9.5 develops interest rate options, caps and floors, swaptions, and the Black model in full, including a worked two-period interest rate call valuation.