A binomial tree is one of the cleanest ways to understand option pricing because it turns a continuous-time idea (“price evolves with uncertainty”) into a small set of possible up/down paths. If you can build the tree correctly, you can price European options, handle early exercise for American options, and compute hedge ratios step by step.
How you use it: Build the stock-price lattice, compute option payoffs at maturity, then work backward using risk-neutral probabilities and discounting to get today’s option price.
Why Binomial Trees Work
In a one-step model, the option can be replicated by holding \(\Delta\) shares and borrowing/lending an amount \(B\). Replication implies a unique no-arbitrage price. The binomial tree extends the same logic across many short steps.
Key idea: price by expected discounted payoff under a risk-neutral probability \(p\), not the real-world probability.
How to Build a Binomial Tree (Step by Step)
Step 1: Choose inputs
- Current stock price \(S_0\)
- Strike \(K\)
- Risk-free rate \(r\) (per period)
- Number of steps \(N\)
- Up and down factors \(u\) and \(d\)
Common choice (CRR): \(u=e^{\sigma\sqrt{\Delta t}}\), \(d=e^{-\sigma\sqrt{\Delta t}}=\frac{1}{u}\).
Step 2: Build the stock-price lattice
After \(i\) steps with \(j\) up moves:
\[ S_{i,j} = S_0\,u^j d^{\,i-j}. \]
Step 3: Compute risk-neutral probability
\[ p=\frac{(1+r)-d}{u-d}. \]
Step 4: Compute option payoff at maturity
For a call:
\[ C_{N,j}=\max(S_{N,j}-K,0). \]
Step 5: Backward induction (discounted expectation)
\[ C_{i,j}=\frac{1}{1+r}\left(p\,C_{i+1,j+1}+(1-p)\,C_{i+1,j}\right). \]
For an American option, compare continuation vs immediate exercise at each node.
Numerical Example (1-Step Call)
Suppose \(S_0=100\), \(K=100\), \(r=0.05\), \(u=1.2\), \(d=0.9\).
Stock outcomes: \(S_u=120\), \(S_d=90\). Call payoffs: \(C_u=20\), \(C_d=0\).
Risk-neutral probability:
\[ p=\frac{1.05-0.9}{1.2-0.9}=\frac{0.15}{0.3}=0.5. \]
Option price:
\[ C_0=\frac{1}{1.05}\left(0.5\cdot 20+0.5\cdot 0\right)=\frac{10}{1.05}\approx 9.52. \]
Common Mistakes
- Using real-world probabilities instead of risk-neutral \(p\).
- Forgetting to discount by \(1+r\) at each step.
- Choosing \(u\) and \(d\) that make \(p\notin[0,1]\) (arbitrage problem).
- Mixing up indices: \(S_{i,j}\) has \(j\) up moves after \(i\) total steps.
- For American options, forgetting to check early exercise at every node.
Why This Matters
Binomial trees are used to price options, teach replication and no-arbitrage, and build intuition for Black–Scholes. They also show you how hedge ratios \(\Delta\) change through time, which is a common exam and interview target.
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