Binomial Option Pricing Model

Q. What is the fair price (=No-Arbitrage) of a European call option at time $t_{0}$ ?

Find the purchase of stock and borrow of money that would have the equivalent payoff as the call option (=reproducing portfolio) Use that to price the stock
Then, according to the No-Arbitrage that must be the same as the current price of the call option
Do this for $n$ periods for a binomial tree

Payoff (=price of call option at time $t_{1}$ )

C (t_{1}) = {max [S (t_{1}) \cdot u_{*} K, 0] max [S (t_{1}) \cdot d_{*} K, 0] probability p probability 1 - p

$Δ (t)$ : number of stocks you purchase
- will increase due to dividend payouts: $Δ (t_{1}) = Δ (t_{0}) e^{q (t_{1} - t_{0})}$
- $q$ is the dividend rate
$b$ : amount you borrow at time $t_{0}$
⇒ Value of portfolio

V (t_{0}) V (t_{1}) = Δ (t_{0}) \cdot S (t_{0}) - b = Δ (t_{1}) \cdot S (t_{1}) - b e^{r (t_{1} - t_{0})}

⇒ Equate $V (t_{1}) = C (t_{1})$ and solve for $Δ (t), b$ to obtain the reproducing portfolio.

Δ (t_{0}) b = \frac{C _{u} ( t _{1} ) - C _{d} ( t _{1} )}{( u - d ) S ( t _{0} )} e^{- q (t_{1} - t_{0})} = \frac{d C _{u} ( t _{1} ) - u C _{d} ( t _{1} )}{u - d} e^{- r (t_{1} - t * 0)}

V (t_{0}) = e^{- q (t_{1} - t_{0})} \frac{C _{u} ( t _{1} ) - C _{d} ( t _{1} )}{( u - d ) S ( t _{0} )} S (t_{0}) - e^{- r (t_{1} - t_{0})} \frac{d C _{u} ( t _{1} ) - u C _{d} ( t _{1} )}{u - d}

C (t_{0}) = e^{- r (t_{1} - t_{0})} [\frac{e ^{(r - q) (t_{1} - t_{0})} - d}{u - d} C_{u} (t_{1}) + \frac{u - e ^{(r - q) (t_{1} - t_{0})}}{u - d} C_{d} (t_{1})]

With the No-Arbitrage we have $E [C (t_{1})] = C (t_{0}) e^{(r - q) (t_{1} - t_{0})}$ .
Then define the risk-neutral probability of an uptick such that

p_{*} : = \frac{e ^{(r - q) (t_{1} - t_{0})} - d}{u - d}

thm. Risk-Neutral Binomial Call Pricing Formula

C (t_{0}; n) = e^{- (nh) r} i = 0 \sum n (i n) p_{*}^{i} (1 - p_{*})^{n - i} C_{u_{i} d_{n - i}} (t_{n})

where

By taking $n \to \infty$ we see

PK's Notes