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# The binomial model ## The risk-neutral pricing methodology ### The binomial model **Setting:** * Stock price today: $S_0$ * Stock price next period: $S_u$ with probability $q$ and $S_d$ with probability $1 − q$ * Payoff of derivative next period: $C_u$ if the stock price is $S_u$ and $C_d...

# The binomial model ## The risk-neutral pricing methodology ### The binomial model **Setting:** * Stock price today: $S_0$ * Stock price next period: $S_u$ with probability $q$ and $S_d$ with probability $1 − q$ * Payoff of derivative next period: $C_u$ if the stock price is $S_u$ and $C_d$ if the stock price is $S_d$ * Risk-free rate: $r$ **Replicating the derivative** We replicate the derivative by holding $\Delta$ shares of stock and $B$ dollars in a risk-free account. We want to choose $\Delta$ and $B$ such that the portfolio has the same payoff as the derivative in both states: $\qquad \Delta S_u + (1 + r)B = C_u$ $\qquad \Delta S_d + (1 + r)B = C_d$ Solving these equations, we obtain: $\qquad \Delta = \frac{C_u - C_d}{S_u - S_d}$ $\qquad B = \frac{C_d S_u - C_u S_d}{(1 + r)(S_u - S_d)}$ **The value of the derivative today** The value of the derivative today is the same as the value of the replicating portfolio: $\qquad C_0 = \Delta S_0 + B = \frac{C_d S_u - C_u S_d}{(1 + r)(S_u - S_d)} + \frac{C_u - C_d}{S_u - S_d} S_0$ Rearranging, we have $\qquad C_0 = \frac{1}{1 + r} \left[ \frac{(1 + r)S_0 - S_d}{S_u - S_d} C_u + \frac{S_u - (1 + r)S_0}{S_u - S_d} C_d \right]$ **Risk-neutral probabilities** Define $\qquad p = \frac{(1 + r)S_0 - S_d}{S_u - S_d}$ Then, $\qquad 1 - p = \frac{S_u - (1 + r)S_0}{S_u - S_d}$ and $\qquad C_0 = \frac{1}{1 + r} [p C_u + (1 - p) C_d]$ $p$ can be interpreted as the probability of an up move in a world where investors are risk-neutral. ### The risk-neutral pricing methodology The expected return on all assets in a risk-neutral world is the risk-free rate. The price of any derivative is its expected payoff in the risk-neutral world, discounted at the risk-free rate. Let $C_T$ be the payoff of the derivative at time $T$. Then, $\qquad C_0 = e^{-rT} \mathbb{E}^*[C_T]$ where $\mathbb{E}^*$ denotes the expectation in the risk-neutral world.