derivatives 10: Valuing a Derivative Using a One-Period Binomial Model

UNFINISHED 10.04 STILL NEEDS TO BE ADDED

What does the binomial model assume about price movement?

Over one period, price either:

• goes up to $S_1^{u}=uS_0$

• goes down to $S_1^{d}=dS_0$

u > 1 = up factor

d < 1 = down factor

What are gross returns in binomial model?

• R_{u}=\frac{S_1^{u}}{S_0}>1

• R_{d}=\frac{S_1^{d}}{S_0}<1

what are the Probabilities q and (1 − q)

q = probability of up move
1 − q = probability of down move

• not needed

  • risk-neutral valuation does not require real-world probabilities.

What does spread between $S_1^{u}$ and $S_1^{d}$ represent?

Volatility of the underlying asset

Call option formula for upper

$$c_1^{u}=\max\left(0,S_1^{u}-X\right)$$

Call option ends up in-the-money

Call option formula for down

$$c_1^{d}=\max\left(0,S_1^{d}-X\right)$$

Call option expires out-of-the-money

What portfolio is used for replication?

• Long h units of stock

• Short 1 call option

Portfolio value at time 0?

$$V_0 = hS_0 - c_0$$

Portfolio value in up/down state?

up: $V_1^{u}=hS_1^{u}-c_1^{u}$

• $$S_1^{u}=R^{u}\cdot S_0$$

• $R^{^{u}}$ ​down multiplier

down: $V_1^{u}=hS_1^{d}-c_1^{d}$

• where $S_1^{d}=R^{d}\cdot S_0$

• $R^d$ ​down multiplier

What is the hedge ratio (h*)?

proportion of the underlying that will offset the risk associated with an option

hedge ratio formula

set V ₁ ^u = V ₁ ^d

solve $h^{*}=\frac{c_1^{u}-c_1^{d}}{s_1^{u}-s_1^{d}}$

up call option - down call option / (up price - down price)