The Binomial Model

Flashcards for the binomial models slides, notes and lecture in PFA

One step Binomial Model:

What is the structure for for pricing options? I.e. underlying assets, up/down move and option payoff

In a Binomial Model we have the following:

• The underlying asset price S₀ can either move:

  • Up: To S_0 *u (u > 1)

  • Down: To S_0 * d (d < 1).

• The payofs are then:

  • Up: V_u

  • Down: V_d

One step model:

1. How do you contruct a risk-free portfolio intuitively?

2. How do you do it formerly?

3. What is the equation?

4. Derive the equation for Delta

1. By making sure that in each possible future state, the payoff is identical.

2. We do this by taking a long position of Delta shares in the underlying, and shorting one option. We the nchoose Delta so make sure that the portfolios payoff is the s ame regardsless of the up or dwon move.

3. S_0*uDelta - V_u = S_0*d*Delta-V_D 

4. Solve for Delta in the above equaation yields:

$\Delta = (V_u - V_d)/(S_0*(u-d)) 

One step model:

1. How is the risk neutral probability defined?

2. How are it defined for a time step Δt?

1. It is defined as p = (exp(rT)-d)/(u-d)

2. p = (exp(rΔt)-d)/(u-d)

1. Write the risk–neutral valuation formula for an option in a one–step model.?

2. What is the intuitive understanding of it?

1. V = exp(-rT)*[pV_u + (1-p)V_D)

2. The value of an option is the discounted weigted average of the risk-neutral probability of each outcome

What is meant by "risk–neutral valuation" in the context of derivative pricing?

Risk-neutral valutation assumes that investors are indifferent to risk so that every insturment earns the risk-free rate. Thus, the derivative´´s price can be determined as the discounted expected payoff computed under the risk-neutral probabilities

How do you compute the value of an option using a two–step binomial tree?

Do the exact same just multiple times. If can however be written as:
f = exp(-2rΔt)[p²V_uu + 2p(1-p)V_ud + (1+p)²v_dd]

What is the meaning of “Delta” in option pricing?

Delta measures the sensitivity of the option’s price to a change in the underlying asset’s price. It is the ratio of the change in option price to the change in stock price and indicates how many shares must be held per option to hedge a position.

How are the parameters u and d chosen to match the asset’s volatility?

To match the volatility over a time Δt, set:

u = exp(sigma * sqrt(Δt))

d = exp(-sigma * sqrt(Δt)) or just d = 1/u

How does Girsanov’s Theorem relate to the binomial model parameters?

Girsanov’s Theorem shows that while switching from the real–world to the risk–neutral measure changes the expected return, the volatility remains unchanged. This justifies using the same volatility–matching parameters in both worlds

What additional step is involved when pricing American options with the binomial model?

For American options, at each node you compare the intrinsic value (the immediate exercise value) to the continuation value (the discounted expected future payoff) and then choose the maximum. This extra step accounts for the possibility of early exercise.

Why is early exercise typically suboptimal for American call options on non–dividend paying stocks?

Early exercise sacrifices the option’s remaining time value. For non–dividend paying stocks, there is no benefit to owning the stock earlier (e.g., dividends), so holding the option (and the flexibility it provides) is usually more valuable.

How are the growth factor adjustments made when pricing European options?

1. with dividends?

2. foreign currencies?

3. On futures contrct?

What is the general theorem for pricing a derivative in a one–period binomial model?

V₀ = exp(-rT)*[qV_u+(1-q)V_d], where q = (exp(rT)-d)/(u-d)