Chapter 12 - The Binomial Model

Question 1

What is the Binominal Model used for?

Answer 1

Value/Price Derivatives

Question 2

What are the 5 assumptions of the Binomial Model?

Answer 2

• The principle of no arbitrage apply
• No trading costs
• No taxes
• No minimum or maximum units of trading
• Stocks and bonds can only be sold at discrete times

Question 3

What is the Expected Return of the underlying stock under the risk-neutral probability Q at time 1? (stock price at t=0 is S0)

Answer 3

E_Q[S1] = S0Uq + S0D(1-q)

q=(exp(r)-d)/(u-d)

E_Q[S1] = S0*exp(r)

Question 4

What is the value of q (the risk-neutral probability)?

Answer 4

q= (exp(r)-d)/(u-d)

Question 5

Why is Q called the risk-neutral probability?

Answer 5

E_Q[S1] = S0Uq + S0D(1-q) = S0*exp(r)

Question 6

How many states in the n-binomial tree at time n?

Answer 6

2^n

Question 7

How does the binomial model allow for different level of volatility in different states?

Answer 7

By allowing different values for u and d in different states.

u & p are price factors.

Question 8

What is the downfall to allowing for different levels of volatility in each state?

Answer 8

the model is limited by the number of states which exists even for relatively low numbers i.e. 2^n.

Question 9

What does the Recombining Binomial tree assume?

Answer 9

That u & d and the same in every states.

Question 10

How many different states do we have for the recombining binomial tree?

Answer 10

N+1

Question 11

What is the formula for the price of a derivative at time t using the recombining binomial tree?

Answer 11

Vt = exp(-r(n-t)) sum[0, n-t] f(S_t* u^k *d^(n-t-k)) * (n-t)Ck * q^k * (1-q)^(n-t-k)

i.e. exp(-r(n-t))sum[payoffnCxq^k(1-q)^(n-t-k))

Question 12

What is a replicating portfolio?

Answer 12

A replicating portfolio is a portfolio that replicates the payoff at time one on a derivative without any risk.

Question 13

In the one period binomial model, what is the portfolio at time 0?

Answer 13

V0= phi*St + Psi

Phi share, psi cash

Question 14

Prove V0= exp(-r)*E_Q[C1]

Where C1 is the payoff

Answer 14

V0=phi*S0+ psi

If share price goes up:
V1= phiS0U+ psiexp(r)
V1=Cu
If share price goes down:
V1= phiS0U+ psiexp(r)
V1=cd

Now to find phi and psi using the two simultaneous equations to get

Phi=cu-cd/So(u-d)
Psi= exp(-r)(cdu-cud)/(u-d)

Substituting this in to V0 we get:
V0=exp(-r)(qcu + (1-q)cd)
Where q=(exp(-r)-d)/(u-d)

V0 = exp(-r)*E_Q[C1]

Question 15

What is St under the recombining binomial tree?

Answer 15

St=SoU^N(u)d^(t-N(u))

N(u) is the number of up-steps

Question 16

What does it mean if a derivative is at the money?

Answer 16

The current share price is the same as the strike price.

Question 17

What is the replicating portfolio method?

Answer 17

V0= phi*S0+Psi

Phi=(Cu-Cd)/(S0(u-d))

Psi = exp(-r)((cdu-cu*d)/(u-d))

Question 18

What is the risk-neutral valuation method?

Answer 18

Vt=exp(-(T-t))*E[C_T|F_T-1]

Question 19

If there is a dividend paid immediately before expiry, how do we factor this into our calcs for the price of the option?

Answer 19

We take off the value of the dividend for the share price.

E.g. if with no dividend, S0=100 and u=1.05
S1=105

If we had a dividend of 3:
S1=102

Question 20

What is the relationship between u, d and exp(r)?

Answer 20

d<exp(r)<u

Question 21

What is the risk-free portfolio and what is its property?

Answer 21

The risk free portfolio consists of
- minus one derivative
- df/ds shares

It must grow at the risk free rate whether share prices go up or down

Question 22

If u and d are not given in the question, what do we use?

Answer 22

u=exp(sigmasqrt((T-t) +q(T-t))
d=exp(-sigmasqrt((T-t) +q(T-t))

(T-t) is the change in time