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Discrete time models
Derivatives 1, professor Alexei Zhdanov
Fall 2010
Handout 2
This handout has benefited from the materials provided by Tony Berrada.
1 / 22
Binomial Model
Definition [Arbitrage]. An arbitrage strategy is a strategy whichrequires no (positive) initial investment, never yields a negativeterminal value and has a strictly positive expected terminal value
2 / 22
One Period Model
The binomial market consists initially of two assets
A risky stockA risk-free bond
We call the beginning of the period time 0 and the end of theperiod time 1
The price of the stock at time 0 is S0
The price of the bond at time zero is B0
The price of the stock at time 1 is a random variableS1 ∈ {Su, Sd}The price of the bond at time 1 is a non random variable B1
3 / 22
One Period Model
Define the risk free interest rate as 1 + r = B1B0
The risky return on stock is u = SuS0
or d = SdS0
Let us call p the probability of an ”up” move and 1− p theprobability of a ”down” moveNormalize the model by setting B0 = 1
4 / 22
Replication
Consider a European call option written on the underlyingasset S with exercise price K and maturity at date 1
The payoff of the option is
cu = (Su −K)+ or cd = (Sd −K)+
The price of the option at time 0 is c0
How can we find it?
5 / 22
Replication
Consider a portfolio V consisting of ∆ shares of the stock andβ shares of the bond
β shares of the bond correspond to an investment of β dollarsat the risk-free rate
The value of the portfolio at time 0 is
V0 = ∆× S0 + β
The value of the portfolio at time 1 is given by
Vu = ∆× Su + β(1 + r) and Vd = ∆× Sd + β(1 + r)
We can construct a portfolio consisting of the stock and thebond such that its value at time 1 is always equal to thepayoff of the option
6 / 22
Replication
We call this portfolio a ”replicating portfolio”
(∆∗, β∗) must be such that
cu = (Su −K)+ = ∆∗ × Su + β∗(1 + r)
cd = (Sd −K)+ = ∆∗ × Sd + β∗(1 + r)
The value of this portfolio today must be equal to the price ofthe call to prevent arbitrage:
c0 = ∆∗ × S0 + β∗
Solving this system of equations yields
∆∗ =cu − cdSu − Sd
β∗ =Sucd − Sdcu
(Su − Sd)(1 + r)
7 / 22
Replication
It follows that the price of the European Call option at time 0is given by
c0 =cu − cdSu − Sd
S0 +Sucd − Sdcu
(Su − Sd)(1 + r)
We can generalize this result to any derivative with payoff(Du, Dd)
D0 =Du −Dd
Su − SdS0 +
SuDd − SdDu
(Su − Sd)(1 + r)
8 / 22
Example
r = 2.5%, S0 = 100, Su = S0u = 125, Sd = S0d = 80,K = 102.5
What is the price of the call option?
What is the price of the put option?
What if the price of the call was $5, $15?
9 / 22
Risk-neutral Valuation
Did we ever use the probability of an ”up” move, p?
What if investors are risk-neutral?
In the risk-neutral world all assets earn the same rate of return- r
Then the ”risk-neutral” probability q :
q =(1 + r)− du− d
and the price of a derivative
D0 =1
1 + r[qDu + (1− q)Dd]
note that it is identical to our no-arbitrage formula
10 / 22
Risk-neutral Valuation
we can write:
D0 =1
1 + rEQ[D1]
where EQ is expectation under the probability measure Q
The probability measure Q is called the risk-neutral measureor martingale measure
Under this measure the expected return on any asset in thebinomial model is the risk-free rate
S0 =1
1 + rEQ[S1]
11 / 22
Multi-period model
We extend the binomial setting to a finer grid
The concepts developed in the one-period model still hold
We still use two assets, the stock and the bond
The bond yields a risk free return of (1 + r) per period
The evolution of the stock price is described by the followingtree
12 / 22
Multi-period model
We will still want to construct a portfolio whose payoffreplicates the payoff of the option at the terminal date
A portfolio starting at time zero with (∆0, β0) can berearranged at any later point in time
We focus our attention on portfolios satisfying the followingcondition
(∆t+1 −∆t)St+1 + (βt+1 − βt)Bt+1 = 0
13 / 22
Multi-period model
We call this portfolio self-financing, as after time 0 they donot require any further cash flow
Self-financing in this case means
∆0Su + β0B1 = ∆uSu + βuB1
∆0Sd + β0B1 = ∆dSd + βdB1
This set of conditions imply
(∆u −∆0)Su + (βu − β0)B1 = 0
(∆d −∆0)Sd + (βd − β0)B1 = 0
We consider a European derivative with maturity at date 2
The payoff is given by some functionD2(S2) ∈ {Duu, Dud, Ddd}
14 / 22
Multi-period model
We solve the replication problem backwards, using aself-financing portfolio strategy
Duu = ∆uSuu + βuB2
Dud = ∆uSud + βuB2
Ddu = Dud = ∆dSdu + βdB2
Ddd = ∆dSdd + βdB2
From this system we can find (∆∗u, β
∗u) :
∆∗u =
Duu −Dud
Suu − Sudβ∗u =
SuuDud − SudDuu
(Suu − Sud)B2
We can find (∆∗d, β
∗d) similarly
15 / 22
Multi-period model
Obtaining the price of the derivative at time 0 is similar to theone period problem of replicating a derivative with payoffgiven by
Du = V ∗u = ∆∗
uSu + β∗uB1
Dd = V ∗d = ∆∗
dSd + β∗dB1
Then
∆∗0 =
V ∗u − V ∗
d
Suu − Sudβ∗0 =
SuV∗d − SdV ∗
u
(Su − Sd)B1
And we have solved the replication problem for any Europeanderivative
D0 = V ∗0 = ∆∗
0S0 + β∗0
Note that it is important that our strategy be self-financing.Why?
16 / 22
Example
r = 2.5%, S0 = 100, u = 1.25, d = 0.8, K = 100, T = 2
What is the price of the call option?
What if the price of the call was $20?
17 / 22
Risk-neutral valuation
We can also interpret the pricing equation as expectation
Du =1
1 + r[qDuu + (1− q)Dud] =
1
1 + rEQ[D2|S1 = Su]
Dd =1
1 + r[qDdu + (1− q)Ddd] =
1
1 + rEQ[D2|S1 = Sd]
18 / 22
Risk-neutral valuation
Then
D0 = V ∗0 =
1
1 + r[qDu + (1− q)Dd] =
1
(1 + r)2[q2Duu + (1− q)2Ddd + 2q(1− q)Dud] =
1
(1 + r)2EQ[D2]
The price of a derivative security is the expected value of thepayoff under the risk neutral measure discounted at the riskfree rate
19 / 22
Hedging
Let us go back to the one period model
Consider the problem of the seller of the call option
The seller wants to hedge his short position in the call byholding an adequate quantity of the underlying stock
If this is possible, we will have another way to determine theprice of the option
The perfectly hedged portfolio must have a return equal tothe risk-free rate
A perfect hedge ∆H requires
∆HSu − cu = ∆HSd − cd
20 / 22
Hedging
It follows that
∆H =cu − cdSu − Sd
And no arbitrage implies
(∆HS0 − c0)(1 + r) = ∆HSu − cu
Solving for c0 yields
c0 =cu − cdSu − Sd
S0 +Sucd − Sdcu
(Su − Sd)(1 + r)
Which is identical to our initial pricing formula (note that ofcourse ∆H = ∆∗)
21 / 22
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