Chapter 5: The binomial model Flashcards
(46 cards)
Arbitrage in one period model with 2 commodities
• If we hold x units of cash at time 0, they become worth x(1 + r) at time 1.
At time t =0, a single unit of stock is worth s units of cash. At time 1, the value of a unit of stock is S_1 =
{sd with probability p_d
{su with probability p_u
Where p_d + p_u =1
And d less than u
If we hold y units of stock (worth ys) at time 0 they become worth yS_1 at time 1
- x and y can be negative =borrowed
PORTFOLIO
Recall also that we use the term portfolio for the amount of cash/stock that we hold at some time.
A portfolio is a pair h = (x,y) ∈ R2, where x is the amount of cash and y is the number of (units of) of stock.
DEF 5.1 value process /price process
Definition 5.1.1 The value process or price process of the portfolio h = (x, y) is the process V h given by
V_0^h = x + y s
V_1^h =x(1+r)+yS_1.
(S_1 is a RV)
DEF 5.2.2 portfolio has ARBITRAGE POSSIBILITY
A portfolio h=(x,y) is an ARBITRAGE POSSIBILITY
V_0^h = 0
P[V_1^h ≥ 0] = 1.
P[V_1^h > 0] > 0.
We say that a market is arbitrage free if there do not exist any arbitrage possibilities.
- initially h is worth nothing e.g neg value x or y, cancel out
- h doesn’t lose money
- h might make money on the initial value
PROP 5.1.3 arbitrage free one period market
Proposition 5.1.3 The one-period market is arbitrage free if and only if d < 1 + r < u.
Sometimes cash outperforms stock lower limit
d less than
Sometimes stock outperforms cash upper limit
bigger than u
Proposition 5.1.3 The one-period market is arbitrage free if and only if d < 1 + r < u.
PROOF
Imply: since d is less than y if this condition fails then
Either: 1) if r less than or equal to d less than u
Choose h=(-S,1)
V_0 ^h = -S_T +1(S) =0
2) d less than u less than or equal to 1+r
V_0^h=S+(-1)S=0
choose n=(s,-1)
V^h_1= s(1+r)-S_1
(S_1 is either Su or Sd)
bigger thn or equal to S(1+r)-Su
hence P[V^h_1 ≥ 0] = 1. Further, with probability pu > 0 we have S1 = su, which means
V_h^1 > s(−(1 + r) + d) ≥ 0. Hence P[V^h_1 > 0] > 0. Thus, h is an arbitrage possibility.
If 0 < d < u ≤ 1 + r then we use the portfolio h’ = (s, −1), which has V^h’_0 = 0 and
V^h’_1 = s(1 + r) − S1 ≥ s(1 + r − u) ≥ 0,
hence P[V^h’_1 ≥ 0] = 1. Further, with probability p_d > 0 we have S_1 = sd, which means
V^h’_1 > s(−(1 + r) + u) ≥ 0. Hence P[V^h’_1 > 0] > 0. Thus, h’ is also an arbitrage possibility.
Remark 5.1.4 In both cases, at time 0 we borrow whichever commodity (cash or stock) will
grow slowest in value, immediately sell it and use the proceeds to buy the other, which we know will grow faster in value. Then we wait; at time 1 we own the commodity has grown fastest in value, so we sell it, repay our debt and have some profit left over.
(⇐) : Now, assume that d < 1 + r < u. We need to show that no arbitrage is possible. To
do so, we will show that if a portfolio has V^h_0 = 0 and V^h_1 ≥ 0 then it also has V^h_1 = 0.
So, let h = (x, y) be a portfolio such that V^h_0 = 0 and V^h_1 ≥ 0. We have
V^h_0 = x + ys = 0.
The value of h at time 1 is
V^h_1 = x(1 + r) + ysZ.
Using that x = −ys, we have
V^h_1 =
{ys(u − (1 + r)) if Z = u,
{ys(d − (1 + r)) if Z = d.
Since P[V^h_1 ≥ 0] = 1 this means that both (a) ys(u − (1 + r)) ≥ 0 and (b) ys(d − (1 + r)) ≥ 0.
If y < 0 then we contradict (a) because 1 + r < u. If y > 0 then we contradict (b) because
d < 1 + r. So the only option left is that y = 0, in which case V^h_0 = V^h_1 = 0.
CONDITION FOR ONE PERIOD MODEL TO BE FREE OF ARBITRAGE
d less than 1+r less than u
introduce q_u and q_d
equiv cond
there exists q_u, q_d ∈ (0, 1) such that both
q_u + q_d = 1 and
1 + r = uq_u + dq_d
*1+r is weighted average of d and u
*solve q_u
=((1+r)-d)/(u-d)
q_d
=(u-(1+r))/(u-d)
IN WORLD Q, RISK-NEUTRAL WORLD we define q_u and q_d
RISK_NEUTRAL PROBABILITIES
Q[S_1=sd]=q_d
Q[S_1=su]=q_u
S_1=
{su w.p q_u
{sd w.p q_d
IN WORLD Q, RISK-NEUTRAL WORLD
EXPECTATIONS USING PROB MEASURE Q
PROB MEASURE P E^P
PROB MEASURE Q E^Q
Expectation does correctly calculate the value (i.e. price) of a single unit of
stock by taking an expectation. The point is that we (1) use E^Q rather than E^P and (2) then
discount according to the interest rate for DISCOUNTED STOCK PRICE
IN WORLD Q, RISK-NEUTRAL WORLD
(1/(1+r))E^Q[S_1]
S_0 and S_1 relation
(1/(1+r))E^Q[S_1] = (1/(1+r))(suQ[S_1=su] +sdQ[S_1=sd)
=
(1/(1+r))(s)(uq_u + dq_d)
=s
The price of the stock at time 0 is S_0=s
The price of stock at time 0 is S_0=s
we have shown price S_1 of unit of stock at time 1 satisfies
S_0=
(1/(1+r))E^Q[S_1]
DISCOUNTED STOCK PRICE
S_0=
(1/(1+r))E^Q[S_1]
This is a formula that is very well known to economists. It gives the stock price today (t = 0) as the expectation under Q of the stock price tomorrow (t = 1), discounted by the rate 1 + r at which it would earn interest.
DEF 5.2.1 CONTINGENT CLAIM
A contingent claim is any random variable of the form X = Φ(S_1), where Φ:R to R is a deterministic function.
“PAYOFF OF THE CONTRACT” dep on price of stock at time 1
FORWARD CONTRACT
example of a contingent claim
wn as the contract function. One example of a contingent
claim is a forward contract, in which the holder promises to buy a unit of stock at time 1 for
a fixed price K, known as the strike price.
In this case the contingent claim would be
Φ(S_1) = S1 − K,
the value of a unit of stock at time 1 minus the price paid for it
EG 5.2.2 EUROPEAN CALL OPTION
its holder the right (but not the obligation) to
buy, at time 1, a single unit of stock for a fixed price K that is agreed at time 0. As for futures,
K is known as the strike price.
Value to the holder at time 1:
Suppose you hold a European call option at time 1
*if S_1 bigger than K
we could exercise our
right to buy a unit of stock at price K, immediately sell the stock for S1 and consequently earn S1 − K > 0 in cash.
*if S_1 K, Alternatively if S1 ≤ K then our option is worthless so don’t exercise
The contingent claim assoc to contract has 2 cases:
we consider when sd less than K less than su
In this case, the contingent claim for our European call option is
Φ(S1) =
{su − K if S_1 = su
{0 if S_1 = sd
In the first case our right to buy is worth exercising; in the second case it is not. A simpler way to write this contingent claim is
Φ(S1) = max(S1 − K, 0)
DEF 5.2.3
REPLICATING PORTFOLIO
We say that a portfolio h is a replicating portfolio or hedging portfolio for the contingent claim Φ(S1) if V^h_1 = Φ(S1).
Finding Φ(S_1)
Finding value Φ(S_1) at time 0 look for a gen way to construct trading strategies ie construct portfolio h=(x,y) st V_1^h= Φ(S_1) and by no arbitrage V_0^h=value of contract at time 0
If a contingent claim Φ(S1) has a replicating portfolio h, then the price of the Φ(S1) at
time 0 must be equal to the value of h at time 0.
A market is COMPLETE
A market is COMPLETE if every CONTINGENT CLAIM can be REPLICATED
If the market is complete we can price
EXAMPLE: 5.2.4
Suppose that s = 1, d =1/2, u = 2 and r =1/4, and that we are looking at the contingent claim
Φ(S_1) =
{1 if S1 = su,
{0 if S1 = sd
We wish to replicate Φ(S_1)
Representing as a tree
[price of contingent claim]
(stock price)
(2) [1] (1) < (0. 5) [0]
we want to replicate Φ(S_1)
ie want portfolio h=(x,y) st V_1^h= Φ(S_1)
*stock goes up
(1 + 1/4)x + 2y =1
*stock goes down
(1 + 1/4)x + 0.5y = 0
so x=-4/15 and y=2/3
Hence the price of our contingent claim Φ(S_1) at time 0 is V_0^h= (-4/15)+(1)(2/3)=2/5
ie we would pay 2/5 cash to hold this contract
IN GENERAL
take an arbitrary contingent claim Φ(S1) and see if we can replicate it.
ie finding a portfolio h such that the value V^h_1 of the portfolio at time 1 is Φ(S1):
V^h_1 =
{Φ(su) if S1 = su,
{Φ(sd) if S1 = sd.
if we write h = (x, y) then we need
(1 + r)x + suy = Φ(su)
(1 + r)x + sdy = Φ(sd),
which is just a pair of linear equations to solve for (x, y). In matrix form,
[1 + r su] [1 + r sd] * [x] [y] = [Φ(su)] [Φ(sd)]
A unique solution exists when the determinant is non-zero, that is when (1+r)u−(1+r)d ≠ 0, or
equivalently when u ≠ d. So, in this case, we can find a replicating portfolio for any contingent claim.
It is an assumption of the model that d ≤ u, so we have that our one-period model is complete if d < u. Therefore prop 5.2.5
PROPOSITION 5.2.5 complete one-period model
If the one-period model is arbitrage free then it is complete.
*when model was arbitrage free we had condition d less than 1+r less than u
RISK_NEUTRAL VALUATION FORMULA’
DERIVATION
solving linear equations for replicating portfolio
to get the price of Φ(S1) at time 0
V^h_0= (1/(1+r))E^Q[Φ(S1) ]
*to find the price of Φ(S1) at 0 we take exp Q and discount one time step of interest by dividing by 1+r
- special case of previous wgere Φ(S1) =S_1 is 1 i.e. pricing the contingent claim corresponding to being given a single unit of stock
x =(1/(1 + r))[uΦ(sd) − dΦ(su)] /(u− d)
y = (1/s)(Φ(su) − Φ(sd))/(u − d)
which tells us that the price of Φ(S1) at time 0 should be
V^h_0 = x + sy
=(1/(1 + r)) [((1 + r)−d)/(u−d)Φ(su) + (u − (1 + r))/(u − d)Φ(sd)]
=(1/(1 + r))(q_uΦ(su) + q_dΦ(sd))
=(1/(1 + r)) E^Q[Φ(S_1)].
PROPOSITION 5.2.6
For a given contingent claim
Then the (unique) replicating portfolio h = (x, y) for Φ(S1) is
Let Φ(S1) be a contingent claim. Then the (unique) replicating portfolio
h = (x, y) for Φ(S1) can be found by solving
V^h_1 = Φ(S1), which can be written as a pair of
linear equations:
(1 + r)x + suy = Φ(su)
(1 + r)x + sdy = Φ(sd).
The general solution is
x =(1/(1 + r))[uΦ(sd) − dΦ(su)] /(u− d)
y = (1/s)(Φ(su) − Φ(sd))/(u − d)
The value (and hence, the price) of Φ(S1) at time 0 is V^h_0 = (1/(1 + r))E^Q[Φ(S1)]
EXAMPLE 5.2.7
European call option value at time 0
For EU call option with strike price K ∈ (sd, su). We found contingent claim Φ(S_T)= max(S_T-K,0)
By prop 5.2.6 replicating portfolio h=(x,y) found by solving V_1^h= Φ(S_1)
(1+r)x + suy = su - K
(1+r)x + sdy = 0
SOL
x = [sd(K−su)]/
[(1+r)(su−sd)],
y = [su−K]/[su−sd] .
By the second part of prop 5.2.6
value of EU call option at time 0 is
1/(1+ r) E^Q[Φ(S1)] =
1/(1 + r)(qu(su − K) + qd(0))
=
(1/(1 + r))[(1 + r) − d]/(u − d)(su − K)
financial derivative
A contract that specifies that buying/selling will occur, now or in the future, is known as a financial derivative, or simply derivative.
