Risk neutral measure -Binomial model
Definition-Binomial model
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Stochastic process Sn in discrete time {0, 1, 2, · · · , N}.
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Sn value of the stock at time n.
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Let Y1 , · · · , YN be i.i.d random variables.
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with probability p
Yi =
0
with probability 1 − p
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0 < p < 1.
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The stock price at time n is:
S0 u
Pn
i=1
Yi
Pn
d n−
i=0
Yi
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Interest rate r . i.e. 1$ at time 1 worth (1 + r )$ at time n + 1.
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0<d <1+r <u
Derivative-Option
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A derivative on a stock is a fnancial instrument whose
value depends on the stock price.
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Example: Call option: The owner of a call option has the
right (but not the obligation) to buy at time N (strike time
)the stock at a fixed value K (srike price). The value of the
call option at time N is
(SN − K )+ = max(SN − K , 0)
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Notation:Vn the value of the option at time n.
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Assumption: VN is known for every realiztion of Y1 , · · · , YN .
Objective
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To obtain the value of the derivative at time 0.
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The "fair" price of the derivative at time 0 will be be
arbitrage free. i.e. A trador can not start with 0, trade with
the derivative and stocks and money market such that she
will not lose with probability 1 and will have a profit with
positive peobability. The theory of pricing derivative under
this assumption is called APT Arbitrage Pricing Theory.
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To construct a portfolio with stocks and money that will
hedge the derivative.
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X (t) the value of the portfolio at time t.
One period model
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The strike time is 1.
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S0 the stock price at time 0 (known).
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V1 at time 1 is known for each realization, i.e. V1 (1), and
V1 (0).
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X0 -value of a portfolio at time 0 containing stocks and
money, whose value at time 1, for each realization is the
value of the option.
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At time 0, ∆0 stocks and X0 − ∆0 S0 in money.
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Want to calculate X0 and ∆0 s.t
V1 (1) = X1 (1), V1 (0) = X1 (0)
The value of the portfolio at time 1:
X1 (1) = ∆0 S0 u + (1 + r )(X0 − ∆0 S0 )
X1 (0) = ∆0 S0 d + (1 + r )(X0 − ∆0 S0 )
But X1 (u) = V1 (u) and X1 (d) = V1 (d), thus
V1 (u) = ∆0 S0 u + (1 + r )(X0 − ∆0 S0 )
V1 (d) = ∆0 S0 d + (1 + r )(X0 − ∆0 S0 )
We got two equations with two unknowns ∆ and X0 .
V1 (u) − V1 (d)
S0 (u − d)
(1 + r − d)V1 (u) + (u − (1 + r ))V1 (d)
X0 =
(1 + r )(u − d)
∆0 =
Notice:
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probability.
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1+r −d
u−d ,
p̃(d) =
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Then p̃ defines a
V1
1+r
The change in the value of X between time 0 and 1 is due
only to the interest and the change in the stock value. Self
financing portfolio.
Example: S0 = 50, u = 2,d = 1/2, r = 1/4, call option
K = 50, T = 1, p̃(u) = p̃(d) = 1/2
X0 = 0.8 ∗ 0.5 ∗ (100 − 50) = 20. Assume price less 20
then e.g. 19.
Buy a call option.
sell 2/3 shares at price of 100/3
invest (100/3-19)=43/3 in the bank.
Stock increas at time 1 the option value is 50, money
43/3*(5/4)=215/12, -2/3 stocks at price of 200/3 over all
50+215/12-200/3=15/12.
if stock decrease you have 215/12-50/3=15/12 If price 20
X0 = Ẽ
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u−(1+r )
u−d .
Assume option price is 21. Sell option, get 21. Buy 2/3 shares
the rest invest in the bank. At time 1 you have
=1.25*(21-100/3)=-185/12. You have 2/3 stocks. If stock
increase your shares worth 200/3,your profit is
200/3-185/12-50=15/12. If decreases 50/3-185/12=15/12.
there is arbirage.
Binomial model-Multi-period case
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VN is known for evry realization of (Y1 , · · · , YN ).
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The goal is to obtain a replicate portfolio such that
XN = VN and at each time unite the value of the portfolio
will be the value of the derivative.
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At time N − 1 portfolio contains ∆N−1 stocks and
XN−1 − ∆N−1 SN−1 in money.
VN (1) − VN (0)
SN−1 (u − d)
(1 + r − d)VN (1) + (u − (1 + r ))VN (0)
=
(1 + r )(u − d)
∆N−1 =
XN−1
where VN (1) = VN (Y1 , · · · , YN−1 , 1), and
VN (0) = VN (Y1 , · · · , YN−1 , 0),
∆N−1 = ∆N−1 (Y1 , · · · , YN−1 ).
VN (1) − VN (0)
S0 (u − d)
(1 + r − d)VN (1) + (u − (1 + r ))Vn (0)
=
(1 + r )(u − d)
∆N−1 =
XN−1
=
1
ẼN−1 [VN ]
1+r
Can continue in the same manner:
VN−1 (1) − VN−1 (0)
SN−2 (u − d)
(1 + r − d)VN−1 (1) + (u − (1 + r ))Vn−1 (0)
XN−2 =
(1 + r )(u − d)
1
1 2
=
) ẼN−2 [VN ]
ẼN−2 [VN−1 ] = (
1+r
1+r
∆N−2 =
VN−1 (1)Vn−1 (Y1 , · · · , Yn−2 , 1), VN−1 (0)Vn−1 (Y1 , · · · , Yn−2 , 0)
ẼN−2 [VN−1 ] = Ẽ[VN−1 |Y1 , · · · , YN−2 ] = ẼN−2 [VN−1 ] =
Ẽ[VN−1 |FN−2 ]
Lemma
1+r −d
u−d ,
1
value St∗ = (1+r
S
)t t
Under the risk neutral measure: P̃(Yi = 1) =
)
P̃(Yi = 0) = u−(1+r
the discounted stock
u−d
a martingale
Proof.
Clearly St∗ is Ft measurable and since it takes only finite
number of values it has a finite mean. Let St∗ = St /(1 + r )t .
∗ |F ) = u S ∗ 1+r −d + d S ∗ u−(1+r ) = S ∗
Ẽ(St+1
t
t
1+r t u−d
1+r t
u−d
Thus E[St∗ ] = S0 , and E[St |Ft−1 ] = (1 + r )St−1
is
Lemma
Let Xt be a self financing portfolio. Under the risk neutral
u−(1+r )
−d
measure: P̃(Yi = 1) = 1+r
the
u−d , P̃(Yi = 0) =
u−d
discounted portfolio value
X ∗ (t) =
X (t)
(1 + r )t
is a matingale
Proof
Ẽ[X ∗ (n + 1)|Fn ]
1
u − (1 + r )
1+r −d
=
+d
)
∆n Sn (u
u−d
u−d
(1 + r )n+1
+(1 + r )(X (n) − ∆n Sn )) = X ∗ (n)
Lemma
Consider the N-periods Binomial modle with the risk neutral
measure defined above. Let VN = VN (Y1 , · · · , Yn ) be the value
of the derivative at time N, and define inductively backwords
Vn (Y1 , · · · , Yn ) =
1
1+r [p̃Vn+1 (Y1 , · · · , Yn , 1) + q̃Vn+1 (Y1 , · · · , Yn , 0)]. Consider a
portfolio with money and stocks such that
∆n (Y1 , · · · , Yn ) =
Vn+1 (Y1 , · · · , Yn , 1) − Vn+1 (Y1 , · · · , Yn , 0)
Sn (Y1 , · · · , Yn )(u − d)
and X0 = V0 , and Xn is defined inductively as follows:
Xn+1 (Y1 , · · · , Yn , Yn+1 ) = ∆n (Y1 , · · · , Yn )Sn+1 (Y1 , · · · , Yn , Yn+1 )
+(1 + r ) (Xn (Y1 , · · · , Yn ) − ∆n (Y1 , · · · , Yn )Sn (Y1 , · · · , Yn ))
Then XN = VN .
The proof is by induction. Assumme that Vj = Xj for
j = 0, · · · , n, and denote Vn+1 (Y1 , · · · , Yn+1 ) = Vn+1 (Yn+1 ),
similarily, Xn+1 (Y1 , · · · , Yn+1 ) = Xn+1 (Yn+1 ) Let Zn = u if
Yn = 1 and d otherwise ∆n (Y1 , · · · , Yn ) = ∆n
Xn+1 (Y1 , · · · , Yn , Yn+1 ) = ∆n Sn Zn+1 + (1 + r )(Xn − ∆n Sn )
Vn+1 (u) − Vn+1 (d)
Zn+1
=
(u − d)
(1 + r − d)Vn+1 (u) + (u − 1 − r )Vn+1 (d)
+(1 + r )
(1 + r )(u − d)
Vn+1 (u) − Vn+1 (d)
−
(u − d)
Vn+1 (0)
Vn+1 (1)
(Zn+1 − d) +
(−Zn+1 + u)
=
u−d
u−d
Thus :
If Yn+1 = 1 we get Vn+1 (1) and otherwise Vn+1 (0)
Notice that by the tower property of the conditional expectation:
Vn =
1
Ẽn [VN ]
(1 + r )N−n
The last lemma states that V0 is the right price for the option:
Theorem
The price V0 =
the option .
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Ẽ[VN ]
(1+r )N
is the unique arbitrage free price for
Proof
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Assume that the price c0 > V0 .Then sell the option and
buy the portfolio at price V0 and put c0 − V0 in bank. At
time N under any scnario your portfolio cover the option
obligations and you will have extra (c0 − V0 )(1 + r )N in the
bank. ]
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Similarly, if c0 < V0 : buy the option pay c0 , and sell the
portfolio. At time N the option value will cover the potfolio
obligation and you will have (V0 − c0 )(1 + r )N in the bank.
Show that if c0 = V0 there is no arbitrage opportunity. Consider
any self financing porfolio in stocks and money, Xt , and
consider a position γ0 in options such that
W0= X0 + γ0 V0 = 0
and assume that WT = XT + γ0 VT ≥ 0 Since self financing
portfolio is a martingale we get that
1
1
Ẽ[X (T )] + γ0
Ẽ[V (T )] = X (0) + γ0 V0 = 0
N
(1 + r )
(1 + r )N
Thus
P̃(W (N) > 0) = 0
Since we assumemd that 0 < p < 1 and 0 < d < 1 + r < u the
risk neutral measure is equivalent to the real measure, we have
that there is no arbirage (with probability 1) also with respect to
the real measure.