Transcript Assicurazioni vita e mercato del risparmio gestito

Financial Products & Markets
Lecture 2
Forward contracts
• The long party in a forward contract defines
at time t the price F at which a unit of the
security S will be purchased for delivery at
time T
• At time T the value of the contract for the
long party will be S(T) - F
Contratti forward: ingredienti
•
•
•
•
Date of the deal 16/03/2005
Spot price ENEL 7,269
Discount factor 16/05/2005: 99,66
Enel forward price:
7,269/0,9966 = 7,293799 ≈ 7,2938
• Long position (purchase) in a forward for 10000 Enel
forward for delivery on May 16 2005 for price 7,2938.
• Value of the forward contract at expiration date
16/05/2005
10000 ENEL(15/09/2005) – 72938
Derivatives and leverage
• Derivative contracts imply leverage
• Alternative 1
Forward 10000 ENEL at 7,2938 €, 2 months
2 m. later: Value 10000 ENEL – 72938
• Alternative 2
Long 10000 ENEL spot with debt 72938 for
repayment in 2 months.
2 m. later: Value 10000 ENEL – 72938
Syntetic forward
• A long/short position in a linear contract (forward)
is equivalent to a position of the same sign and
same amount and a debt/credit position for an
amount equal to the forward price
• In our case we have that, at the origin of the deal,
16/03/2005, the value of the forward contract
CF(t) is
CF(t) = 10000 x 7,269 – 0,9966 x 72938 ≈ 0
• Notice tha at the origin of the contract the forward
contract is worth zero, and the price is set at the
forward price.
Futures
• Asssume that a forward contract is closed and
settled every day (mark-to-market)
• You have obtained a futures market
– Margin (buyers and sellers post a deposit to guarantee
their performance on the contract
– Prices are marked-to-market every day and profits and
losses are settled on the margin (margin call)
– Products traded are standardized, and in some cases are
adjusted for “grade”. The seller has a “delivery option”
(if the contract is for “physical delivery”)
A binomial model for options
• The simplest way to reprsent risk is to work with a
finite set of scenarios.
• Risky assets take different values in different
states of the world, while the risk free asset take
the same value in all states.
• In the simplest example, that is the binomial
model, we assume two dates and two states. We
assume that the first date is the valuation date,
while the second date is the maturity date of the
option.
A binomiam model of a call
option
Time t
Time T
Price
H
L
Y(t)
Y(H)
Y(L)
CALL(Y,t;T,K) Max(Y(H)-K,0) Max(Y(L)-K,0)
v(t,T)
1
1
Arbitrage relationships
• Assume the a portfolio:
Long a position in one call option C
Short position in  units of the underlying Y
• Set
 =[max(Y(H) –K,0)–max(Y(L)–K,0))]/(Y(H)–Y(L))
• At time T the portfolio is riskless
Max(Y(H) – K,0) –  Y(H) = W
Max(Y(L) – K, 0) –  Y(L) = W
• Notice that W can also be recovered.
Arbitrage and price
• If the portfolio C(T) –  Y (T) = W is riskless its return
from t to T must be equal to that of the risk-free asset.
• So, it must be
C(t) = Y(t) + v(t,T)W
• In general, whatever derivative can be expressed as a
position in the underlying asset and in the risk free asset.
• A peculiarity of derivative contracts is that these positions
are of opposite sign, so there is a long position in one asset
funded by a short position on the other (leverage).
Arbitrage relationships
• If in the arbitrage relationship
CALL(t) = Y(t) + P(t,T)W
• …we substitute
 = (CALL(H) – CALL(L))/(Y(H) – Y(L))
W= – (CALL(H) Y(L) – CALL(L) Y(H))/(Y(H) – Y(L))
and collect terms X(H) and X(L)
CALL(t) = v(t,T)[Q(H)C(H) +Q(L)C(L)]
Q(H) = (Y(t)/v(t,T) – Y(L))/(Y(H) – Y(L))
Q(L) = (Y(H) - Y(t)/v(t,T))/(Y(H) – Y(L))
Risk-adjusted measure
•
•
•
•
•
•
•
•
Notice that
Y(L) < Y(t)/P(t,T) < Y(H)  Q(H), Q(L) > 0
Q(H) + Q(L) = 1
Q is a probability measure and Call(t) is
Call(t) = v(t,T) E Q [Call(T)]
It may also be easily verified that
Y(t) = P(t,T) E Q[Y(T)]
Notice: measure Q represents no-arbitrage (no link
to forecast)
Equivalent Martingale Measure
•
•
•
•
From v(T,T) =1
CALL(t) /v(t,T) = E Q[CALL(T)/v(T,T)]
Y(t) /v(t,T) = E Q[Y(T)/v(T,T)]
If we define the prices of forward prices y(t) =
Y(t) /v(t,T) and call(t) = CALL(t) /v(t,T), these are
endowed with the martingale property.
• For this reason the risk neutral measure is called
the equivalent martingale measure (EMM)
Fundamental theorem of finance
• Equivalent Martingale Measure: no arbitrage opportunities
exist if and only if there exists a probability measure
equivalent to the historical one such that under that
measure the prices of all the risky assets, measured using
the risk free asset as numeraire, are martingale.
• Risk-neutral measure: under the martingale measure every
financial product X is expected to yield lthe same return as
the risk free asset.
E Q[X(T)/X(t) - 1] = 1/v(t,T) – 1 = rf
Extension to multiple periods
• Assume that in every period the price of assets
could move only up or down to specific values
(binomial model)
• Backward induction: starting from maturity of the
derivative contract we can move back building the
replicating portfolio at the earlier step, and so on
until we reach the current value of the replicating
portfolio.
Two-period example
Y(HH)
Y(H)
C(H) = HY(H) + BWH
Y(HL)
Y(0)
C(0) = Y(0) + BW
Y(LH)
Y(L)
C(L) = LY(L) + BWL
Y(LL)
Enel(H) = 7,5
∆(H) = 1, W(H) = – 7,4
Call(H) = 1x7,5 – v(t,,T)x 7,4
Enel(HH) = 7,7
=7,5 – 0,99x7,4 = 0,174
Call(HH) = 0,3
Enel(t) = 7,269
∆ = 0,435 W = – 3,0855
Call(t) = 0,435x7,269 – 0,9966x3,0855
Enel(HL) = 7,4
Call(HL) = 0
Enel(LH) = 7,3
= 0,084016
Enel(L) = 7,1
Call(LH) = 0
∆(L) = 0, W(L) = 0
Enel(LL) = 7,0
Call(H) = 0
Call(LL) = 0
Self-financing portfolios
• From the definition of replicating portfolio
CALL(H) = Y(H) + W = HY(H) + BWH
CALL(L) = Y(L) + W = LY(L) + BWL
where B = 1/v is the value of one unit of risk free
asset invested for one period.
• This feature is called self-financing property: once
the replicating portfolio is built, it can be
rebalanced at no cost, and with no dividends.
Bushy trees/Recombining trees
• After n periods (steps) the tree has 2n nodes
(states). A tree with 100 steps generates
1267650600228230000000000000000 nodes
• Since this kind of tree raises computational
problems, often we assume that paths with the
same numbers of increases or decreases lead to the
same node, no matter what the sequence of
changes is. These are called recombining trees or
lattices.
• After 100 steps a recombining tree has 101 nodes
Evaluation of measure Q on a lattice
Enel(HH) = 7,7
Enel(H) = 7,5
Q = 48,4497%
QH = [7,5/0,99 – 7,3]/[7,7 – 7,3]
Enel(HL) = 7,3
Enel = 7,269
QL = [7,1/0,99 – 7,0]/[7,3 – 7,0]
Enel(L) = 7,1
Enel(LL) = 7,0
Again on arbitrage: APT
• Statistical assumptions
– Holding period returns of every financial asset i from
time t to time T are generated by
ri = ai + bi f , con E(f) = 0, Var(f) = 1
• Financial assumptions
– There exists a risk-free asset that from t to T yields the
rate of return r
– No frictions or transaction costs
– No arbitrage opportunity
APT one factor model
• Build a portfolio with two bonds i and j to
eliminate the dependence on f
• In practice, choose wi = bj /(bj - bi)
• The no-arbitrage assumption assume
wi ri + (1- wi ) rj = r …from which
(ai - r)/ bi = (aj - r )/ bj = 
for any i and j
Risk premium
• For every asset i it must be
E( ri ) = ai = r + bi  = rf +i 
E(Return) = risk-free + risk-premium
• Risk premium = i  where
i :risk of asset i
 : market price of risk
• Notice. Market price of risk must be the
same for all financial products.
N-factor APT model
• The natural extension of the APT model
consists in the assumption that the holding
period return of all securities be a linear
function of N risk factors. The DGP of the
returns isN
ri  ai    ij f j
j 1
   1, E  f , f   0 per k  j
E  f j   0, E f
2
j
k
j
The immunized portfolio
• Since with a single risk factor we use 2 bonds in
the immunized portfolio, it is natural to assume
that in a N factor model we use N + 1 risk
securities. The portfolio must be immunized to all
the N risk factors. This portfolio implies that, for
every risk factor j,
N 1
w 
i 1
i
ij
0
j
The no-arbitrage principle
• In oder to rule out arbitrage opportunity we
must have that
N 1
wa
i 1
i i
r
N 1
N 1
w a  w r
i i
i 1
N 1
i 1
 w a  r   0
i 1
i
i
i
In matrix form
a 1  r
 
 11
 12

 .
 1N

a2  r . . a N 1  r   1  0





 21 . .  N 1,1   2
0

  
 22 . .  N 1, 2   .    . 

  
.
. .
.  .   . 
 2 N . .  N 1, N   N 1  0
The solution
• The homogeneous linear system has non trivial
solution only if the matrix has reduced rank. This
implies that each row/column could be recovered
as linear combination of the others, so me may
write
N
ai  E ri   r   ij j
j 1
Risk premia
• The extension of model to N risk factors is immediate
• The expected return of every asset must be equal to
– The risk-free rate r
– The risl premium
• The risk premium is the inner product of
– Market prices of risk
– Sensitivity of each asset to the risk factors (factor loading)
• Notice that ruling out arbitrage opportunities requires that
the market price of risk of each risk factor be the same for
all assets.
The extension of the APT model
• A natural extension is to add a shock that is independent
of the risk factors. This represents the idiosyncatic shock
that is elimated if the portfolio is perfectly diverisfied.
N
ri  ai    ij f j   i
j 1
   1, E  f , f   0 per k  j
E    0, E     , E  , f   0
E  f j   0, E f
i
2
j
2
i
k
2
i
j
k
j
Historical probability and EMM
• From APT model we know that
E(Y(T)/Y(t) – 1) = r + Y
• …while under the risk-adjusted measure
EQ(Y(T)/Y(t) – 1) = 1/P(t,T) = r
• In a binomial model…
Y(H) - Y(L)
Y =
Y(0)
p1 - p 
…with p historical measure
The relationship between measures
• Computing the difference between measures
E(Y(T)/Y(t)) - EQ(Y(T)/Y(t)) = Y
E(Y(T)/Y(t)) - EQ(Y(T)/Y(t)) = (p-Q)(Y(H)-Y(L))/Y(t)
and the volatility under the historical measure
p-Q =  p1 - p 
• The risk adjustment is then obtained changing the
the probability measure.
Historical and implied information
• Historical information •
• “Objective” probability •
• Holding period return •
E[ri] = rf + ’i
•
• Capital budgeting
Y t  
E p Y T 
1  r  
Implied information
Risk-neutral probability
Holding period return
E[ri] = rf
Derivatives evalutation

EQ Y T 
1 r
A special tree
• Assume a tree in which
Y(t+1) can take two values Y(t)*u (state H) or Y(t)*d (state L).
u and d are the same on every node (indipendent of time and state)
u*d = 1
• The tree is recombining (or lattice), meaning that same
numbers of up and down moves lead to the same node. For
a very large number of steps the model converges to a
geometric brownian motion, used in the Black and Scholes
model.
Towards a continuous time
• Given an investment horizon h
Y(t+h) – Y(t) = rYY(t)
is the holding period return.
• The rate of return rY is a random variable
rY =  + , with   N (0,1) from which
Y(t+h) – Y(t) = Y(t) + Y(t)
• In continuous time (h0)
dY(t) =  Y(t) dt +  Y(t) dz(t)
Wiener process
• A diffusive process
z(t+h) – z(t)  N(0, h)
…is called Wiener process
• The process has continuous trajectories, but it is
not differentiable in any point.
• In continuous time:
dz(t) = lim h0 E[z(t+h) – z(t)]
• Diffusive processes: ds(t) = dt + dz(t)
Conditional probability
• Ex. ds(t) =  dt +  dz(t)
• The probability distibution at time  > t of s is
gaussian with mean ( - t) and variance 2 ( - t)
• Ex. dY(t)/Y(t) =  dt +  dz(t) represents
Geometric Brownian Motion, in our case the
instantaneous return of a risky asset.. The
instantaneous return is normally distributed, while
Y(t) is not.
Ito’s lemma
• If s(t) is a diffusive process and p = f(s,t) is a function,
then p(t) is also a diffusive process with
lim h0 E[p(t+h) – p(t)] = (f t+ s f s + ½ s2 f ss)dt
lim h0 Var[p(t+h) – p(t)] = (s f s)2dt
• Ex. From dY(t)/Y(t) =  dt +  dz(t) e f(Y,t) = log Y
we obtain…
d log Y(t) = ( - ½ 2)dt +  dz(t)
… and Y( | t) ha lognormal distribution.
Derivative pricing application
• Assume the underlying follows a GBM
dY(t) = Y(t) dt +  Y(t) dz(t)
• The value of the derivative contract C(Y,t)
follows, for Ito’s lemma
E(dC(t)) = (C t+ Y CY + ½ 2Y2 CYY)dt
Var[dC(t)] = (YCY)2dt
Application: delta hedging
• As in the discrete case, consider a portfolio
– Long one unit of derivative C
– Short  = CY units of Y
• The dynamics of Y(t) – CYY(t) is
E(dC(t) - CY dY(t)) = (C t + ½ 2Y2 CYY)dt
Var[dC(t) - CY dY(t)] = 0
Black & Scholes
• No arbitrage implies that risk free portfolio earn
the risk free rate
C t + ½ 2Y2 CYY = r(C(t) - CYY(t))
• …from which fundamental PDE
½ 2Y2 CYY + C t + rCYY(t) - rC(t) = 0
• …and the value of the derivative contract must be
the solution of the PDE under the boundary
condition
C(Y(T),T) = pay-off function
Girsanov theorem
• For Girsanov theorem, given a Wiener z(t) defined
under a probability measure P and defined a new
process
z*(t) = z(t) + dt
it is possible, under regularity conditions on the
drift term , to find a new probability measure, i.e.
Q, under which z*(t) is a Wiener process.
• Girsanov says that it is possible to change the
probability measure by changing the drift.
Girsanov theorem: application
• From APT model
dY(t)) = (r + )Ydt +  Ydz(t)
Define a new meaure Q such that
EQ(d(z(t) + dt) = 0
We can write
dY(t)) = rYdt +  Ydz*(t)
with dz*(t) = dz(t) + dt and z*(t) is a Wiener
process under probability measure Q.
Risk-neutral probability
• Under probability Q
EQ(dY(t)) = rYdt
• Apply now Ito’s lemma to get the expected return
of the derivative C
EQ(dC(t)) = (C t+ rY CY + ½ 2Y2 CYY)dt
• Notice that the fundamental PDE gives
EQ(dC(t)) = r Cdt
that confirms Q as the risk neutral measure.
Risk-neutral valuation
C(Y,t) = exp(-r(T-t)) EQ(C(Y,T))
• We obtained that also in continuous time the same
valuation rule based on computing the expected
value of the product under the risk neutral measure
and discounting with the risk free rate holds. The
EMM theorem then holds.
• This also says that PDE functions such as that
obtained in these models can be solved computing
expected values under some probability measure
(Feynman-Kac solution).
Black & Scholes model
• Assuming log-normal distribution for prices
(corresponding to GBM) we can explicitly obtain
Black & Scholes formula for call options
callY , t; K , T   Y t N d1   e T t KN d 2 
d1 


ln Y t / K      2 / 2 T  t 
 T t
d 2  d1   T  t
Put options
… and put options. Use put-call parity:
(Y +
put = v(t,T)K + call) and simmetry of standard
normal (1 – N(a) = N(– a) )
put Y , t ; K , T   callY , t ; K , T   Y t   e  T t K
 Y t N d1   1  e  T t K N d 2   1
 Y t 1  N d1   e  T t K 1  N d 2 
 Y t N  d1   e  T t KN  d 2 
Replicating portfolios
• Long call
long N(d1) underlying
debt K N(d2)
• Continuous rebalancing
• For Y going to infinity
N(d1) and N(d2) converge
to 1
• Long forward
• Loing put
short N(– d1) underlying
credit K N(– d2)
• Continuous relbalancing
• For Y goingto zero , N(–
d1) and N(– d2) congerge
to – 1
• Short forward
Interest rate derivatives
• Let us write the pricing function for a derivative
contract whose payoff is a function of the interest
rate observed at time T. Define this pay-off G(r(T),
T). The price will be
g (r(t),t) =
 T


EQ[ exp   r u du G(r(T),
 t

T)]
• Notice. The discount factor and the pay-off cannot
be factorized because they are both function of r(t)!
Term structure models
• Solving the derivative pricing equations
with condition G(r(T),T) = 1 gives the set of
zero-coupon-bond prices from which we
can compute the term structure.
• Notice that using the term structure as a risk
factor requires a model that specifies its
dynamics.
Term structure models
• Different ways of modelling the dynamics of the
term structure
• Affine factor models: the curve is determnined
endogenously based on a set of risk factors with
prescribed dynamics
• HJM: the whole curve of forward rates is
considered as a risk factor and its dynamics is
taken exogenously as a function of the volatility
curve
• LMM, SMM, & co: the discrete tenor forward rate
is modelled, using the GBM assumption.
Affine models
• Assume the term structure is completely determined from
the dynamics of the instantaneous interest rates (interest
rate intensity) r
• Assume the dynamics is:
dr = k( – r)dt + rdz
with  = 0,0.5.
• The parameter  denotes the long run mean of the interest
rate, k denotes the mean reversion force with which the
process converges toward the mean.
• The parameter  discriminates between two models. For
both model we have an affine solution, meaning that the
interest rate curve is a linear function of r.
Affine models
• For  = 0 we have the Vasicek model (and more generally
the Ornstein-Uhlembeck process).
– The transition probabilities and the steady state
probability are gaussian.
– Interest rate may be negative
• For  = 0.5 we have the CIR model (square root process)
– The transition probability is non central chi square and
the steady state probability is gamma.
– Interest rate cannot be negative. Under a condition
(Feller) zero barrier is unaccessible.
Affine models: ZCB dynamics
• In the one factor model: P(r;t,T)
 P
1  2 P 2 2 
P 
dP   k   r  
 r  dt  r dz
2
2 r
r
 r

  P Pdt   P Pdt
2

1 P
1  P 2 2 
1 P 
 P   k   r  
 r  P 
r
2
P  r
2 r
P r

Affine models: no arbitrage
• Remember the APT condition:
P – rf = (r )P
1  P
1  2 P 2 2 
1
P 
 r   rf   r  r
 k   r  
2
P  r
2 r
P
r



1  2 P 2 2 P






r

k


r


r

r
 rf P  0
2
2 r
r
Leibniz rule for Ito processes
• For Ito processes X, Y and Z we have
d(XY/Z) = (dX/X + dY/Y – dZ/Z +
cov[dX/X – dZ/Z, dY/Y – dZ/Z])XY/Z
• In particular, given processes
dg(t) = Ggdt + Ggdz
dP(t,T) = TP(t,T) dt + TP(t,T)dz
we have, for F  g/P(t,T)
dF(t) = FFdt + FFdz with
F = G – T e F = G – T – T F
Forward price dynamics…
• Given a derivative on interest rates g(t), with exercize date
T and pay-off function G(r(T),T), define forward price F(t)
 g(t)/P(t,T).
• Notice that
F(T)  G(r(T),T)/P(T,T) = G(r(T),T).
• Given risk neutral dynamics
dg(t) = rgdt + Ggdz
dP(t,T) = rP(t,T) dt + TP(t,T)dz*
we obtain the dynamics of forward price F(t)
dF(t) = -T F Fdt + FFdz*
Forward martingale measure
• A look at the forward rate dynamics suggests a change
of measure under which they could be martingale. In
fact, we can change the measure Q into a new measure
Q(T) so that if z*(t) is a Wiener process under measure
Q, then
dz**=dz* – P dt
is a Wiener process under measure Q(T)
• Notice that under the new measure Q(T)
dF(t) = FF(– P dt + dz*) = FF dz**
the forward is a martingale.
Interest rate derivatives
• Let us go back to the valuation problem
g (r(t),t) = EQ[P(r; t, T)G(r(T), T)]
• We saw that g(r(t),t)/P(t,T) is a martingale under
measure Q(T), that is
• g(r(t),t)/P(t,T) = EQ(T)[G(r(T), T)/ P(T,T)]
• Obviously, it is P(T,T) = 1, and we have
g(r(t),t) = P(t,T) EQ(T)[G(r(T), T)]
...and the factorization of the discount factor and
expected pay-off is now correct, thanks to measure
Q(T)!
Derivative pricing under FMM
• The receipe for the evaluation of interest
derivative contracts is simple
• Step 1: consider forward price, instead of spot
price, as the under underlying value
• Step 2: compute the expected pay-off under the
forward measure
• Step 3: disxcount the payoff with the zero-couponbond discount factor.
Black formula and LMM
• Given the FMM one can assume that the LIBOR
rate, or the reference rate in general, follows a lognormal process.
• This is called Libor Market Model
• Remember that the forward is defined as F(Y,t) =
Y(t)/v(t,T) and the lognormal assumption gives
Call = v(t,T)[F(Y,t)N(d1) – KN(d2)]
Put = v(t,T)[– F(Y,t)N(– d1) + KN(– d2)]
Cap/Floor
• Options on interest rates are used to set an upper
limit (cap) or lower limit (floor) to the value of an
indexed bond.
• The cap/floor is a portfolio of call/put options on
interest rates, typically defined on the set of terms
of the indexed coupons.
• Each option of the portfolio is called
caplet/floorlet. The use is
Ref. Rate – max(Ref. Rate – Strike, 0)
Ref. Rate + max(Strike – Ref. Rate, 0)
Call – Put = v(t,)(F – Strike)
• Remember the put-call parity and apply it to the
cap/floor
Caplet(strike) – Floorlet(strike)
=v(t,)[exp.coupon – strike]
=v(t,)[f(t,,T) – strike]
• This confirms that the underlying asset of caplet
and floorlet must be the forward rate and the
volatility must be that of that rate.
Cap/Floor: replicating portfolio
• Using Black formula, we obtain
Caplet = (v(t,tj) – v(t,tj+1))N(d1) – v(t,tj+1) KN(d2)
Floorlet =
(v(t,tj+1) – v(t,tj))N(– d1) + v(t,tj+1) KN(– d2)
• The formula suggests the replicating portfolio based
on long and short positions in floating and fixed
rates.
Swaption
• Swaptions are options that allow to enter a
payer or receiver swap.
• A payer swap corresponds to call option
while a receiver-swaption gives right, but
not the obligation, to enter a receiver swap
and corresponds to a put option.
Swaption
• Assume a swap at a future da tn with swap rate Rs
and a set of reset dates {tn , tn+1,……tN} with
payments {tn +1 , tn +2,……tN + 1}
• Define i = ti +1– ti the daycount and the value of
one annuity paying 1 euro
N
At; n, N     i vt , ti 1 
i n
Pay-off of a swaption…
• The pay-off of a swaption is with strike Rs gives
right to a stream of payments equal to the
difference between floating and fixe payments
i max[R(tn;n,N) - Rs ,0]
where R(tn;n,N) is the swap rate that will be
observed at the time of exercize
• The stream of pay-off is
A(tn;n,N) max[R(tn;n,N) - Rs ,0]
Black formula and SMM
• Assuming the value of the annuity as numeraire, one
could prove a martingale result:
Swaption = A(t;n,N) EA{max[R(tn;n,N) - Rs ,0]}
• Assume now that the swap rate distribution is lognormal. Then, we have again a Blac formula
Swaption = A(t;n,N) Black[S(t;n,N),K,tn,(n,N)]
Swaption: replicating portfolio
• Notice that the Black formula
Swaption =
(v(t,tj) – v(t,tN))N(d1) – iv(t,ti) KN(d2)
suggests a replicating strategy based on
– A long position on a zero coupon bond on the exercize
date of the swaption for a nominal amount N(d1)
– A short position on a zero coupon bond on the last
payment date of the swaption for a nominal amount
N(d1)
– A short position in a portfolio of bonds for the swap
maturities for an amount.