Transcript Assicurazioni vita e mercato del risparmio gestito

Advanced methods of insurance
Lecture 3
Exotic options: tools
• Pay-off
– Digital, chooser, compound
• Barriers
– Up/Down-In/Out, parigine
• Path dependent
– Asian, look-back, ladder, shout
• Investment period
– Forward start, ratchet.
• Early exercize
– Bermuda, American
• Multivariate
– Basket, rainbow, exchange
• Foreign exchange
– Quantos, compos
Pay-off
• Chooser: at an intermediate date can be
chosen to be call or put
• Digital: fixed pay-off if a trigger condition,
i.e. underlying price above some threshold,
zero otherwise
• Compound options: underlying is an option
Chooser options
• At date  the holder of the option may be call or
put. If the strike is the same the option is called
“simple”
• At time  the chooser will ne
chooser() = max(call(S,;K,T), put(S,;K,T))
= call(S,;K,T) + max(KP(,T) – S(),0)
• To avoid arbitrage, at time t the value of the
chooser must be
chooser(t) = call(S,t;K,T) + put(S,t;KP(,T),)
Digital…
Digital CoN
1,2
1
0,8
0,6
Digital CoN
0,4
0,2
0
48
48,5
49
49,5
50
50,5
51
51,5
52
…and vertical spreads (super-replication)
Spread verticali e opzioni digitali
1.2
1
0.8
0.6
0.4
0.2
0
48
48.5
49
49.5
50
50.5
51
51.5
52
Digital call…
• Remember that a cash-or-nothing (CoN) option is
given by
Digital Call CoN = P(t,T)Q( S(T) > K)
• We know that the payoff can be approximated by
Call K - h   Call K 
Call K 
Digital Call CoN  lim

h 0
h
K
…from which
QS T   K   
1 CallK 
Pt , T  K
…and put
• Sama analysis for a digital put paying one unit of
cash if S(T)  K
Digital Put CoN = P(t,T)Q( S(T)  K)
• Same approximation
Put K   Put K - h  Put K 
Digitale Put CoN  lim

h 0
h
K
…from which
1 PutK 
QS T   K 
Pt , T  K
Barrier (quasi-path dependent)
• The option is activated (knocked-in) or knocked
(out) if the price of the underlying crosses a
barrier that can be upper (up) or lower (down)
• In case the contract is knocked out a fixed sum
can be payed (rebate)
• Parisian options are knocked-in or out if the price
remains beyond the barrier beyond a given period.
• Discrete barrier options: monitoring at discrete
time.
• Digital options with barrier: one-touch, no-touch.
Simmetry of barrier options
• It is immediate to verify that to exclude arbitrage
opportunities
Plain vanilla option =
Down(Up)-and-in(H) + Down(Up)-and-out(H)
• Every option with barrier can be represented with
– A long position in a plain vanilla option
– A short position in the symmetric barrier option
Path-dependent
• The value of the reference price or of the strike to
be used for the pay-off depends on the prices of
the underlying asset in a period of time.
• Asian: use the average of the underlying as
reference rate (average rate) or strike (average
rate).
• Lookback: the strike is set at the
maximum/minimum on a reference period
• Ladder: the strike is updated on a grid of values,
whenever the underlying crosses the value.
Asian options
• The problem of most Asian options is that they are
written on arithmetic averages.
• In the Black and Scholes model, in which prices
are log-normal, the sum of them is not lognormale.
• Evaluation techniques:
– Moment matching (Turnbull e Wakeman): the
distributon is approximated by a log-normal
distribution
– Monte Carlo: pay-offs generated for every path and
averaged
Ladder options
• In ladder options the strike is set at a level H if
the underlying during the life of the option
crosses the value H. Each of these levels H
represents a step of a stairs (ladder).
• Ladder options allow to reset the strike
accroding to movements of the market.
• In the extreme case of a continuous grid we get a
lookback.
Ladder options: evalutation
• Take a single step H.
• Assume an option with strike price K and date
T. We have that:
Ladder (K, H) =
Down(Up)-and-Out(K,H)
+ Down(Up)-and-In(H,H)
Forward start options
• Assume that at time  the strike is K = S().The value of the option at
time  will be
B & S(S(); S(),T–)
• Notice that given a constant c we have
B & S(S(); S(),T – ) =c B & S(S()/c; S()/c ,T – )
• Setting c = S()/S(t) we have
B & S(S(); S(),T–)= S() B & S(S(t); S(t),T – )/S(t)
• The value of the forward start option is equivalent to buying N = B &
S(S(t); cS(t),T–)/S(t) units of the underlying at time t. At time  we
have
B & S(S(); S(),T–) = S() N
• At time t we compute
Forward Start =exp(– r( – t) EQ[S()]N
= exp(– r( – t))S(t) exp((r –d)( – t))N
= exp(– d( – t)) B & S(S(t); S(t),T – )
Multivariate options
• Basket: the value of the underlying is computed as
a weighted average of a basket of stocks
• Rainbow: use different aggregating functions:
– Options on maximum or minimum of a basket of stocks
(option on the max/min)
– Options to exchange a financial asset with another
(exchange option),
– Options written on the price difference of assets (spread
option),
– Options with different strikes for every stock in the
basket (multi-strike).
Reverse convertible
• Period: 1° Febbraio-1° settembre 2000
• Fixed coupon 22%, paid 01/09/2000
• Repayment of principal in cash or Telecom
stocks if the following conditions occur
– On 25/08/2000 Telecom is below 16.77 Euro
– Between 28/01/2000 and 25/08/2000 the price
of Telecom stock touches 13.416 Euro
• Reverse convertible = ZCB – put
(with barrier)
Index-Linked Bond
• Period: July 31 2000 – July 31 2004
• Coupon and principal: paid at maturity
• Coupon defined as the maximum between 6% and
the average growth of end of quarter equally
weighted portfolio of : Nikkei 225, Eurostoxx 50 e
S&P 500.
• Index-Linked Bond = zero coupon + option
(Asian basket call)
Quanto structured note
• Period 13 March 2000 - 10 October 2000
• Fixed coupon 23% paid at maturity
• Principal repaid in cash at maturity if the pound
value of Vodaphone stock is not below 3.775 on
3/10/2003; alternatively Vodaphone stocks are
delivered
• Quanto structured note = zero coupon bond quanto put
Equity linked structure
•
•
•
•
Consider the following policy.
Five year maturity
Repayment of principal at maturity
Coupon paid at maturity as the maximum
between
– The appreciation of a stock market index for a
given percentage (participation rate)
– A guaranteed return
Structuring choices
• How to make the product less riskhy? If the product is considered
much too risky, the speculative content can be reduced in two
ways
– Reducing leverage
– Reducing volatility
• Risk can be reduced by increasing the strike price (the guaranteed
return) or reducing the participation rate
• Volatility can be reduced choosing an Asian option for the
investment on the index:
– Reduced risk for the investor: smoothing
– Reduced dependence on long term volatility
Crash protection
• The investment horizon of a product like this
could be considered too long. If the market
decreases by a relevant amount, the vlaue of the
option can decrease to zero, and the investor
remains locked in a low return investment.
• For this, we may consider a crash protection
clause. Under this clause, if the value falls
below a percentage h of the initial value the new
strike is set at that level.
Crash protection: evaluation
• The value of the product is determined as
ZCB + Call Ladder (S(t)/S(0), t; 1, h)
• We can isolate the value of the crash protection clause using
– The replicating portfolio of the latter option
– The symmetry of in and out options.
• We compute
ZCB + Call(S(t)/S(0), t; 1, h) +
Down-and-In(S(t)/S(0), t;h, h) – Down-and-In(S(t)/S(0), t;
1, h)
• The value of the crash protection clause is then given by the
difference between a Down-and-In with strike equal to the barrier
and that with the initial strike.
Callability/putability
• A callable bond at time  can be decomposed in terms of a
exchangeable compund option.
• Payoff example
Max[1, S(T)/S(0)] =
= 1 + max[S(T)/S(0) – 1,0]
callable at time  at par. At time  the value is
Min[1,P(,T) +Call(S()/S(0);1,T)] =
= P(,T) +Call(S()/S(0);1,T) –
max[Call(S()/S(0);1,T) – (1 – P(,T)), 0]
A different product
• Assume that investors prefer a product giving a
stream of payments indexed to equity.
• We can think of a sequence of coupons of the kind
Coupon (t + i) = max[S(t + i)/S(t + i – 1 ) – 1,0]
• This way the product produces the cash flows that
an investor would earn by investing every year on
the stock market, while being protected by losses.
Ratchet index-linked
• The new product can be represented as a
coupon bond whose flow of interest is
represented by a sequence of forward start
options, which define what is called a rachet
or cliquet option.
• This amounts, if we rule out dividends, to N
one year options.
Ratchet index-linked
• If we consider a constant dividend yield q
we have
Coupons = (1 – vN )/(1 – v)AtM Call
with
v = exp(– q )
Vega bond
• Assume a product that in N years pay a coupon
defined as
Coupon = max[0, D + imin(S(t+i)/S(t+i–1) – 1,0)]
• In other terms, the coupon is given by an initial
endowment D, expressed in percentage of the
initial principal, from which negative movements
of the market are subtracted.
Vega bond
• Rewrite the pay-off as,
Coupon = max[0,D – imax(1 – S(t+i)/S(t+i–
1),0)]
• Interest payment is a put option written on a
ratchet put.
Multivariatedigital notes
(Altiplano)
• Assume a coupon defined (reset date) and paid at time tj.
• Assume a basket of n stocks, whose prices are Sn(tj).
• Denote Sn(t0) the reference prices, typically taken at the
beginning of the contracts and used as strike price.
• Denote Ij indicator function taking value 1 if Sn(tj)/Sn(t0) > 1
and 0 otherwise for all the securities.
• The coupon is a multivariate option, and, given the coupon rate
c
Cedola  c*I j
Altiplano with memory
• Assume a coupon defined (reset date) and paid at time
tj, and a sequence of dates {t0,t1,t2,…,tj – 1}.
• Assumiame a basket of n = 1,2,…N assets, whose
prices are Sn(ti).
• Denote B a barrier and Ii an indicator function taking
value 1 if Sn(ti)/ S(t0) > B and zero otherwise.
• The memory feature implies that the first time ti when
the indicator function is taking value 1 all coupons up
to ti are paid.
Everest
• Assume a coupon defined and paid at time T.
• Assume a basket of n = 1,2,…N bonds, whose prices are
Sn(T).
• Denote Sn(t0) the reference prices, typically recoorded at
the origin of the contract and used as strike prices
• The coupon of an Everest is
max[min(Sn(T)/Sn(0),1+k] =
= (1 + k) + max[min(Sn(T)/Sn(0) – (1+k),0]
with n = 1,2,…,N and a guaranteed return equal to k.
Equity-linked bond
• Assume a coupon defined and paid at time T.
• Assume a basket n = 1,2,…N stocks, whose prices are
Sn(T).
• Denote Sn(t0) the reference prices, typically recorded
at the origin of the contract, and used as strike prices.
• The coupon of the basket option is
max[Average(Sn(T)/Sn(0),1+k] =
= (1 + k) + max[Average(Sn(T)/Sn(0) – (1+k),0]
with n = 1,2,…,N and minimum guaranteed return k.
Long/short correlation
• For a multivariate option it is crucial to
determine the sign of exposure to changes
in correlation.
• The sign of the exposure to correlation is
linked to the presence of AND or OR in the
contract.
• Example: a call on min is long correlation,
while a a call on max is short.