Finance Products and Markets

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Finance Products and Markets

• Lecture 3

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Menu

• Structured finance problems and products
• Exotic options
• Univariate structured products
• Long/short volatility
• Multivariate structured products
• Long/short correlation

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Structured finance the questions

• A structured finance product is a security that
  includes one or more derivative contracts.
• The first question is whether the derivative
  contract is in the repayment plan or in the
  coupon plan
• The second question is who is writing the
  derivative, and who is buying it
• The third question is the sign of exposure to i)
  underlying ii) volatility iii) correlation

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A general rule of thumb

• If in the prospect of the security there is a
  clause stating that at some future time the value
  of a cash flow will be max(y, K), with K a given
  value, then that cash flow has an option in
  favour of the investor (the receiver). In this
  case the option increases the value of the
  security
• If the there is a clause stating that a cash flow
  will be min(y, K), then that cash flow has an
  option in favor of the issuer (the payer). In
  this case the value of the option decreases the
  value of the security.

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Decomposition via put-call parity

• Option in favour of the receiver (investor)
• Max( y, K) y max(K y, 0)
• Max ( y, K) K max(y K, 0)
• Long underlying, long volatility
• Option in favour of the payer (issuer)
• Min( y, K) y max(y K, 0)
• Min ( y, K) K max(K y, 0)
• Long underlying, short volatility

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Putable bonds

• Traditional example is the CTO (certificati del
  Tesoro con opzione) Italian government
  securities issued in the 80s with 6 year maturity
  retractable at 3 years
• Three years after issuance (time ?) the value of
  the bond was
• CTO(?,Tc) max(P(?,Tc), 1)
• P(?,Tc) max(1 P(?,Tc),0)
• 1 max(P(?,Tc) 1,0)
• At time t, the value can be seen as a bond
  expiring at time T and a put option with exercise
  at ? (retractable) or a bond expiring a time ?
  and a call on a three year bond (extendible).

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Callable bonds

• Callable bonds are so traditional and well known
  that they are not considered structured finance
  products
• At the call date (time ?) the value of the bond
  is
• CALLABLE(?,Tc) min(P(?,Tc), 1)
• P(?,Tc) max(P(?,Tc) 1,0)
• 1 max(1 P(?,Tc),0)
• At time t, the value can be seen as a bond
  expiring at time T minus a call option with
  exercise at ? (retractable) or a bond expiring a
  time ? and the sale of o put option on a bond
  expiring at T (extendible).

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Bermudan clauses

• In many callable bonds the callability clause is
  bermudan, meaning that it can be exercised on a
  set of given dates.
• Bermudan options are the intermediate case
  between European options (exercize at maturity)
  and American option (exercise by the date of
  maturity).
• Typical example corporate bond with 15 year
  maturity callable every six months a10 years
  after issuance.

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Convertibles

• In convertible bonds the repayment of the
  principal can be done in terms of cash or stocks
  (equity).
• In convertibles the choice of repayment is with
  the investor.
• In reverse convertibles the choice of repayment
  is with the issuer.

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Convertible

• The repayment plan can be decomposed as
• max(100, nS(T))
• 100 n max(S(T) 100/n, 0)
• The product includes n call options on the
  underlying with strike 100/n.
• In many cases, the options involved are endowed
  with the Bermudan clause.

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Example

• The underlying ENEL
• Number of stocks n 12,5
• Strike n 100/12,5 8
• Repayment
• max(100, 12,5 Enel(T))
• 100 12,5 max(Enel(T) 8, 0)
• The value of the bond includes 12,5 call call
  options with strike at 8 euro

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Reverse convertible

• The repayment plan
• min(100, nS(T))
• 100 n max(100/n S(T), 0)
• Il product includes a short position in n put
  options with strike 100/n.
• In many cases includes a very high coupon to
  attract investors and a barrier.

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Example

• The underlying ENEL
• Number of stocks n 12,5
• Strike n 100/12,5 8
• Repayment
• min(100, 12,5 Enel(T))
• 100 12,5 max(8 Enel(T), 0)
• The value of the bond includes a short position
  in 12,5 options with strike at 8 euros

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Parallel interview

• Convertibile
• Call option
• To the investitor
• Long call
• Long underlying
• Long gamma
• Long vol

• Reverse convertible
• Put option
• To the issuer
• Short put
• Long underlying
• Short gamma
• Short vol

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CoCos (Contingent Convertible)

• CoCos (contingent convertible) convertible bonds
  for which the investor can choose the payment
  only if the price has grown above a barrier
  before the maturity.
• CoCos for banks convertible bonds issued by
  banks that can be converted into equity if the
  RWA (risk weighted assets) are falling below a
  given level. This kind of bonds is allowed as
  regulatory capital.

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Reverse convertible

• Period 1 Feb-1 Sept 2000
• Coupon 22, paid 01/09/2000
• Repayment of principal in cash or in Telecom
  stocks if two conditions apply
• On 25/08/2000 Telecom stock quotes below 16.77
  Euros
• Btw 28/01/2000 and 25/08/2000 the price has
  reached Telecom the threshold of 13.416 Euros
• Reverse convertible ZCB put (with barrier)

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Barrier options

• Barrier options include a threshold, so that the
  final value of the optiion depends on whether or
  not the underlying asset (or some other
  underlying) has reached that value (called
  barrier) during the lifetime of the option (or in
  subperiod of it)
• Barrier options can be divided into
• Up barrier
• Down barrier
• Knock-in barrier
• Knock-out barrier

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Barrier options Knock-in
Down-and-in option The option is activated
Up-and-in option The option is acticvated
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Barrier option Knock-out
Down-and-out option The option disappears
Up-and-out option The option disappears
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Parisian, parasian co

• In order to avoid market abuse, barrier options
  can be made more difficult to manipulate.
• Parisian options are activated if the price of
  the underlying remains below (or above) the
  barrier for more than a given period time without
  interruption or cumulated (cumulative parisian)
• Alternatively, the barrier can be compared with
  average values instead of point in time values
  (parasian).

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Example Parisian up
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Equity linked note

• Assume a structure like this
• Investment with guarantee of the principal for 5
  years
• Coupon paid at maturity, linked to the
  performance of a stock index
• Questions
• Who would buy it? And which options to include?
• What is the value of the product, both in the
  host bond and the derivative part
• What are the risk exposures? And how can it be
  handled?

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Alternative investments

• The product allows to invest in the stock market
  over a long time horizon. The perspectives of
  profit are lined to i) market and ii) volatility
• Alternatives
• Fund management with guarantee on the principal
  (CPPI)
• Options and warrants
• Long term options (private banking)
• Dynamic management with ETF or futures.

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Structuring choices

• The degree of risk of the equity linked note can
  be altered in two ways.
• Increase of the strike
• Reduction of volatility
• The change of strike can be obtained in two ways
  
• Including a guaranteed return rg
• Including a participation rate ?
• The payment at maturity will be max(?S(T),1
  rg)
• The payoff can be decomposed as
• 1 rg ? maxS(T) (1 rg )/ ?, 0

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Structuring choices

• Il order to reduce risk, volatility can be
  reduced by
• Using the average price as underlying asset
  (smoothing)
• Using the average price of different markets
  (diversification)
• This can be achieved by introducing exotic
  options asian options and basket

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Index-Linked Bond

• Consider the bond
• Period 31 July 2000 31 July 2004
• Coupon and principal paid at marturity
• Fixed principal, coupon computed as the higher
  between 6 and the average increase of end of
  quarter of a equally weighted portfolio of
  Nikkei 225, Eurostoxx 50 e SP 500.
• Index-Linked Bond zero coupon option
• (asian call basket)

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Asian options

• Asian options use average of the price for the
  underlying (average rate) or for the strike
  (average strike). In some cases averages are
  computed in discrete time with different
  frequencies.
• Valuation techniques
• Moment matching (Turnbull e Wakeman) the
  distribution of the average is approximated with
  a log-normal distribution with same mean and
  variance
• Monte Carlo method scenarios are generated for
  the sampling dates, the pay-offs for every path
  and the average is computed

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Crash protection

• The investment horizon of this product can be
  perceived as too long. If the market decreases by
  a relevant amount, the option value gets to zero
  and the investor can remain locked-in in a low
  return investment.
• For this reason, the production could be enhanced
  by including the so-called crash protection
  clause. For this clause, if the value of the
  underlying decreases below a given percentage of
  the initial value, the new strike of the option
  is reset at that level.

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Crash protection valuation

• The value of the product is now
• 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 ladder option
• The symmetry between in and out options
• We compute
• ZCB Call(S(t)/S(0), t 1, h)
• Down-and-In(S(t)/S(0), th, 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 Down-and-In
  option with strike equal to the barrier and that
  with the original strike,

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A different product

• We can think that investors would be more allured
  by a product that could generate income and cash
  flows through time more than by a product the
  paid the coupon in the end.
• We could think of a sequence of coupons like
• Coupon (t i) maxS(t i)/S(t i 1 )
  1,0
• This way, the product would produce a cash flow
  of interest equal to the appreciation of teh
  market, excluding losses.

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Cliquet index-linked

• The new product can be represented as a sequence
  of coupons that are determined as a sequence of
  forward start options, that is a ratchet
  (cliquet)
• If one rules out the presence of dividends, the
  value of N coupons amounts to the sum of N
  at-the-money options with one year exercise.
• In the product with a single coupon at maturity
  the interest payment is a single at-the-money
  option with five year maturity.

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Reverse cliquet (Vega bond)

• Assume a product that in N years pay a coupon
  defined as
• Coupon max0, D ?imin(S(ti)/S(ti1) 1,0)
• In other terms, the coupon the coupon is made by
  an initial endowment D, from which negative
  changes of the market are subtracted in every
  period.
• This product is called vega bond for the exposure
  to volatility.

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Multivariate digital notes(Altiplano)

• Assume that a coupon is determined (reset date)
  and paid at time tj.
• Assume a basket of n 1,2 bonds, whose prices
  are Sn(tj).
• Denote Sn(t0) the reference prices (strike),
  typically recorded at the beginning of the
  contract.
• Denote Ij the indicator function taking value 1
  if Sn(tj)/Sn(t0) gt 1 for both assets and 0
  otherwise.
• The coupon is a bivariate digital option, paying
  c

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Bivariate digital note

• Investment horizon March 2000 - March 2005
• Principal repaid at maturity
• Coupon paid March 15 every year.
• Coupon 10 if (i 1,2,3,4,5)
• Nikkei (15/3/200i) gt Nikkei (15/3/2000) and
• Nasdaq 100 (15/3/200i) gt Nasdaq 100 (15/3/2000)
• Coupon 0 otherwise
• Digital Note ZCB bivariate digital calls

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Altiplano with memory

• Assume a coupon defined (reset date) and paid at
  time tj, and a sequence of dates t0,t1,t2,,tj
  1.
• Assume a set of n 1,2,N assets, whose prices
  are Sn(ti).
• Denote B a barrier and Ii the monitor taking
  value 1 if Sn(ti)/ S(t0) gt B for all bonds and 0
  otherwise.
• The coupon of a Altiplano bond
• wher c is a coupon and k is the guaranteed
  return.

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Everest

• Assume a coupon defined and paid at time T.
• Assume a basket ofi n 1,2,N bonds, whose
  prices are Sn(T).
• Denote Sn(t0) the reference prices (strike),
  typically recorded at inception of the contract
  allorigine del contratto, e usati come prezzi
  strike.
• The payoff is
• maxmin(Sn(T)/Sn(0),1k
• (1 k) maxmin(Sn(T)/Sn(0) (1k),0
• with n 1,2,,N and minimun guaranteed return k.

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Basket bond

• Assume a coupon defined and paid at time T.
• Assume a basket ofi n 1,2,N bonds, whose
  prices are Sn(T).
• Denote Sn(t0) the reference prices (strike),
  typically recorded at inception of the contract
  allorigine del contratto, e usati come prezzi
  strike.
• The payoff is
• maxAverage(Sn(T)/Sn(0),1k
• (1 k) maxAverage(Sn(T)/Sn(0) (1k),0
• with n 1,2,,N and minimun guaranteed return
  k.

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Long/short correlation

• The sign of the exposure to correlation is linked
  to the presence in the product of clauses AND or
  OR for the pricing kernel of the derivative
  contracts embedded in the product.
• In Everest the sign is clear it is a long
  position in correlation.
• Assume a product that pays a coupon given by the
  maximum of a set of appreciation rates if this is
  greater than a guaranteed return. The product is
  short correlation.