1. Advanced methods of insurance Lecture 3

2. 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

3. 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

4. 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),  )

5. Digital…

6. …and vertical spreads (super-replication)

7. 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 …from which

8. …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 …from which

9. 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.

10. 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

11. 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.

12. 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 log- normale. 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

13. 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.

14. 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)

15. 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) E Q [S(  )]N = exp(– r(  – t))S(t) exp((r –d)(  – t))N = exp(– d(  – t)) B & S(S(t);  S(t),T –  )

16. 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).

17. 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)

18. 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)

19. 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

20. 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

21. 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

22. 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.

23. 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.

24. 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]

25. 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.

26. 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.

27. Ratchet index-linked If we consider a constant dividend yield q we have Coupons = (1 – v N )/(1 – v)AtM Call with v = exp(– q )

28. Vega bond Assume a product that in N years pay a coupon defined as Coupon = max[0, D +  i min(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.

29. Vega bond Rewrite the pay-off as, Coupon = max[0,D –  i max(1 – S(t+i)/S(t+i– 1),0)] Interest payment is a put option written on a ratchet put.

30. Multivariatedigital notes (Altiplano) Assume a coupon defined (reset date) and paid at time t j. Assume a basket of n stocks, whose prices are S n (t j ). Denote S n (t 0 ) the reference prices, typically taken at the beginning of the contracts and used as strike price. Denote I j indicator function taking value 1 if S n (t j )/S n (t 0 ) > 1 and 0 otherwise for all the securities. The coupon is a multivariate option, and, given the coupon rate c

31. Altiplano with memory Assume a coupon defined (reset date) and paid at time t j, and a sequence of dates {t 0,t 1,t 2, …,t j – 1 }. Assumiame a basket of n = 1,2, … N assets, whose prices are S n (t i ). Denote B a barrier and I i an indicator function taking value 1 if S n (t i )/ S(t 0 ) > B and zero otherwise. The memory feature implies that the first time t i when the indicator function is taking value 1 all coupons up to t i are paid.

32. Everest Assume a coupon defined and paid at time T. Assume a basket of n = 1,2, … N bonds, whose prices are S n (T). Denote S n (t 0 ) the reference prices, typically recoorded at the origin of the contract and used as strike prices The coupon of an Everest is max[min(S n (T)/S n (0),1+k] = = (1 + k) + max[min(S n (T)/S n (0) – (1+k),0] with n = 1,2,…,N and a guaranteed return equal to k.

33. Equity-linked bond Assume a coupon defined and paid at time T. Assume a basket n = 1,2, … N stocks, whose prices are S n (T). Denote S n (t 0 ) 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(S n (T)/S n (0),1+k] = = (1 + k) + max[Average(S n (T)/S n (0) – (1+k),0] with n = 1,2,…,N and minimum guaranteed return k.

34. 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.