1. Advanced methods of insurance Lecture 2

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

3. 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

4. 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

5. 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.

6. Non linear contracts: options Call (put) European: gives at time t the right, but not the obligation, to buy (sell) at time T (exercise time) a unit of S at price K (strike or exercise price). Payoff of a call at T: max(S(T) - K, 0) Payoff of a put at T: max(K - S(T), 0)

7. Example inspired to Enel Enel = 7,269 v(t,T) = 0,9966 Call(Enel,t;7,400,T) = ? Enel (H) = 7,500 v(T,T) = 1 Call (H) = 0,100 Enel(L) = 7,100 v(T,T) = 1 Call (L) = 0 T = 16 May 2005 t = 16 March 2005

8. Arbitrage relationship among prices Consider a portfolio with Long  units of Y Funding/investment W Set  =[max(Y(H) –K,0)–max(Y(L)–K,0))]/(Y(H)– Y(L)) At time T Max(Y(H) – K,0) =  Y(H) + W Max(Y(L) – K, 0) =  Y(L) + W

9. Call(Enel,16/03/05;7,400, 16/05/05) Consider un portfolio with  = (0,100 – 0)/(7,500 – 7,100) = 0,25 Enel W = – 0,25 x 7,100 = – 1,775 (leverage) At time T C(H) = 0,100 = 0,25 x 7,500 – 1,775 C(L) = 0 = 0,25 x 7,100 – 1,775 The no-arbitrage implies that at date 16/03/05 Call(Enel,t) = 0,25 x 7,269 – 0,9966 x 1,775 = 0,048285 A call on 10000 Enel stocks for strike price 7,400 is worth 4828,5 € and corresponds to A long position in 2500 ENEL stocks Debt (leverage) for 17750 € face value maturity 16/05/05

10. Alternative derivation Take the value of a call option and its replicating portfolio Call(Y,t;K,T) =  Y(t) + v(t,T)W Substitute  and W in the replicating portfolio Call(Y,t;K,T) = v(t,T)[Q Call(H) +(1 – Q) Call(H)] with Q = [Y(t)/v(t,T) – Y(L)]/[Y(H) – Y(L)] a probability measure. Notice that probability measure Q directly derives from the no-arbitrage hypothesis. Probability Q is called risk-neutral.

11. Enel example Enel = 7,269 v(t,T) = 0,9966 Call(Enel,t;7,400,T) = = 0,9966[Q 0,1 + (1 – Q) 0] = 0,048285 Enel (H) = 7,500 v(T,T) = 1 Call (H) = 0,100 Enel(L) = 7,100 v(T,T) = 1 Call (L) = 0 T = 16 May 2005 t = 16 March 2005 Q = [7,269/0,9966 – 7,1]/[7,5 – 7,1 ] = 48,4497%

12. Q measure and forward price Notice that by construction F(S,t) =Y(t)/v(t,T)= [Q Y(H) +(1 – Q) Y(H)] and the forward price is the expected value of the future price Y(T). In the ENEL case 7,239799 = 7,269/0,9966 = = 0,484497 x 7,5 + 0,515503 x 7,1 Notice that under measure Q, the forward price is an unbiased forecast of the future price by construction.

13. Extension to more periods Assume in every period the price of the underlying asset could move only in two directions. (Binomial model) Backward induction: starting from the maturity of the contract replicating portfolios are built for the previous period, until reaching the root of the tree (time t)

14. Enel(t) = 7,269 ∆ = 0,435 W = – 3,0855 Call(t) = 0,435x7,269 – 0,9966x3,0855 = 0,084016 Enel(H) = 7,5 ∆(H) = 1, W(H) = – 7,4 Call(H) = 1x7,5 – v(t, ,T)x 7,4 =7,5 – 0,99x7,4 = 0,174 Enel(HH) = 7,7 Call(HH) = 0,3 Enel(HL) = 7,4 Call(HL) = 0 Enel(LL) = 7,0 Call(LL) = 0 Enel(LH) = 7,3 Call(LH) = 0 Enel(L) = 7,1 ∆(L) = 0, W(L) = 0 Call(H) = 0

15. Self-financing portfolios From the definition of replicating portfolio C(H) =  Y(H) + W =  H Y(H) + v(t, ,T) W H C(L) =  Y(L) + W =  L Y(L) + v(t, ,T) W L This feature is called self-financing property Once the replicating portfolio is constructed, no more money is needed or generated during the life of the contract.

16. Measure Q Enel = 7,269 Enel(H) = 7,5 Enel(L) = 7,1 Enel(HH) = 7,7 Enel(HL) = 7,3 Enel(LL) = 7,0 Q H = [7,5/0,99 – 7,3]/[7,7 – 7,3] Q = 48,4497% Q L = [7,1/0,99 – 7,0]/[7,3 – 7,0]

17. Black & Scholes model Black & Scholes model is based on the assumption of normal distribution of returns. The model is in continuous time. Recalling the forward price F(Y,t) = Y(t)/v(t,T)

18. Put-Call Parity Portfolio A: call option + v(t,T)Strike Portfolio B: put option + underlying Call exercize date: T Strike call = Strike put At time T: Value A = Value B = max(underlying,strike) …and no arbitrage implies that portfolios A and B must be the same at all t < T, implying Call + v(t,T) Strike = Put + Undelrying

19. Put options Using the put-call parity we get Put = Call – Y(t) + v(t,T)K and from the replicating portfolio of the call Put = (  – 1)Y(t) + v(t,T)(K + W) The result is that the delta of a put option varies between zero and – 1 and the position in the risk free asset varies between zero and K.

20. Structuring principles Questions: Which contracts are embedded in the financial or insurance products? If the contract is an option, who has the option?

21. Who has the option? Assume the option is with the investor, or the party that receives payment. Then, the payoff is: Max(Y(T), K) that can be decomposed as Y(T) + Max(K – Y(T), 0) or K + Max(Y(T) – K, 0)

22. Who has the option? Assume the option is with the issuer, or the party that makes the payment. Then, the payoff is: Min(Y(T), K) that can be decomposed as K – Max(K – Y(T), 0) or Y(T) – Max(Y(T) – K, 0)

23. Convertible Assume the investor can choose to receive the principal in terms of cash or n stocks of asset S max(100, nS(T)) = 100 + n max(S(T) – 100/n, 0) The contract includes n call options on the underlying asset with strike 100/n.

24. Reverse convertible Assume the issuer can choose to receive the principal in terms of cash or n stocks of asset S min(100, nS(T)) = 100 – n max(100/n – S(T), 0) The contract includes a short position of n put options on the underlying asset with strike 100/n.

25. Interest rate derivatives Interest rate options are used to set a limit above (cap) or below (floor) to the value of a floating coupons. A cap/floor is a portfolio of call/put options on interest rates, defined on the floating coupon schedule Each option is called caplet/floorlet Libor – max(Libor – Strike, 0) Libor + max(Strike – Libor, 0)

26. Call – Put = v(t,  )(F – Strike) Reminding the put-call parity applied to cap/floor we have Caplet(strike) – Floorlet(strike) =v(t,  )[expected coupon – strike] =v(t,  )[f(t, ,T) – strike] This suggests that the underlying of caplet and floorlet are forward rates, instead of spot rates.

27. Cap/Floor: Black formula Using Black formula, we have Caplet = (v(t,t j ) – v(t,t j+1 ))N(d 1 ) – v(t,t j+1 ) KN(d 2 ) Floorlet = (v(t,t j+1 ) – v(t,t j ))N(– d 1 ) + v(t,t j+1 ) KN(– d 2 ) The formula immediately suggests a replicating strategy or a hedging strategy, based on long (short) positions on maturity t j and short (long) on maturity t j+i for caplets (floorlets)