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Advanced methods of insurance

Advanced methods of insurance. 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.

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Advanced methods of insurance

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

  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 x7,100 – 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 T = 16 May 2005 t = 16 March 2005 Enel (H) = 7,500 v(T,T) = 1 Call (H) = 0,100 Q = [7,269/0,9966 – 7,1]/[7,5 – 7,1 ] = 48,4497% 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(L) = 7,100 v(T,T) = 1 Call (L) = 0

  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(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(t) = 7,269 ∆ = 0,435 W = – 3,0855 Call(t) = 0,435x7,269 – 0,9966x3,0855 = 0,084016 Enel(HL) = 7,4 Call(HL) = 0 Enel(LH) = 7,3 Call(LH) = 0 Enel(L) = 7,1 ∆(L) = 0, W(L) = 0 Call(H) = 0 Enel(LL) = 7,0 Call(LL) = 0

  15. Self-financing portfolios • From the definition of replicating portfolio C(H) = Y(H) + W = HY(H) + v(t,,T) WH C(L) = Y(L) + W = LY(L) + v(t,,T) WL • 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(HH) = 7,7 QH = [7,5/0,99 – 7,3]/[7,7 – 7,3] Enel(H) = 7,5 Enel(HL) = 7,3 Q = 48,4497% Enel = 7,269 QL = [7,1/0,99 – 7,0]/[7,3 – 7,0] Enel(L) = 7,1 Enel(LL) = 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,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 immediately suggests a replicating strategy or a hedging strategy, based on long (short) positions on maturity tj and short (long) on maturity tj+i for caplets (floorlets)

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