Advanced methods of insurance
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Transcript of 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
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
16/05/200510000 ENEL(15/09/2005) – 72938
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
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.
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)
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 2005t = 16 March 2005
Arbitrage relationship among prices
• Consider a portfolio withLong units of YFunding/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
Call(Enel,16/03/05;7,400, 16/05/05)• Consider un portfolio with
= (0,100 – 0)/(7,500 – 7,100) = 0,25 EnelW = – 0,25 x 7,100 = – 1,775 (leverage)
• At time TC(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 toA long position in 2500 ENEL stocksDebt (leverage) for 17750 € face value maturity 16/05/05
Alternative derivation• Take the value of a call option and its replicating
portfolioCall(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.
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 2005t = 16 March 2005
Q = [7,269/0,9966 – 7,1]/[7,5 – 7,1 ]
= 48,4497%
Q measure and forward price
• Notice that by constructionF(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 case7,239799 = 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.
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)
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) = 0Enel(L) = 7,1
∆(L) = 0, W(L) = 0
Call(H) = 0
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.
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
QH = [7,5/0,99 – 7,3]/[7,7 – 7,3]
Q = 48,4497%
QL = [7,1/0,99 – 7,0]/[7,3 – 7,0]
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)
tTdd
tT
tTKtYFd
dKNTtvdNtYTKtYcall
12
2
1
21
2/1/,ln
,,;,
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, implyingCall + v(t,T) Strike = Put + Undelrying
Put options
• Using the put-call parity we getPut = Call – Y(t) + v(t,T)K
and from the replicating portfolio of the callPut = ( – 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.
Structuring principles
• Questions:
• Which contracts are embedded in the financial or insurance products?
• If the contract is an option, who has the option?
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)
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)
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.
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.
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)
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.
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)