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)