Advanced Risk Management I

Lecture 6

Non-linear portfolios

Non linearities in portfolios: options

• Call (put) option: gives the right, but not the obligation, to buy (sell) at time T (exercise time) one unit of S at price K (strike or exercise price).

• Payoff of the call at T: max(S(T) - K, 0)

• Payoff of the put at T: max(K - S(T), 0)

Black & Scholes model

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

tTdd

tT

tTKtYFd

dKNTtvdNtYTKtYcall

12

2

1

21

2/1/,ln

,,;,

Prices of put options

• From the put-call parity and the property of the standard normal distribution: 1 – N(a) = N(– a) we get

tTdd

tT

tTKtYFd

dKNTtvdNtYTKtYput

12

2

1

21

2/1/,ln

,,;,

Greek letters

• Delta: first derivative of the contract with respect to the price of the underlying asset.

• Gives the quantity of the underlying asset to be bought or sold to yield a portfolio “locally” risk neutral. Notice that the delta changes with the underlying asset and the time to exercize.

• The delta of the call option is N(d1) and that of the put is N(d1) – 1.

Delta

-1.000

-0.800

-0.600

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

60 80 100 120 140 160 180 200

Sottostante

De

lta Call

Put

Gamma• Since the delta changes with the underlying asset,

we have to take into account the second order effect, called gamma.

• Notice that the no-arbitrrage relationship between call and put prices implies that

delta(put) = delta(call) – 1,

and gamma is the same for call and put options. In the Black & Scholes model

tTtY

dn

tY

ddn

tY

dN

111

1

Gamma

0

0.005

0.01

0.015

0.02

0.025

0 20 40 60 80 100 120 140 160 180 200

Sottostante

Ga

mm

a

Theta

• The value of the option changes as time elapses

• The theta value is obtained observing that the Black & Scholes PDE equatin

02

1 22 tgtYY

Theta

-8

-6

-4

-2

0

2

4

6

30 80 130 180 230

Sottostante

Th

eta Call

Put

A Taylor series expansion

• Remember that a derivative contract is function of the underlying price and of time. For this reason, delta, gamma and theta are the only “greek letters” that make sense, and every derivative g can be approximated by a Taylor expansion

2

2

1),(),( dYdYdtttYgdttdttYg

Sensitivity analysis

• It is usual to analyze the behavior of the value of the derivative contract with changes of the parameters, such as for example the interest rate (and the dividend yield) and volatility

• The sensitivity to interest rate is called “rho”

…but the sensitivity to the volatility, called vega, is much more relevant…

dKNtTTKtYput

rho dKNtTTKtYcall

rho PC 22),;,(),;,(

Volatility risk

• Many traders build their strategies on sensitivity of a “book” of options to forecasts of volatility, and use a sensitivity measure called vega

• Others, more sophisticated, use the second derivative and the cross derivative (vomma e vanna)

1),;,(),;,(

dntTtYTKtYputTKtYcall

vega

Vega

0

5

10

15

20

25

30

35

40

0 50 100 150 200 250

Sottostante

Ve

ga

Implied volatility

• The volatility that in the Black and Scholes formula gives the option price observed in the market is called implied volatility.

• Notice that the Black and Scholes model is based on the assumption that volatility is constant.

The Black and Scholes model

• Volatility is constant, which is equivalent to saying that returns are normally distributed

• The replicating portfolios are rebalanced without cost in continuous time, and derivatives can be exactly replicated (complete market)

• Derivatives are not subject to counterpart risk.

Beyond Black & Scholes

• Black & Scholes implies the same volatility for every derivative contract.

• From the 1987 crash, this regularity is not supported by the data– The implied volatility varies across the strikes

(smile effect)– The implied volatility varies across different

maturities (volatility term structure)• The underlying is not log-normally distributed

Smile, please!Smiles in the equity markets

0

0,5

1

1,5

2

2,5

3

3,5

4

0,8 0,85 0,9 0,95 1 1,05 1,1 1,15 1,2 1,25 1,3

Moneyness

Imp

lied

Vo

lati

lity

Mib30

SP500

FTSE

Nikkei

Delta-Gamma approximation

• Assume to have a derivative sensitive to a single risk factor identified by the underlying asset S.

• Using a Taylor series expansion up to the second order

22

2

2

2

1

2

1SSS

S

VS

S

VV

22

2

2

2

1

2

1SSS

S

VS

S

VV

22

2

1

S

S

V

S

S

S

V

S

V

V 22

2

1

S

S

V

S

S

S

V

S

V

V

• Since the distribution is not known, statistical approximations can be used

• These methods are based on the computation of moments of the distribution and matching moments with those of a known distribution

• Methods

Johnson familyCornish-Fisher expansion

Delta-Gamma approximation

-2

-1

0

1

2

3

4

5

15.2 16.2 17.2 18.2 19.2 20.2 21.2 22.2

Black & Scholes Delta Delta-Gamma Payoff a scadenza

-2

-1

0

1

2

3

4

5

6

15.2 16.2 17.2 18.2 19.2 20.2 21.2 22.2

Black & Scholes Delta Delta-Gamma Payoff a scadenza

Monte Carlo

• Monte Carlo method is a technique based on the

simulation of a number of possible scenarios

representative of the evolution of the risk factors

driving the price of the securities

• This technique is based on the idea of approximating

the expected value of a function computing the

arithmetic average of results obtained from the

simulations.

Monte Carlo methods in finance

• In finance the Monte Carlo method is used both for

the valuation of options or the loss at a given

probability level.

• The key input consists in the definition of the

dynamic process followed by the underlying asset.

• The typical assumption is that the underlying

follows a geometric Brownian motion.

Random data generation• Several methods can be used to extract data from a

distribution H(.). • Given value x, the integral transform H(x), defined as

the probability of extracting a value lower or equal to x has uniform distribution in the interval from zero to one.

• Then, it is natural to use the algorithm– Extract the variable u from the uniform distribution in [0,1]– Compute the inverse of H(.): x = H –1(u)

• The variable x is distributed according to H(.)

Monte Carlo: uses• Compute the prices of derivative contracts.

– Call fT the value of the option at maturity T, the current value, f, will be

given by

TrT fEef ˆ T

rT fEef ˆ

Monte Carlo: uses

• VaR evaluation of a portfolio of non linear derivatives

VaRLPE &1 VaRLPE &1

A process for equity pricesProcesses for the undelrying

Scenario generation

Probsability distributioon of prices

Computation of mean and error

Monte Carlo

• The mean square error of the estimator decreases as the dimension of sample increases, with law

• Notice that this is independent of the dimension of the system.

n/1 n/1

tzt2S

S 2

0

ln tzt

2S

S 2

0

ln

tzt

2SS

2

0 exp

tzt

2SS

2

0 exp

A process for equity prices

Notice: in these formulas z is a variable generated from a standard normal distributon N(0,1).

Notice: in these formulas z is a variable generated from a standard normal distributon N(0,1).