advanced risk management i lecture 6 non-linear portfolios
TRANSCRIPT
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).