1. Advanced Risk Management I Lecture 6 Non-linear portfolios

2. 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)

3. 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)

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

5. 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(d 1 ) and that of the put is N(d 1 ) – 1.

6. Delta

7. 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

8. Gamma

9. Theta The value of the option changes as time elapses The theta value is obtained observing that the Black & Scholes PDE equatin

10. Theta

11. 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

12. 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…

13. 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)

14. Vega

15. 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.

16. 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.

17. 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

18. Smile, please!

23. 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

24. 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 family Cornish-Fisher expansion Delta-Gamma approximation

27. 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.

28. 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.

29. 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(.)

30. Monte Carlo: uses Compute the prices of derivative contracts. –Call f T the value of the option at maturity T, the current value, f, will be given by

31. Monte Carlo: uses VaR evaluation of a portfolio of non linear derivatives

32. A process for equity prices Processes for the undelrying Scenario generation Probsability distributioon of prices Computation of mean and error

33. 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.

34. A process for equity prices Notice: in these formulas z is a variable generated from a standard normal distributon N(0,1).