2 f. f t S 2. Delta measures the sensitivityof the portfolio value to changes in the price of the underlying

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1 Sensitivity analysis Simulating the Greeks Meet the Greeks he value of a derivative on a single underlying asset depends upon the current asset price S and its volatility Σ, the risk-free interest rate r, and the time to maturity t. hat is, V f S, r, Σ, t. (It also depends upon constants like the strike price K.) aking a aylor series expansion, the change in value over a small time period can be approximated by dv f f f f dr dσ S r Σ t dt 1 2 f 2 S 2 2 other second order terms higher order terms he partial derivatives in this expansion are known collectively as "the Greeks". hey measure the sensitivity of a portfolio to changes in the underlying parameters. Specifically f S f Ρ r f v Σ f t Delta measures the sensitivityof the portfolio value to changes in the price of the underlying Rho measures the sensitivityof the portfolio value to changes in the interest rate Vega measures the sensitivityof the portfolio value to changes in the volatility of the underlying heta measures the sensitivityof the portfolio value to the passage of time 2 f Gamma measures the sensitivity of delta to changes in the price of the underlying, or the curvature of the S V curve. S 2 S Substituting in (1), the change in value of the portfolio can be approximated by dv Ρ dr v dσ dt Because differentiation is a linear operator, the hedge parameters of a portfolio are equal to a weighted average of the hedge parameters of its components. In particular, the hedge parameters of a short position are the negative of the hedge parameters of a long position. Consequently, (2) applies equally to a portfolio as to an individual asset. he sensitivity of a portfolio to the risk factors (S, r, Σ) can be altered by changing the composition of the portfolio. It can be reduced by adding assets with offsetting parameters. he Greeks are not independent. Any derivative (or portfolio of derivatives) V f S, r, Σ, t must satisfy the Black-Scholes differential equation f t r S f Substituting t S 1 2 Σ2 S 2 2 f S S 2 r V 2 S 2 it follows that the Greeks must satisfy the following relationship

2 2 SensitivityAnalysis.nb r S 1 2 Σ2 S 2 r V Computing the Greeks Formulae for vanilla European options he Greeks of vanilla European options have straightforward formulae, which can be derived from the Black- Scholes formula. he Black-Scholes formulae for European options are c S q Nd 1 K r Nd 2 p K r Nd 2 S q Nd 1 where d 1 lns K r q Σ2 2, d 2 lns K r q Σ2 2 Σ Σ he partial derivatives ("the Greeks") are Call Put Delta q Nd 1 q Nd 1 1 d 1 s Gamma q N ' d 1 S Σ q N ' d 1 S Σ Rho K r Nd 2 K r Nd 2 Vega q S N ' d 1 q S N ' d 1 heta q S Σ N ' d 1 2 q q S Nd 1 r K r Nd 2 As an example of the derivation, for a call option S q N ' d 1 d 1 S q N ' d 1 q S Σ N ' d 1 2 q q S Nd 1 r K r Nd 2 S Σ 1 Calculating vega from the Black-Scholes formula is an approximation, since the formula is derived under the assumption that volatility is constant. Fortunately, it can be shown that it is a good approximation to the vega calculated from a stochastic volatility model (Hull 2003: 318). Some exotic options (e.g. barrier options) have analogous formulae. However, for most exotic options, the Greeks must be estimated by numerical techniques. Since these are the type of options for which institutions require such information, this motivates are interest in the accurate computation of option values and sensitivities. Numerical differentiation in general In principle, the Greeks for general options can be estimated by numerical differentiation. For example Delta, which measures the sensitivity of the option value to changes in the price of the underlying, is defined as

3 SensitivityAnalysis.nb 3 VS VS VS lim S 0 An obvious method to evaluate is to compute VS VS for small. his is known as the forward difference. A better alternative (though more costly to compute) is VS VS 2 which is known as the central difference. he other first-order Greeks (rho, theta and vega) can be estimated similarly. Gamma is the derivative of delta, or the second derivative of VS. Using central differences, gamma can be estimated by S 1 S VS 2 VS VS 2 Simulating the Greeks V SV S V SV S V SV S V SV S Finite differences he delta of a derivative is V With simulation, this can be estimated by the average of the finite-differences over a sample n of replications. 1 n V V If we use different random samples for estimating V and V, the variance of becomes very large as becomes small. A better estimate of is generally obtained by using the same random numbers in estimating both V and V. he variance of is Var 1 2 VarV VarV 2 CovV, V Provided that V and V are positively correlated, the estimate obtained from common random numbers will have a lower variance. Even with common random numbers, the finite difference estimator is biased. In general, the bias increases with, while the variance decreases with. Consequently, choice of the optimal value of involves a tradeoff between bias and variance. Except in the simplest cases, it is not possible to determine the optimal tradeoff analytically. he alternative techniques, the pathwise derivative and the likelihood ratio method are both unbiased, though they do not necessarily offer lower variance.

4 4 SensitivityAnalysis.nb Pathwise derivative he pathwise derivative decomposes the derivative into two components. V V S S Assuming geometric Brownian motion and the risk-neutral distribution S so that Σ2 rq Σ 2 (1) S rq Σ 2 Σ 2 S he discounted payoff of a vanilla European call option is r max S K, 0 V d r max S K, 0 r, S K S d 0, S K If the option is in-the-money, a small increase in the terminal price increases the discounted payoff by r. If the option is out-of-the money, a small increase in the terminal price leaves the payout unchanged. he function is not differentiable at S K, but this is a zero-probability event. Combining the two factors, the pathwise estimator of delta for a vanilla European call option is V V S S r I S K S where I S K equals one if S K and zero otherwise. his estimator can easily be computed by simulation. Similarly, differentiating (1) by Σ, S Σ S Σ Solving (1) for 1 Σ Log S r q Σ2 2 and substituting in the previous equation gives (2) S Σ S Σ Log S r q Σ2 2 Σ S Σ Σ2 Log S r q Σ2 2 S Σ Log S r q Σ2 2 Combining this with (2) gives the pathwise estimator of vega for a European call option V Σ V S S Σ S r Σ I S K Log S r q Σ2 2 Again, this can easily be estimated by simulation.

5 SensitivityAnalysis.nb 5 he pathwise estimator of delta for an arithmetic Asian option is V V S S Analogous to the vanilla option (2) V d r max S K, 0 r I S K S d and (3) S 1 m m 1 St i i0 1 m St i m 1 S i0 Combining the two factors, the pathwise estimator of delta for an Asian call option is V V S S r I S K S he pathwise method requires continuity in the discounted payoff function as a function of the parameter. For this reason, it is generally not applicable to estimating second derivatives (gamma). Likelihood ratio he likelihood ratio method provides an alternative approach, differentiating the probabilities rather than the payoff. Consider a derivative the payoff Y of which depends upon a random variable X whose density g Θ x depends upon some parameter Θ. Its expected value is E Θ Y E Θ f X f x g Θ x x Assuming that we can interchange differentiation and integration Θ E ΘY f x g Θx x, g Θx Θ g Θx which can be rewritten as Θ E ΘY f x g Θx x f x g Θx herefore, the expression f X g ΘX g Θ X g Θ x g Θx x E Θ f X g is an unbiased estimator of the derivative of E Θ Y. Note that g ΘX g Θ X logg Θ Θ ΘX g Θ X is sometimes known as the score function. he score function for delta is g ΘX g Θ X S0 Σ and therefore the likelihood ratio estimator for delta (Θ of a European call is

6 6 SensitivityAnalysis.nb f X g ΘX g Θ X r S K S0 Σ Since the score function is the derivative of the density function of S, it does not depend on the specific payoff function. herefore, the likelihood ratio of delta for any option in which the payoff depends only on S would take the same form, albeit with a different payoff function. Similarly the likelihood ratio estimator for vega (Θ Σ of a European call is vega f X g ΘX g Θ X r S K 2 1 Σ Σ he Markov property of GBM implies that only affects the density of S 1, and has no effect on the density of S 2, S 3..., S m. It follows that the likelihood ratio estimator of delta (Θ for an Asian call option is r S K 1 Σ t 1 where t 1 is the time to the first observation, and 1 is the random variable generating this simulated value. In contrast to the pathwise method, the likelihood ratio method can be effective in estimating second derivatives, using the estimator f X ġ. ΘX g Θ X he likelihood ratio estimator for gamma for a European option is f X g ΘX g Θ X r S K Σ 2 2 Σ Hybrid estimators We can also combine the pathwise and likelihood ratio methods in a hybrid estimator for gamma. he likelihood ratio estimator for delta for a vanilla call option is r S K S0 Σ Differentiating with respect to, we obtain the LR-PW estimator for gamma r S K Σ r IS K S 2 0 Σ Reversing the order, the PW-LR estimator for gamma is r IS K S 2 Σ 1 K