A VegaGamma Relationship for EuropeanStyle or Barrier Options in the BlackScholes Model


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1 A VegaGamma Relationship for EuropeanStyle or Barrier Options in the BlackScholes Model Fabio Mercurio Financial Models, Banca IMI Abstract In this document we derive some fundamental relationships between the Greeks of a barrier option under the BlackScholes model. A Europeanstyle option can be considered as a limit case. Besides the classical Black and Scholes P.D.E., a barrier option price is shown to also satisfy: i) a time homogeneity; ii) a scale invariance of time; iii) an interest rates symmetry; iv) a VegaGamma relationship. Notation t: the running time. S t : the price at time t in domestic currency of one unit of foreign currency. r: the (constant) domestic instantaneous riskfree rate. q: the (constant) foreign instantaneous riskfree rate. σ: the exchange rate (constant) percentage volatility. T : the maturity of a given derivative. τ: the timetomaturity of this given derivative, namely τ = T t. V T : the derivative s payoff at time T. V (t, S t, r, q, σ, τ): noarbitrage price at time t of a given derivative security with timetomaturity τ, when the underlying exchange rate is S, the domestic instantaneous riskfree rate is r, the foreign instantaneous riskfree rate is q and the exchange rate percentage volatility is σ. A shorthand notation for it is V t. Q d : the domestic riskneutral measure. E d : expectation under Q d. F t : the σalgebra generated by S up to time t.. Assumptions The exchange rate S is assumed to evolve under the domestic riskneutral measure Q d according to: ds t = S t (r q) dt + σ dw t ] where W is a standard Brownian motion under Q d. We are given a derivative security with payoff V T at time T. This derivative is either a Europeanstyle option or a (single) barrier option. We actually restrict our analysis to
2 Greeks of FX options out barrier options, which become worthless if the barrier is hit prior to maturity. In fact, a Europeanstyle payoff can be obtained by letting the barrier level either go to zero or to infinity, whereas an in barrier option is simply the difference between the corresponding Europeanstyle and out barrier options.. The pricing of a Barrier Option Standard theory in derivatives pricing implies that the noarbitrage price at time t of the payoff V T at time T is V t = e rτ E d V T F t ] () Set ν := r q σ and X t := νt + σw t M X t := max u 0,t] (ωx u) where ω = for maximum and ω = for minimum. Then, see for instance Musiela and Rutkowski (998), for ωy max(0, ωx), Q d{ X T dx, MT X ωy } ( ) ( ) ] = ω σ x νt πt e σ T e νy σ σ x y νt πt e σ T Assuming that the option payoff, if the barrier is not hit prior to maturity, is a function h(s T ) of the underlying at maturity, the barrier option price at time 0 can be written as V 0 = e rt h(e z )f(z; S 0, r, q, σ, T ) dz where neither h nor A IR depend on S 0, r, q, σ, T and ( ) f(z; S, r, q, σ, T ) = ω σ z ln S νt πt e σ T e ν ln(h/s) σ σ πt e ( z ln S ln(h/s) νt where H is the barrier level. By the timehomogeneity of the exchange rate dynamics, we have that } Q {X d T dx, max (ωx u) ωy F t = Q d{ X τ dx, Mτ X ωy } u t,t ] Therefore, the barrier option price at time t, given that the barrier has not been hit before, is V (t, S t, r, q, σ, τ) = e rτ h(e z )f(z; S t, r, q, σ, τ) dz (3) which depends on t only through τ. σ T ) ] ()
3 .3 The Black and Scholes P.D.E. Greeks of FX options The derivative s price is assumed to be a smooth function of time and exchange rate. From the standard Black and Scholes theory, the function V then follows t + (r q)s S + σ S V = rv (4) S with terminal condition V T = h(s T )..4 The Time Homogeneity From (3), we immediately see that the derivative s price depends on t only implicitly through the timetomaturity τ = T t. This leads to t = τ (5).5 The Scale Invariance of Time From (3) we also see that replacing r, q, σ and τ as follows r rα q qα σ σ α (6) τ τ α where α is any (strictly) positive real number, the derivative price does not change, namely V (t, S t, rα, qα, σ α, τ/α) = V (t, S t, r, q, σ, τ) Since this relation holds for any α, we can differentiate both sides with respect to α and then set α =. We get σ σ + r r + q τ τ = 0 or equivalently, by (5), σ σ + r r + q + τ t = 0 (7) This relationship has been derived by Reiss and Wystup (00). Equivalently, by () and the FeynmanKac formula. 3
4 .6 The Interest Rates Symmetry Differentiating (3) with respect to r and q, we get = τv + e rτ h(e z ) r r f(z; S t, r, q, σ, τ) dz = e rτ h(e z ) f(z; S t, r, q, σ, τ) dz Noting that f(z; S t, r, q, σ, τ) = r f(z; S t, r, q, σ, τ) Greeks of FX options the sensitivities of the derivative s price with respect to the two instantaneous rates then satisfy r + + τv = 0 (8).7 The VegaGamma Relationship From (4) we obtain = rv (r q)s t S σ S V S which substituted into (7) yields σ σ + r r + q + τ rv (r q)s S σ S V ] = 0 S Rearranging terms we get σ σ = τ σ S V ( ) + (r q)τs S S r r + τv Applying (8) and collecting terms, we finally obtain q σ σ = τ σ S V S + (r q) τs S + ] (9) Remark.. The VegaGamma relationship (9) can also be proved by noting that an analogous relationship also holds for the density f(z; S, r, q, σ, τ), i.e. σ f σ = τ σ S f S + (r q) τs f S + f ] This simply follows by the calculation of the related derivatives. 4
5 Greeks of FX options Remark.. In the case of a Europeanstyle payoff, using a property of the lognormal distribution, we can also prove that = τs S so that the VegaGamma relationship for Europeanstyle options reads as References σ σ = τ σ S V S ] Black, F. and Scholes, M. (973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy 8, ] Musiela, M. and Rutkowski, M. (998) Martingale Methods in Financial Modelling. Springer. Berlin. 3] Reiss, O. and Wystup, U. (00) Computing Option Price Sensitivities Using Homogeneity and Other Tricks, The Journal of Derivatives 9(),
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