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Calculation of the Greeks Using Malliavin Calculus Calculation of the Greeks using Malliavin Calculus Christian Bayer University of Technology, Vienna 09/25/2006 – SCS Seminar 09/26/2006 – Mathematics Seminar Christian Bayer University of Technology, Vienna Calculation of Greeks Outline: Part I 1 Hedging in Mathematical Finance The Model A Reminder on Mathematical Finance Hedging in the Model 2 Calculation of the Greeks Introduction Selected Results from Malliavin Calculus The Bismut-Elworthy-Li Formula 3 Summary Christian Bayer University of Technology, Vienna Calculation of Greeks Outline: Part II 4 Universal Malliavin Weights Stochastic Taylor Expansion Step-m Nilpotent, Free Lie Groups m Iterated Stratonovich Integrals on Gd,0 Universal Malliavin Weights on Nilpotent, Free Lie Groups 5 Two Approaches to Approximate Universal Weights Calculating the Greeks by Cubature Formulas Approximation of the Heat Kernel 6 Summary Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging Calculation of the Greeks Summary Part I Greeks and Malliavin Calculus in Finance Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Model 1 d Let Bt = (Bt ,..., Bt ), t ∈ [0, T ], be a d-dimensional Brownian motion on the Wiener space (Ω, F, P). Ft denotes the filtration generated by B and we assume that F = FT . We model a financial market, in which n + 1 assets are traded, n ≤ d. 0 St , t ∈ [0, T ], is the “bank account” earning a risk free interest rate r > 0 (continuous compounding). 1 n St = (St ,..., St ) gives the risky assets (“stocks”). n n For bounded, measurable functions a : [0, T ] × R → R and n n×d σ : [0, T ] × R → R we set 0 0 dSt = rSt dt i i i Xd ij i j dS = a (t, St )S dt + σ (t, St )S dB , i = 1,..., n. t t j=1 t t Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Strategies A strategy describes the amount of money invested in each available asset at any given time t ∈ [0, T ]. Definition n A (self-financing) strategy is a predictable, R -valued process π such that “all of the following integrals are well-defined”. Remark The strategy is self-financing, because we neither allow consumption nor external money entering the financial market. Thus, strategies are determined by n coordinates. All positions are allowed to be negative (short selling). Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Strategies A strategy describes the amount of money invested in each available asset at any given time t ∈ [0, T ]. Definition n A (self-financing) strategy is a predictable, R -valued process π such that “all of the following integrals are well-defined”. Remark The strategy is self-financing, because we neither allow consumption nor external money entering the financial market. Thus, strategies are determined by n coordinates. All positions are allowed to be negative (short selling). Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Wealth Process of a Portfolio The wealth process X x,π with initial capital x associated to the x,π strategy π is defined by X0 = x and its dynamics n x,π n X πi X − P πi dX x,π = t dSi + t i=1 t dS0. t Si t S0 t i=1 t t Definition A strategy is admissible if the corresponding wealth process is bounded from below by some fixed real number. The restriction to admissible strategy is economically sensible and disallows “doubling strategies”. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model The Wealth Process of a Portfolio The wealth process X x,π with initial capital x associated to the x,π strategy π is defined by X0 = x and its dynamics n x,π n X πi X − P πi dX x,π = t dSi + t i=1 t dS0. t Si t S0 t i=1 t t Definition A strategy is admissible if the corresponding wealth process is bounded from below by some fixed real number. The restriction to admissible strategy is economically sensible and disallows “doubling strategies”. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model No Arbitrage Definition An arbitrage opportunity is an admissible strategy π such that 0,π 0,π P XT ≥ 0 = 1 and P XT > 0 > 0. Definition Let Q be a probability measure on (Ω, F) equivalent to P. Q is called equivalent (local) martingale measure if the discounted price −rt process Set = e St , t ∈ [0, T ], is a (local) martingale under Q. Remark The Fundamental Theorem of Asset Pricing roughly says that the existence of (local) martingale measures is equivalent to the non-existence of arbitrage opportunities. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model No Arbitrage – 2 Proposition d Assume there exists a predictable, R -valued process θ such that i Pd ij j (i) a (t, St ) − r = j=1 σ (t, St )θ (t), i = 1,..., n, R T 2 (ii) 0 kθ(t)k dt < ∞ a. s., R T 1 R T 2 (iii) E exp − 0 hθ(t),dBt i − 2 0 kθ(t)k dt = 1. Then our model is free of arbitrage and the probability measure Q R T 1 R T 2 with density ZT = exp − 0 hθ(t),dBt i − 2 0 kθ(t)k dt is a martingale measure. Consequently, the dynamics of the model under Q are given by i i Xd ij i j dS = rS dt + σ (t, St )S dW , i = 1,..., n, t t j=1 t t where W denotes a Brownian motion under Q. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Complete and Incomplete Markets Definition A contingent claim is an FT -measurable, Q-absolutely integrable random variable. A contingent claim Y is attainable or replicable if there is an admissible, self-financing portfolio π and a number x x,π such that XT = Y a. s. π is called replicating portfolio for Y . Definition A financial market is complete, if every contingent claim is replicable. Otherwise, it is called incomplete. Proposition Our market is complete if and only if n = d and σ(t, St (ω)) is invertible for dt ⊗ P - a. e. (t, ω) ∈ [0, T ] × Ω. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Complete and Incomplete Markets Definition A contingent claim is an FT -measurable, Q-absolutely integrable random variable. A contingent claim Y is attainable or replicable if there is an admissible, self-financing portfolio π and a number x x,π such that XT = Y a. s. π is called replicating portfolio for Y . Definition A financial market is complete, if every contingent claim is replicable. Otherwise, it is called incomplete. Proposition Our market is complete if and only if n = d and σ(t, St (ω)) is invertible for dt ⊗ P - a. e. (t, ω) ∈ [0, T ] × Ω. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Complete and Incomplete Markets – 2 Remark Under realistic conditions, completeness is a very strong property. Many experts agree that realistic models are not complete. Assumptions From now on, we assume that our model is arbitrage-free and complete. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Pricing Contingent Claims Definition x ∈ R is an arbitrage-free price of a contingent claim Y if there is x,π an admissible strategy π such that Y = XT a. s. Proposition In a complete, arbitrage-free model, any claim Y has a unique −rT arbitrage-free price x = EQ e Y , where Q is the unique e. m. m. Remark In incomplete markets, there is, in general, no replicating portfolio, only super-replicating ones. Typically, there is an interval of arbitrage-free prices bounded by the super replication prices of the buyer and of the seller. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Pricing Contingent Claims Definition x ∈ R is an arbitrage-free price of a contingent claim Y if there is x,π an admissible strategy π such that Y = XT a. s. Proposition In a complete, arbitrage-free model, any claim Y has a unique −rT arbitrage-free price x = EQ e Y , where Q is the unique e. m. m. Remark In incomplete markets, there is, in general, no replicating portfolio, only super-replicating ones. Typically, there is an interval of arbitrage-free prices bounded by the super replication prices of the buyer and of the seller. Christian Bayer University of Technology, Vienna Calculation of Greeks Hedging The Model Calculation of the Greeks A Reminder on Mathematical Finance Summary Hedging in the Model Black-Scholes PDE For simplicity, we pass to the discounted model by setting 0 i −rt i Set = 1 and Set = e St , i = 1,..., n.
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