Risk Neutral Valuation

copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012

copyright 2012 Christian Fries 2 / 51 Outline Notation Differential Equations Monte-Carlo Simulation Modeling Risk Neutral Valuation Change of Measure / Drift Example: Black Scholes Model Example: Black-Scholes Model Monte-Carlo Option Pricer

NOTATION DIFFERENTIAL EQUATIONS

copyright 2012 Christian Fries 4 / 51 Notation I Integral: tn 0 f (t) dt n 1 f (t i ) t i i=0 Discrete Interpretation: Integral is approximately the sum of f (t i ) t i. For piecewise constant function f, constant on [t i,t i+1 ) we have " =" above.

copyright 2012 Christian Fries 5 / 51 Notation II Differential Equation: Define the function t X(t) by specifying its change: T dx = f (t) dt : X(T ) = X(0) + f (t) dt 0 Discrete Interpretation: f is the change of X per unit time (rate of change): X(t i ) = f (t i ) t i X(t i ) / t i = f (t i ).

copyright 2012 Christian Fries 6 / 51 Notation III Differential Equation - Special Case: Define the function X by specifying its relative change. T dx = f (t) X(t) dt X(T ) = X(0) + f (t) X(t) dt 0 Discrete Interpretation: f is the relative change of X per time (percentage rate of change): X(t i ) = f (t i ) X(t i ) t i X(t i )/ X(t i ) ti = f (t i ).

copyright 2012 Christian Fries 7 / 51 Notation IV Exercise: Excel Sheet with Discretization of Differential Equation

Notation V State, Pobability, Pobability Measure: Ω = { ω 1,ω 2,...,ω n } (probability space) P({ω i }) (probability that we are in state ω i ) P (probability measure) Probability of an State Configuration (Event): P({ω 1,ω 2,...,ω k }) = P({ω 1 }) + P({ω 2 }) +... + P({ω k }) Random Variable: X : Ω IR Example: X(ω i ) payment that depends on the state ω i. Expectation: E P (X) := X(ω i ) P({ω i }) ω i Ω copyright 2012 Christian Fries 8 / 51

copyright 2012 Christian Fries 9 / 51 Notation VI Conditional Expectation: E P (X F) := 1 P(F) X(ω i ) P({ω i }) ω i F

NOTATION MONTE-CARLO SIMULATION

copyright 2012 Christian Fries 11 / 51 Notation VI Monte-Carlo Simulation Expectation: E P (X) := X(ω i ) P({ω i }) ω i Ω Monte-Carlo Simulation: Numerical Approximation of Expectation: Let Ω := { ω 1, ω 2,..., ω m } denote elements from Ω - a drawing from Ω, i.e. a set of samples. Some ω i s may be the same and we may have more ω i s than Ω has elements. Our set of sample paths should have the following property: A path ω Ω occures in Ω approximately m P({ω}) times. Then we have E P (X) := X(ω i ) P({ω i }) 1 ω i Ω m X( ω i ) ω i Ω

copyright 2012 Christian Fries 12 / 51 Outline Notation Differential Equations Monte-Carlo Simulation Modeling Risk Neutral Valuation Change of Measure / Drift Example: Black Scholes Model Example: Black-Scholes Model Monte-Carlo Option Pricer

copyright 2012 Christian Fries 13 / 51 Modeling Random Variable and Stochastic Processes Financial Product: A stream of payments depending on events described in a contract. Modeling: Value / Payment Random Variable X : Ω IR. Value of an asset (aka. underlying) at a fixed time t dep. on the states of the world. Payment of a financial product at a fixed time t dep. on state of underlying assets. Value of the financial product at a fixed time t depending on states of the assets.

copyright 2012 Christian Fries 14 / 51 Modeling Random Variable and Stochastic Processes Modeling: Value Process / Payoff Stream Stochastic Process := family of random variables over time S : [0, ) Ω IR S(ω) with ωεω is a map [0, ) IR: the path of S in state ω. Note: All random variables (at all times t) are given over the same space (Ω,F ) ω Ω path / history / chain of events

copyright 2012 Christian Fries 15 / 51 Modeling Random Variable and Stochastic Processes Modeling: Information is modeled through the filtration: Family of σ algebras F t. F t F s, t < s. Elements of F t are the events, which may be known at time t. Measurable: X is F T measurable X is known in time T Natural Condition (for stochastic process): S(T ) is F T measurable.

copyright 2012 Christian Fries 15 / 51 Modeling Random Variable and Stochastic Processes Modeling: Information is modeled through the filtration: Family of σ algebras F t. F t F s, t < s. Elements of F t are the events, which may be known at time t. Measurable: X is F T measurable X is known in time T Natural Condition (for stochastic process): S(T ) is F T measurable.

copyright 2012 Christian Fries 15 / 51 Modeling Random Variable and Stochastic Processes Modeling: Information is modeled through the filtration: Family of σ algebras F t. F t F s, t < s. Elements of F t are the events, which may be known at time t. Measurable: X is F T measurable X is known in time T Natural Condition (for stochastic process): S(T ) is F T measurable.

copyright 2012 Christian Fries 16 / 51 Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Modeling (e.g. a bet): Ω = {(h,h,h),(h,h,t),(h,t,h),...,(t,t,t)} X : Ω IR S : {T 1,T 2,T 3 } Ω IR S(T k ) paid at time T k. - Prob space (head or tail). - Bet with a single payoff Bet with payments - or evolution of value (stochastic process) Natural Condition (*): S(T k ) may depend on events known on or before T k only. S(T k,(e 1,...,e n )) = const. e i {h,t} mit i > k

copyright 2012 Christian Fries 16 / 51 Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Modeling (e.g. a bet): Ω = {(h,h,h),(h,h,t),(h,t,h),...,(t,t,t)} X : Ω IR S : {T 1,T 2,T 3 } Ω IR S(T k ) paid at time T k. - Prob space (head or tail). - Bet with a single payoff Bet with payments - or evolution of value (stochastic process) Natural Condition (*): S(T k ) may depend on events known on or before T k only. S(T k,(e 1,...,e n )) = const. e i {h,t} mit i > k

Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Define family of σ algebras (filtration): F 0 = {0,Ω} F 1 = σ({{(h,, )},{(t,, )}}) F 2 = σ({{(h,h, )},{(h,t, )},{(t,h, )},{(t,t, )}}) F 3 = σ({{(h,h,h)},{(h,h,t)},{(h,t,h)},...,{(t,t,t)}}) h ω 1 ω 2 ω 3 ω 4 t ω 5 ω 6 ω 7 ω 8 T 0 T 1 T 2 T 3 T 4 Condition (*) S(T k ) is F k -measurable : S is adapted to {F k }. copyright 2012 Christian Fries 17 / 51

Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Define family of σ algebras (filtration): F 0 = {0,Ω} F 1 = σ({{(h,, )},{(t,, )}}) F 2 = σ({{(h,h, )},{(h,t, )},{(t,h, )},{(t,t, )}}) F 3 = σ({{(h,h,h)},{(h,h,t)},{(h,t,h)},...,{(t,t,t)}}) h ω 1 ω 2 ω 3 ω 4 t ω 5 ω 6 ω 7 ω 8 T 0 T 1 T 2 T 3 T 4 Condition (*) S(T k ) is F k -measurable : S is adapted to {F k }. copyright 2012 Christian Fries 17 / 51

Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Define family of σ algebras (filtration): F 0 = {0,Ω} F 1 = σ({{(h,, )},{(t,, )}}) F 2 = σ({{(h,h, )},{(h,t, )},{(t,h, )},{(t,t, )}}) F 3 = σ({{(h,h,h)},{(h,h,t)},{(h,t,h)},...,{(t,t,t)}}) h ω 1 ω 2 ω 3 ω 4 t ω 5 ω 6 ω 7 ω 8 T 0 T 1 T 2 T 3 T 4 Condition (*) S(T k ) is F k -measurable : S is adapted to {F k }. copyright 2012 Christian Fries 17 / 51

Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Define family of σ algebras (filtration): F 0 = {0,Ω} F 1 = σ({{(h,, )},{(t,, )}}) F 2 = σ({{(h,h, )},{(h,t, )},{(t,h, )},{(t,t, )}}) F 3 = σ({{(h,h,h)},{(h,h,t)},{(h,t,h)},...,{(t,t,t)}}) h ω 1 ω 2 ω 3 ω 4 t ω 5 ω 6 ω 7 ω 8 T 0 T 1 T 2 T 3 T 4 Condition (*) S(T k ) is F k -measurable : S is adapted to {F k }. copyright 2012 Christian Fries 17 / 51

Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Define family of σ algebras (filtration): F 0 = {0,Ω} F 1 = σ({{(h,, )},{(t,, )}}) F 2 = σ({{(h,h, )},{(h,t, )},{(t,h, )},{(t,t, )}}) F 3 = σ({{(h,h,h)},{(h,h,t)},{(h,t,h)},...,{(t,t,t)}}) h ω 1 ω 2 ω 3 ω 4 t ω 5 ω 6 ω 7 ω 8 T 0 T 1 T 2 T 3 T 4 Condition (*) S(T k ) is F k -measurable : S is adapted to {F k }. copyright 2012 Christian Fries 17 / 51

Modeling Random Variable and Stochastic Processes Example: (Time discrete) stochastic process of coin toss at T 1, T 2, T 3. Define family of σ algebras (filtration): F 0 = {0,Ω} F 1 = σ({{(h,, )},{(t,, )}}) F 2 = σ({{(h,h, )},{(h,t, )},{(t,h, )},{(t,t, )}}) F 3 = σ({{(h,h,h)},{(h,h,t)},{(h,t,h)},...,{(t,t,t)}}) h ω 1 ω 2 ω 3 ω 4 t ω 5 ω 6 ω 7 ω 8 T 0 T 1 T 2 T 3 T 4 Condition (*) S(T k ) is F k -measurable : S is adapted to {F k }. copyright 2012 Christian Fries 17 / 51

copyright 2012 Christian Fries 18 / 51 Modeling Random Variable and Stochastic Processes Modeling: Prototype of a stochastic process - building block of Itô processes: Brownian Motion: W W (t) defined over (Ω,F,P). W (0) = 0. W (t) normal distribution with mean 0 and standard deviation t. W (t 2 ) W (t 1 ) normal distribution with mean 0 and standard deviation t 2 t 1 (i.i.d). W (, ω) continuous (but nowhere differentiable) function [0, ) IR (P-a.s.).

copyright 2012 Christian Fries 18 / 51 Modeling Random Variable and Stochastic Processes Modeling: Prototype of a stochastic process - building block of Itô processes: Brownian Motion: W W (t) defined over (Ω,F,P). W (0) = 0. W (t) normal distribution with mean 0 and standard deviation t. W (t 2 ) W (t 1 ) normal distribution with mean 0 and standard deviation t 2 t 1 (i.i.d). W (, ω) continuous (but nowhere differentiable) function [0, ) IR (P-a.s.).

Modeling Random Variable and Stochastic Processes Construction of realizations at discrete times: k 1 W (t k ) := where i=0 W (t i ) (0 = t 0 < t 1 <...), W (t 0 ) := 0, W (t i ) = (W (t i+1 ) W (t i )) N (0, t i+1 t i ) (i.i.d). W(t) ΔW(T 1 ) = W(T 2 )-W(T 1 ) ΔW(T 2 ) = W(T 3 )-W(T 2 ) ΔW(T 3 ) = W(T 4 )-W(T 3 ) ω - (path) T 0 T 1 T 2 T 3 T copyright 2012 Christian Fries 4 19 / 51

copyright 2012 Christian Fries 19 / 51 Modeling Random Variable and Stochastic Processes Construction of realizations at discrete times: k 1 W (t k ) := where i=0 W (t i ) (0 = t 0 < t 1 <...), W (t 0 ) := 0, W (t i ) = (W (t i+1 ) W (t i )) N (0, t i+1 t i ) (i.i.d). Infinitesimal Notation: t W (t) =: dw (τ). 0

Modeling Random Variable and Stochastic Processes Stochastic Differential Equation: Euler Discretization (e.g.): i.e. ds = µ(t,s(t)) dt + σ(t,s(t))dw (t) S(t i ) = µ(t }{{} i, S(t i )) t i + σ(t }{{} i, S(t i )) W i, }{{} S(t i+1 ) S(t i ) t i+1 t i N (0, t i ) S(t i+1 ) = S(t i ) + µ(t i, S(t i )) (t i+1 t i ) + σ(t i, S(t i )) W i, with S(0) = S(0). { S(t i ) 0 = t 0 < t 1 <...} is the Euler discretization of {S(t) t 0}. copyright 2012 Christian Fries 20 / 51

copyright 2012 Christian Fries 21 / 51 Modeling Random Variable and Stochastic Processes Example: log normal process for stock value S: ds(t) = µ(t)s(t)dt +σ(t)s(t)dw (t) ds(t) S(t) = µ(t)dt +σ(t)dw (t) Euler Scheme: S(t j+1 ) = S(t j ) + µ(t j ) S(t j ) t j + σ(t j ) S(t j ) W j, }{{} N (0, t j ) Log Euler Scheme: S(t j+1 ) = S(t ( j ) exp (µ(t j ) 1 2 σ(t j) 2 ) t j + σ(t j ) W j }{{} ), N (0, t j )

copyright 2012 Christian Fries 21 / 51 Modeling Random Variable and Stochastic Processes Example: log normal process for stock value S: ds(t) = µ(t)s(t)dt +σ(t)s(t)dw (t) ds(t) S(t) = µ(t)dt +σ(t)dw (t) Euler Scheme: S(t j+1 ) = S(t j ) + µ(t j ) S(t j ) t j + σ(t j ) S(t j ) W j, }{{} N (0, t j ) Log Euler Scheme: S(t j+1 ) = S(t ( j ) exp (µ(t j ) 1 2 σ(t j) 2 ) t j + σ(t j ) W j }{{} ), N (0, t j )

copyright 2012 Christian Fries 21 / 51 Modeling Random Variable and Stochastic Processes Example: log normal process for stock value S: ds(t) = µ(t)s(t)dt +σ(t)s(t)dw (t) ds(t) S(t) = µ(t)dt +σ(t)dw (t) Euler Scheme: S(t j+1 ) = S(t j ) + µ(t j ) S(t j ) t j + σ(t j ) S(t j ) W j, }{{} N (0, t j ) Log Euler Scheme: S(t j+1 ) = S(t ( j ) exp (µ(t j ) 1 2 σ(t j) 2 ) t j + σ(t j ) W j }{{} ), N (0, t j )

Modeling Random Variable and Stochastic Processes Exercise: Excel Sheet with Monte-Carlo Simulation copyright 2012 Christian Fries 22 / 51

copyright 2012 Christian Fries 23 / 51 Outline Notation Differential Equations Monte-Carlo Simulation Modeling Risk Neutral Valuation Change of Measure / Drift Example: Black Scholes Model Example: Black-Scholes Model Monte-Carlo Option Pricer

copyright 2012 Christian Fries 24 / 51 Risk Neutral Valuation Product A: Weather Derivative At time T 2 > 0 measure amount of rain R(T 2 ) (in mm) fallen at a predetermined place for a predetermined period. Pay the Euro amount A(T 2 ) := (R(T 2 ) X) C mm. What determines the value A(T 1 ) of this contract at time T 1 < T 2? Note: R(T 2 ) is stochastic Value depends on assessment of the probability of rain, risk,... Product B: Equity Derivative At time T 2 measure the quoted value of IBM S(T 2 ) (in C). Pay the Euro amount B(T 2 ) := (S(T 2 ) X) What determined the value B(T 1 ) of this contract at time T 1 < T 2? Product looks most similar to the first one.

copyright 2012 Christian Fries 25 / 51 Risk Neutral Valuation Replication: The payoff of B can be replicated through products traded in T 1 : Let P(T 2 ;T 1 ) denote the value of a credit which has to be payed back in T 2 by an amount of 1 ( Zero Bond). In T 1 : Then Buy stock S(T 1 ) and X times a credit P(T 2 ;T 1 ) (stock long, bond short). Value in T 2 : S(T 2 ) X = B(T 2 ) Value in T 1 : S(T 1 ) X P(T 2 ;T 1 )! = B(T 1 ) Value of replication portfolio is a fair value for the derivative.

copyright 2012 Christian Fries 26 / 51 Risk Neutral Valuation Discrete Time (T 1,T 2 ), Two Assets (S,B), Two States (ω 1,ω 2 ) Given: In T 2 we have two disjoint events (states) ω 1 and ω 2, the payoff of a derivative product V (T 2,ω i ) and two traded assets S und B. Wanted: Solution: A portfolio αs + βb, which replicates the value V (T 2 ) in T 2 for any state. Its value in T 1 determines the (fair) value V (T 1 ) of V cost of replication. α S(T 1 ) + β B(T 1 ) α S(T 2 ;ω 1 ) + β B(T 2 ;ω 1 )! = V (T 2 ;ω 1 ) α S(T 2 ;ω 2 ) + β B(T 2 ;ω 2 )! = V (T 2 ;ω 2 )

Risk Neutral Valuation Linear equation, two equations, two unknowns. Solvable α S(T 1 ) + β B(T 1 ) α S(T 2 ;ω 1 ) + β B(T 2 ;ω 1 )! = V (T 2 ;ω 1 ) α S(T 2 ;ω 2 ) + β B(T 2 ;ω 2 )! = V (T 2 ;ω 2 ) S(T 2 ;ω 1 ) B(T 2 ;ω 2 ) S(T 2 ;ω 2 ) B(T 2 ;ω 1 ) 0 B 0 Overhead: S(T 2 ;ω 1 ) B(T 2 ;ω 1 ) S(T 2;ω 2 ) B(T 2 ;ω 2 ) For every derivative product V, the replication portfolio α(t), β(t) has to be calculated (though every time step t). copyright 2012 Christian Fries 27 / 51

copyright 2012 Christian Fries 28 / 51 Risk Neutral Valuation Same example again - but consider everything divided by B α S(T 1) B(T 1 ) + β 1 p 1 p α S(T 2;ω 1 ) B(T 2 ;ω 1 ) + β 1 =! V (T 2;ω 1 ) B(T 2 ;ω 1 ) α S(T 2;ω 2 ) B(T 2 ;ω 2 ) + β 1 =! V (T 2;ω 2 ) B(T 2 ;ω 2 )

copyright 2012 Christian Fries 28 / 51 Risk Neutral Valuation Same example again - but consider everything divided by B α S(T 1) B(T 1 ) + β 1 p 1 p α S(T 2;ω 1 ) B(T 2 ;ω 1 ) + β 1 =! V (T 2;ω 1 ) B(T 2 ;ω 1 ) α S(T 2;ω 2 ) B(T 2 ;ω 2 ) + β 1 =! V (T 2;ω 2 ) B(T 2 ;ω 2 ) In general ( ) V (T 1 ) V B(T 1 ) (T2 ) EP B(T 2 ) and thus ( ) S(T 1 ) B(T 1 ) S(T2 ) EP. B(T 2 )

copyright 2012 Christian Fries 28 / 51 Risk Neutral Valuation Same example again - but consider everything divided by B α S(T 1) B(T 1 ) + β 1 q 1 q α S(T 2;ω 1 ) B(T 2 ;ω 1 ) + β 1 =! V (T 2;ω 1 ) B(T 2 ;ω 1 ) α S(T 2;ω 2 ) B(T 2 ;ω 2 ) + β 1 =! V (T 2;ω 2 ) B(T 2 ;ω 2 )

copyright 2012 Christian Fries 28 / 51 Risk Neutral Valuation Same example again - but consider everything divided by B α S(T 1) B(T 1 ) + β 1 q 1 q α S(T 2;ω 1 ) B(T 2 ;ω 1 ) + β 1 =! V (T 2;ω 1 ) B(T 2 ;ω 1 ) α S(T 2;ω 2 ) B(T 2 ;ω 2 ) + β 1 =! V (T 2;ω 2 ) B(T 2 ;ω 2 ) ( ) ( ) However, if sign S(T2 ;ω 2 ) B(T 2 ;ω 2 ) S(T 1) B(T 1 ) sign S(T2 ;ω 1 ) B(T 2 ;ω 1 ) S(T 1) B(T 1 ) there exists a prob.-measure Q s.th. ( ) S(T 1 ) B(T 1 ) = S(T2 ) EQ B(T 2 ) and thus ( ) V (T 1 ) V B(T 1 ) = (T2 ) EQ. B(T 2 )

copyright 2012 Christian Fries 29 / 51 Risk Neutral Valuation Equivalent Martingale Measure The measure Q is the so called equivalent martingale measure. It does not depend on V (!) - this does not hold if one considers V (T 1 )! = E(V (T 2 )) in place of V (T 1) B(T 1 )! = E ( V (T2 ) B(T 2 ) Numéraire The measure Q depends on the reference quantity (here B), the so called numéraire. Here: ). q S(T 2;ω 1 ) B(T 2 ;ω 1 ) + (1 q) S(T 2;ω 2 )! = S(T 1) B(T 2 ;ω 2 ) B(T 1 ) q = S(T 2 ;ω 1 ) B(T 2 ;ω 1 ) S(T 1) B(T 1 ) S(T 2 ;ω 1 ) B(T 2 ;ω 1 ) S(T 2;ω 2 ) B(T 2 ;ω 2 )

copyright 2012 Christian Fries 30 / 51 Risk Neutral Valuation Universal Pricing Theorem: V (0) N(0) = EQN ( ) V (Tn ) F 0 N(T n ) Assume V consists of finite number of payments ( X X i paid in T i, N-relative payment value: E QN i ) F 0. N(T i ) Value E QN ( V N ) F 0 = n i=1 E QN ( Xi N(T i ) ) F 0

copyright 2012 Christian Fries 31 / 51 Risk Neutral Valuation Monte-Carlo Simulation: Sample space Ω = {ω 1,...,ω m }, e.g. m 10000 V (0) ) ( ) F 0 = n i=1 E QN X i N(T i ) F 0 N(0) = ( V EQN ( N ) E QN X i F 0 1 m ω Ω N(T i ) X i (ω) N(T i ;ω) V (0) = N(0) E QN ( V N ) F 0 1 m ω Ω n i=1 N(0) X i (ω) N(T i ;ω) }{{} state price deflator / discount factor

copyright 2012 Christian Fries 32 / 51 Risk Neutral Valuation Monte-Carlo Simulation Advantages Pricing of path dependent options in a natural way. High dimensions possible (but slower convergence in high dimensions). Straight forward to implement. Challenges Sensitivities (partial derivatives of option price w.r.t. model parameters) tend to be unstable (harder to get them stable). Pricing of Bermudan option needs additional effort. Calibration.

copyright 2012 Christian Fries 33 / 51 Risk Neutral Valuation Monte-Carlo Simulation Alternatives Analytic pricing formulas - only for simple models and simple payoffs. Numerical integration - only for simple models and European style options. Lattice models (PDE, Tree, Markov Functional) - only for low dimensional models

RISK NEUTRAL VALUATION CHANGE OF MEASURE / DRIFT

copyright 2012 Christian Fries 35 / 51 Change of Drift Trick I Risk Neutral Pricing: Find a measure Q such that for all underlying assets S we have ( ) E Q S(Ti+1 ) N(T i+1 ) F T i = S(T i) N(T i ), then for a derivative the cost of replication V satisfies ( ) V E Q (Ti+1 ) N(T i+1 ) F T i = V (T i) N(T i ). Martingale Property: (in place of S N or V N we simple write X) E Q( ) X(T i+1 ) F Ti = X(Ti ) E Q( ) X(T i+1 ) X(T i ) F }{{} Ti = 0 =: X(T i ) How to calculate the measure Q?

copyright 2012 Christian Fries 36 / 51 Change of Drift Trick II Never need to (explicitly) calculate the measure Q: Conditional Expectation: E Q ( X(T i ) F Ti ) = X(T i,ω) Q({ω}) ω Ω interpretation 1: the measure has changed from P to Q: interpretation 2: the values have changed: = X(T i,ω) Q({ω}) ω Ω P({ω}) P({ω}) }{{} measure changed = X(T i,ω) Q({ω}) P({ω}) ω Ω P({ω}) }{{} value changed

copyright 2012 Christian Fries 37 / 51 Change of Drift Trick III Question: How does the process look like, if it has to satisfy the martingale property? Answer: The drift is zero. dx(t) = µdt + σdw (t) and E Q (X(T i+1 ) X(T i ) F Ti ) = 0 µ = 0 Note: X(T i+1 ) X(T i ) = X = T i+1 T i dx(t).

copyright 2012 Christian Fries 38 / 51 Change of Drift Trick IV Conclusion: Instead of calculating the pricing measure (martingale measure), simply write down the process and calculate the correct drift. change of measure change of drift A Monte-Carlo simulation of that process corresponds to a simulation under the pricing measure.

Change of Drift Trick V Itô Lemma: Main Tool to Calculate the Drift of Processes Let X denote an Itô prozess with dx(t) = µ dt + σ dw. Let g(t,x) denote some function g C 2 ([0, ] R). The we have that is an Itô process with Y (t) := g(t,x(t)) dy t = g t (t,x t) dt + g x (t,x t) dx t + 1 2 2 g x 2 (t,x t) (dx t ) 2, where (dx t ) 2 = (dx t ) (dx t ) is given by formal expansion with dt dt = 0, dt dw = 0, dw dt = 0, dw dw = dt, copyright 2012 Christian Fries 39 / 51

copyright 2012 Christian Fries 40 / 51 Change of Drift Trick VI i.e. (dx t ) 2 = (dx t ) (dx t ) = (µ dt + σ dw ) (µ dt + σ dw ) = µ 2 dt dt + µ σ dt dw + µ σ dw dt + σ 2 dw dw = σ 2 dt

copyright 2012 Christian Fries 41 / 51 Change of Drift Trick VII Itô Lemma: Example: dx(t) = µx(t) dt + σx(t) dw. Consider logarithm of X Y (t) := log(x(t)) then dy (t) = (µ 1 2 σ 2 ) dt + σ dw.

RISK NEUTRAL VALUATION EXAMPLE: BLACK SCHOLES MODEL

copyright 2012 Christian Fries 43 / 51 Example: Black-Scholes Model Black-Scholes Model: ds = µsdt + σsdw ds = µdt + σdw S Interpretation: Consider discrete times t 0,t 1,t 2,t 3,... Then: where t i = t i+1 t i S(t i ) S S(t i ) S(t i ) = S(t i+1) S(t i ) S(t i ) µ t i = µ (t i+1 t i ) σ W (t i ) = σ (W (t i+1 ) W (t i )) }{{} Normal distributed with variance t i = µ t i + σ W (t i ) A time period. Relative change of the stock over the period t i = t i+1 t i. Risk less part: A deterministic drift. µ = annualized risk less rate of return. Risky part: A normal distributed random variable. σ = annualized standard deviation of the return.

copyright 2012 Christian Fries 43 / 51 Example: Black-Scholes Model Black-Scholes Model: ds = µsdt + σsdw ds = µdt + σdw S Interpretation: Consider discrete times t 0,t 1,t 2,t 3,... Then: where t i = t i+1 t i S(t i ) S S(t i ) S(t i ) = S(t i+1) S(t i ) S(t i ) µ t i = µ (t i+1 t i ) σ W (t i ) = σ (W (t i+1 ) W (t i )) }{{} Normal distributed with variance t i = µ t i + σ W (t i ) A time period. Relative change of the stock over the period t i = t i+1 t i. Risk less part: A deterministic drift. µ = annualized risk less rate of return. Risky part: A normal distributed random variable. σ = annualized standard deviation of the return.

copyright 2012 Christian Fries 44 / 51 Example: Black-Scholes Model Monte-Carlo Implementation Implementation: (Monte-Carlo) Simulation: Consider time-discrete version of Black-Scholes Model i.e. S(t i ) S = µ t i + σ W (t i ) S(t i+1 ) = S(t i ) + µ S(t i ) t i + σ S(t i ) W (t i ). For a single path (scenario) we have S(t i+1,ω) = S(t i,ω) + µ S(t i,ω) t i + σ S(t i,ω) W (t i,ω) (Euler Sch Remark: Much more accurate numerical simulation is given by S(t i+1,ω) = S(t i,ω) exp ( (µ 1 2 σ 2 ) t i + σ W (t i,ω) ) (Log-Euler

copyright 2012 Christian Fries 45 / 51 Example: Black-Scholes Model Binomial Tree Implementation Implementation: (Binomial) Tree: Consider time-discrete version of Black-Scholes Model S(t i+1 ) = S(t i ) + µ S(t i ) t i + σ S(t i ) W (t i ). Approximate the normal distributed W (t i ) by a binomial distributed random variable B(t i ) where S(t i+1 ) S(t i ) + µ S(t i ) t i + σ S(t i ) B(t i ). B(t i ) = { + t i with probability 1 2 t i with probability 1 2 Justification: Repetitive binomial experiments converge to a normal distribution. n i=0 B(t i ) n i=0 W (t i ) for n large.

copyright 2012 Christian Fries 46 / 51 Example: Black-Scholes Model Under Martingale Measure Back-Scholes Model: Evolution of Money Market Account: Evolution of Stock........ : db = r B dt ds = µ Sdt + σ SdW (t) Choose Numéraire and consider Martingale Measure: B chosen as Numéraire Change of Drift Trick: S B is Q-martingale d S B = (µ r)s B dt + σ S dw (t) B drift has to be zero µ = r

RISK NEUTRAL VALUATION EXAMPLE: BLACK-SCHOLES MODEL MONTE-CARLO OPTION PRICER

Example: Black-Scholes Model Example: Monte-Carlo European Option Pricing Example: European option value: European option value is a known function of the underlying process(es) at some future time T n : V (T n ) = max(s(t n ) K,0). Choose a Model for Underlyings: e.g. Black-Scholes Evolution of Money Market Account: Evolution of Stock........ : db = r B dt ds = µ Sdt + σ SdW (t) Choose Numéraire and consider Martingale Measure: B Numéraire S B Q-martingale µ = r Monte-Carlo Simulation (of the Q dynamics): B(t i+1,ω) = B(t 0 ) exp(r t i+1 ) S(t i+1,ω) = S(t i ) + r S(t i ) + σ S(t i ) W (t i,ω) for ω = ω 1,ω 2,ω 3,ω 4,ω 5,ω 6,...,ω N (some 1000 s samples). copyright 2012 Christian Fries 48 / 51

Example: Black-Scholes Model Example: Monte-Carlo European Option Pricing Example: European option value: European option value is a known function of the underlying process(es) at some future time T n : V (T n ) = max(s(t n ) K,0). Choose a Model for Underlyings: e.g. Black-Scholes Evolution of Money Market Account: Evolution of Stock........ : db = r B dt ds = µ Sdt + σ SdW (t) Choose Numéraire and consider Martingale Measure: B Numéraire S B Q-martingale µ = r Monte-Carlo Simulation (of the Q dynamics): B(t i+1,ω) = B(t 0 ) exp(r t i+1 ) S(t i+1,ω) = S(t i ) + r S(t i ) + σ S(t i ) W (t i,ω) for ω = ω 1,ω 2,ω 3,ω 4,ω 5,ω 6,...,ω N (some 1000 s samples). copyright 2012 Christian Fries 48 / 51

Example: Black-Scholes Model Example: Monte-Carlo European Option Pricing Example: European option value: European option value is a known function of the underlying process(es) at some future time T n : V (T n ) = max(s(t n ) K,0). Choose a Model for Underlyings: e.g. Black-Scholes Evolution of Money Market Account: Evolution of Stock........ : db = r B dt ds = µ Sdt + σ SdW (t) Choose Numéraire and consider Martingale Measure: B Numéraire S B Q-martingale µ = r Monte-Carlo Simulation (of the Q dynamics): B(t i+1,ω) = B(t 0 ) exp(r t i+1 ) S(t i+1,ω) = S(t i ) + r S(t i ) + σ S(t i ) W (t i,ω) for ω = ω 1,ω 2,ω 3,ω 4,ω 5,ω 6,...,ω N (some 1000 s samples). copyright 2012 Christian Fries 48 / 51

Example: Black-Scholes Model Example: Monte-Carlo European Option Pricing Example: European option value: European option value is a known function of the underlying process(es) at some future time T n : V (T n ) = max(s(t n ) K,0). Choose a Model for Underlyings: e.g. Black-Scholes Evolution of Money Market Account: Evolution of Stock........ : db = r B dt ds = µ Sdt + σ SdW (t) Choose Numéraire and consider Martingale Measure: B Numéraire S B Q-martingale µ = r Monte-Carlo Simulation (of the Q dynamics): B(t i+1,ω) = B(t 0 ) exp(r t i+1 ) S(t i+1,ω) = S(t i ) + r S(t i ) + σ S(t i ) W (t i,ω) for ω = ω 1,ω 2,ω 3,ω 4,ω 5,ω 6,...,ω N (some 1000 s samples). copyright 2012 Christian Fries 48 / 51

copyright 2012 Christian Fries 49 / 51 Example: Black-Scholes Model Example: Monte-Carlo European Option Pricing Calculate Payoff for each Sample Path: V (T n,ω j ) = max ( S(T n,ω j ) K,0 ) Calculate Price: ( ) V V (0) = B(0) E Q (Tn ) B(T n ) 1 N = 1 N max ( S(T n,ω j ) K,0 ) N B(T j=1 n,ω j ) N V (T n,ω j ) B(T j=1 n,ω j )

copyright 2012 Christian Fries 50 / 51 Example: Black-Scholes Model Example: Monte-Carlo European Option Pricing Exercise: Excel Sheet: Monte-Carlo Simulation with Option Pricing

copyright 2012 Christian Fries 51 / 51 Further Reading I FRIES, CHRISTIAN P.: Mathematical Finance. Theory, Modeling, Implementation. John Wiley & Sons, 2007. ISBN 0-470-04722-4. http://www.christian-fries.de/finmath/book.