STOCHASTIC INTEGRALS

Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1 <... < t n = t: P/L t = i<n θ ti (X ti+1 X ti ) }{{} X ti GRID BECOMES DENSE : max i t i P/L t θ u dx u INTEGRAL DEFINED AS LIMIT OF SUMS Winter 27 1 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 PROPERTIES MOSTLY FROM SUMS: (aθ ti + bη ti ) X ti i<n }{{} = a θ ti X ti + b ηt i X ti i<n i<n }{{} }{{} (aθ u + bη u )dx u a θ u dx u + b η u dx u LINEARITY OK TIME VARYING INTEGRAL: θ u dx u = limit of θ ti (X ti+1 X ti ) t i+1 t Limit in probability Winter 27 2 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 MARTINGALE PROPERTY: If X t = M t = MG: U (n) t = t i+1 t θ ti X ti : U (n) t i+1 = θ ti X ti If on grid t, t 1,...: E( U (n) t i+1 F ti ) = E(θ ti X ti F ti ) = θ ti E( X ti F ti ) = U (n) t is F ti MG Taking limits: E(U t F s ) = U s Winter 27 3 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 QUADRATIC VARIATION (Q. V.) U (n) t = t i+1 t θ ti X ti so: U (n) t i = U (n) t i+1 U (n) t i = θ ti X ti Aggregate: [U (n), U (n) ] t }{{} = = ( U (n) t i ) 2 = θ 2 t i ( X ti ) 2 ( U (n) t i ) 2 t i+1 t t i+1 t [U, U] t = θ 2 t i ( X ti ) 2 = If X t = W t = B.M. : d[x, X] t = dt IT FOLLOWS THAT [U, U] t = θ2 udu t i+1 t θ 2 t i [X, X] ti }{{ } θ 2 ud[x, X] u Winter 27 4 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 INTEGRAL: U (n) t i becomes DIFFERENTIAL NOTATION = θ ti X ti vs. U t = U + du t = θ t dx t vs. U t = U + QUADRATIC VARIATION: t i+1 t θ ti X ti θ s dx s ( U (n) t i ) 2 = θ 2 t i ( X ti ) 2 vs. [U (n), U (n) ] t = θ 2 t i [X, X ti ] [U (n), U (n) ] ti = θ 2 t i [X, X] ti becomes: (du t ) 2 = θ 2 t (dx t ) 2 vs. [U, U] t = θ 2 ud[x, X] u BROWNIAN MOTION: d[u, U] t = θ 2 t d[x, X] t (dw t ) 2 = dt AND d[u, U] t = θ 2 t dt Winter 27 5 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 QUADRATIC COVARIATION: U, Z : [U, Z] t = limit of t i+1 t U ti Z ti CASE OF TWO INTEGRALS: U t = θ s dx s, Z t = η s dy s THEN: [U, Z] t = θ s η s d[x, Y ] s BECAUSE or U ti Z ti = θ ti η ti X ti Y ti du t dz }{{} t = θ t η t dx t dy }{{} t d[u, Z] t d[x, Y ] t IF: X t = Y t = W t THE SAME B.M.: d[u, Z] t = θ t η t dt Winter 27 6 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 DETERMINISTIC INTEGRAND IF θ t IS NONRANDOM: t i+1 t θ t i W ti : θ s dw s = limit of t i+1 t θ ti W ti LINEAR COMBINATION OF NORMAL RANDOM VARIABLES IS A NORMAL RANDOM VARIABLE MEAN: E t i+1 t θ t i W ti = VARIANCE: Var ( t i+1 t θ t i W ti ) = t i+1 t θ2 t i Var ( W ti ) = t i+1 t θ2 t i t i θ sdw s : IN THE LIMIT: NORMAL RANDOM VARIABLE MEAN IS ZERO VARIANCE: Var ( θ sdw s ) = E[ θ sdw s, θ sdw s ] t = θ2 sds Winter 27 7 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 ITÔ s FORMULA X t : CONTINUOUS PROCESS (SOME RESTRICTIONS): ξ: TWICE CONTINUOUSLY DIFFERENTIABLE ξ(x t ) = ξ(x ) + ξ (X u )dx u + 1 2 ξ (X u )d[x, X] t DIFFERENTIAL NOTATION: dξ(x t ) = ξ (X t )dx t + 1 2 ξ (X t )d[x, X] t EX: X t = W t = BROWNIAN MOTION: dξ(w t ) = ξ (W t )dw t + 1 2 ξ (W t )dt EX: dx t = ν t dt + σ t dw t ITÔ PROCESS or: X t = X + ν sds + σ sdw s First question: What is d[x, X] t? Winter 27 8 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 FIRST: CASE OF EXPLICIT INTEGRATION: W t = B.M. WHAT IS W sdw s? U t = W 2 t = ζ(w t ), ζ(x) = x 2 du t = ζ (W t )dw t + 1 2 ζ (W t )dt = 2W t dw t + dt so: W t dw t = 1 2 du t 1 2 dt >: W s dw s = 1 2 (U t U ) 1 2 ds = 1 2 W 2 t 1 2 t DIFFERENT FROM ORDINARY INTEGRAL: If X t = g(t) g exists, continuous, g(t) = X s dx s = g(s)g (s)ds = 1 2 g(t)2 = 1 2 X2 t Winter 27 9 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 ITÔ PROCESS: Grid: X t = X + ν s ds+ }{{} Z t σ s dw s } {{ } U t so: X ti = Z ti + U ti ( X ti ) 2 = ( Z ti ) 2 + ( U ti ) 2 + 2 Z ti U ti ( Xti ) 2 = ( Z ti ) 2 + ( U ti ) 2 + 2 Z ti U ti Z ti = i+1 t i sup ( Zti ) 2 (sup s s (sup s ν s ds i+1 t i ν s (t i+1 t i ) = sup s ν s ) 2 i ( t i ) 2 ν s ) 2 sup t i }{{} : [Z, Z] t = ALSO: [Z, U] t = ν s ds t i i }{{} t ν s t i ONLY: [U, U] t = σ2 sds Winter 27 1 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 X t = X + ITÔ PROCESS ν s d s + }{{} Z t σ s dw s } {{ } U t d[z, Z] t = d[z, U] t = d[u, U] t = σt 2 dt (dz t ) 2 dz t du t USING DIFFERENTIALS: ANY dt-term HAS ZERO Q.V.: (dz t ) 2 = ν 2 t (dt) 2 = ETC COMBINING TERMS: (dx t ) 2 = (dz t + du t ) 2 = (dz t ) 2 + 2dZ t du t + (du t ) 2 = (du t ) 2 = σt 2 dt RIGOROUS: ( X ti ) 2 = Zt 2 i + 2 Z ti U ti + ( U ti ) 2 SUM OVER t i, TAKE LIMITS, GET SAME RESULT Winter 27 11 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 INTEGRALS WITH RESPECT TO AN ITÔ PROCESS X t = X + ν s ds + σ s dw s CAN SHOW THAT: θ s dx s = θ s ν s ds + θ s σ s dw s A NEW ITÔ PROCESS Winter 27 12 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 BACK TO ITÔ S FORMULA: dξ(x t ) = ξ (X t )dx t + 1 2 ξ (X t )d[x, X] t ( ) ITÔ PROCESS: dx t = ν t dt + σ t dw t SO d[x, X] t = σ 2 t dt PLUG IN: dξ(x t ) = ξ (X t )(ν t dt + σ t dw t ) + 1 2 ξ (X t )σt 2 dt = (ξ (X t )ν t + 1 2 ξ (X t )σt 2 )dt + ξ (X t )σ t dw t EASIER TO REMEMBER (*)... Winter 27 13 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 PROOF OF ITÔ S FORMULA: U t = ξ(x t ): U ti = ξ(x ti+1 ) ξ(x ti ) = ξ(x ti + X ti ) ξ(x ti ) sum up: = ζ (X ti ) X ti + 1 2 ζ (X ti ) X 2 t i + 1 3! ζ (X ti ) X 3 t i + U t U = ζ (X ti ) X ti }{{ } + ζ (X s )dx s + 1 ζ (X ti ) Xt 2 2 i }{{ } 1 2 ζ (X s )d[x, X] s OTHER PROOF : du t = ζ(x t + dx t ) ζ(x t ) = ζ (X t )dx t + 1 2 ζ (X t ) (dx t ) 2 + }{{} d[x,x] t Winter 27 14 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 MULTIVARIATE FORMULA etc. U t = ζ(x t, Y t ) du t = ζ x(x t, Y t )dx t + ζ y(x t, Y t )dy t + 1 { ζ 2 xx(x t, Y t )d[x, X] t + ζ yy(x t, Y t )d[y, Y ] t } + 2ζ xy(x t, Y t )d[x, Y ] t Winter 27 15 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 EXAMPLE: GEOMETRIC BROWNIAN MOTION S t = S exp{ σ s dw s + ( r s 1 ) 2 σ2 s ds} SET X t = σ sdw s + ( ) rs 1 2 σ2 s ds S t = f(x t ) (f(x) = S exp{x}) USE ITÔ S FORMULA ds t = f (X t )dx t + 1 2 f (X t )d[x, X] t f (x) = f (x) = f(x) AND d[x, X] t = σt 2 dt, SO: ds t = f(x t )dx t + 1 2 f(x t)σt 2 dt = S t dx t + 1 2 S tσt 2 dt = S t (dx t + 1 ) 2 σ2 t dt = S t (σ t dw t + r t dt) = S t σ t dw t + S t r t dt DIFFERENTIAL REPRESENTATION OF S t Winter 27 16 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 VASICEK MODEL dr t = (α βr t )dt + σdw t STEP 1: SET U t = R t α β EQUATION BECOMES: du t = βu t dt + σdw t STEP 2: NOTE THAT (FROM ITO S FORMULA) d(exp{βt}u t ) = exp{βt}du t + U t d exp{βt} = exp{βt}du t + U t β exp{βt}dt = exp{βt} (du t + U t βdt) = exp{βt}σdw t SO exp{βt}u t = U + exp{βs}σdw s OR U t = exp{ βt}u + exp{β(s t)}σdw s Winter 27 17 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 IN OTHER WORDS: U t IS NORMAL MEAN IS exp{ βt}u VARIANCE IS (exp{β(s t)}σ) 2 ds = exp{2β(s t)}σ 2 ds [ 1 = exp{2β(s t)}σ2 2β = 1 (1 exp{ 2βt}) σ2 2β ] s=t s= DEDUCE FOR R t = U t + α β THAT R t IS NORMAL E(R t ) = exp{ βt}u + α β Var (R t ) =Var (U t ) Winter 27 18 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 LEVY S THEOREM IF M t IS A CONTINUOUS (LOCAL) MARTINGALE, M =, [M, M] t = t FOR ALL t, THEN M t IS A CONTINUOUS BROWNIAN MOTION PROOF: SET f(x) = exp{hx} ITO: df(m t ) = f (M t )dm t + 1 2 f (M t )d[m, M] t = f (M t )dm t + 1 2 f (M t )dt. SINCE dm t TERM IS MG, AND f (x) = h 2 f(x): E(f(M t ) s ) = f(m s ) + 1 ( ) 2 h2 E f(m u )du s s = f(m s ) + 1 2 h2 E(f(M u ) s )du Set g(t) = E(exp{h(M t M s )} s ): s g(t) = 1 + 1 2 h2 s g(u)du Winter 27 19 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 SOLUTION: g(t) = exp{ 1 2 h2 (t s)} IN OTHER WORDS: E(exp{h(M t M s )} s ) = exp{ 1 2 h2 (t s)} CHARACTERISTIC FUNCTION ARGUMENT GIVES: M t M s IS INDEPENDENT OF s M t M s IS N(, t s) Winter 27 2 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 X t = X + Decomposition unique: ITÔ PROCESSES dw t term is martingale dt term is drift a s ds+ }{{} dt term (Doob-Meyer decomposition) b s dw s }{{} dw t term UNDER RISK NEUTRAL MEASURE P Discounted securities only have dw term: d S t = µ t S t dt+σ }{{} t S t dw t = Winter 27 21 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 UNDISCOUNTED SECURITIES UNDER P : d S t = σ t S t dw t 1) Numeraire = B t = exp( r udu) properties: db t = r t B t dt, [B, B] t =, [B, S] t = S t = S t B t ds t = B t d S t + S t db t = B t σ t St dw t }{{} + S t B t r t }{{} dt տր ր = σ t S t dw t + r t S t dt 2) Other numeraire: Λ t B t, Λ t has dw (2) t term d W, W (2) t = ρ t dt d S t Λ t = full use of Itô s formula Not same P!!! Winter 27 22 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 UNDISCOUNTED SECURITIES UNDER P : ds t = r t S t dt + σ t S t dw t LOG SCALE: ITO S FORMULA d log(s t ) = 1 ds t + 1 S t 2 ( 1 St 2 )d[s, S] t = 1 (r t S t dt + σ t S t dw t ) + 1 S t 2 ( 1 St 2 )σ 2 t S t dt = (r t 1 2 σ2 t )dt + σ t dw t Winter 27 23 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 PAYOFF: f(s T ) OPTIONS PRICES: PDE S DISCOUNTED PAYOFF: e rt f(s T ) = f( S T ) S T = e rt S T f( s) = e rt f(e rt s) CALL: f(s) = (s K) + f( s) = ( s e rt K) + CANDIDATE PRICE: DISCOUNTED: C( S t, t) satisfies: C( S, T) = f( S) AND (UNDER P ): d C( S t, t) { Hedge BS PDE = }MG term = C s( S t, t)d S t d S = σ t St dw t + C t( S t, t)dt + 1 2 C ss( S t, t) d[ S, S] t }{{} dt terms σt 2 S t 2 dt Winter 27 24 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 2 APPROACHES THE BS PDE: Solve (*) C t( s, t) + 1 C 2 ss( s, t)σ 2 s 2 = C( s, T) = f( s) THE MARTINGALE APPROACH: Set C( s, t) = E [ f( S T ) S t = s] Markov: C( St, t) = E [ f( S T ) F t ] = price under P This C either 1) Market is complete: C automatically satisfies (*) 2) Otherwise: check if C satisfies (*): yes: solution OK no: try something else Winter 27 25 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 REVERSAL OF DISCOUNTING Numeraire: B t = exp{rt} C(S t, t) = B t C( S t, t) ( ) St = B t C, t B t = e rt C(e rt S t, t) Hence: C(s, t) = e rt C(e rt s, t) = e rt E [ f( S T ) S t = e rt s] = e r(t t) E [f(s T ) S t = s] since f( s) = e rt f(e rt s) Winter 27 26 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 COMPUTATION OF EXPECTED VALUES log S T = log S t + ν(t t) }{{} ν = σ2 2 +σ (W T W t ) }{{} T t Z Z N(, 1) and so : S T = S t exp(ν(t t) + σ T t Z) C(s, t) = E[ f( S T ) S t = s] = E[ f( s exp(ν(t t) + σ T t Z)] = + f( sexp(ν(t t) + σ T t z)φ(z)dz where φ(z) = 1 2π exp( 1 2 z2 ) For non-discounted S t : ν = r 1 2 σ2 Winter 27 27 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 MORE GENERAL EUROPEAN CONTINGENT CLAIMS (ECC) η = PAYOFF AT TIME T }{{} FIXED LOOKBACK: η = ( max S t K) + t T ( 1 T ASIAN: η = S u du K T (S T K) + BARRIER η = UNLESS min S t X t T ) + ONLY THE IMAGINATION IS THE LIMIT... Winter 27 28 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 OPTION PRICES: GENERAL SCHEME SELF FINANCING STRATEGIES: η = C T dc t = θ () t db t + C t = θ () t B t + K i=1 K i=1 Same as (by numeraire invariance): η = 1 B T η : η = C T d C t = K θ (i) t d S (i) t i=1 C t = θ () t + K i=1 θ (i) t S (i) t ] θ (i) t ds (i) t θ (i) t S (i) t Key requirement ] Defines θ () t PROOF: ITÔ S FORMULA Winter 27 29 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 ON THE DISCOUNTED SCALE η = C T d C K t = θ (i) t d S (i) t UNDER P : SAME AS i=1 η = c + K T i=1 θ (i) t d S (i) t ( ) BY TAKING C t = E ( η F t ) If B = 1: c = C = PRICE AT (*): MARTINGALE REPRESENTATION THEOREM Winter 27 3 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 WHEN DOES THE REPRESENTATION THEOREM HOLD? Theorem: IF W (1),...,W (K) INDEPENDENT B.M. S F t = F W(1),...,W (K) t η @ F T, E η < : η = c + P i=1 T f t dw (i) t Brownian motions are like binomial trees Winter 27 31 Per A. Mykland

Stat 391/FinMath 346 Lecture 8 FROM BROWNIAN MOTION TO STOCK PRICE or: d S t = σ S t dw t log S t = log S 1 2 σ2 t + σ W t η @ F S T η @ F W T Get: η = c + = c + T T f t dw t f t σ S t }{{} θ t More complex if σ t or r t random... d S t Winter 27 32 Per A. Mykland