MFE6516 Stochastic Calculus for Finance

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1 MFE6516 Stochastic Calculus for Finance William C. H. Leon Nanyang Business School December 11, / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance 1 Symmetric Random Walks Scaled Symmetric Random Walks 2 2 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

2 A Symmetric Random Walk Symmetric Random Walks Scaled Symmetric Random Walks Suppose a fair coin toss outcome is ω =ω 1,ω 2,ω 3,...). Let { 1 if ωi = Head with probability 1 2, X i = Define M t = 1 if ω i = Tail with probability 1 2. { 0 if t =0, t X i for t =1, 2, 3,... The process M =M t ) is a symmetric random walk. 3 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Martingale Symmetric Random Walks Scaled Symmetric Random Walks The process M =M t ) is a martingale. 1 M is adapted. 2 M t t for all t. 3 For 0 s t, EM t F s )= EM t M s + M s F s ) = E M t M s )+M s = M s. 4 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

3 Increment Symmetric Random Walks Scaled Symmetric Random Walks Consider a time partition Π = { t 0, t 1, t 2,...,t m 1, t m } where 0=t 0 < t 1 < t 2 < < t m 1 < t m. The r.v. are independent. In addition, M t1 M t0, M t2 M t1,...,m tm M tm 1 E M ti M ti 1 ) =0, Var M ti M ti 1 ) = t i j=t i 1 +1 Var X j )=t i t i 1. 5 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Symmetric Random Walks Scaled Symmetric Random Walks Quadratic Variation of a Discrete Time Process The quadratic variation of the symmetric random walk M up to time T is defined to be [M, M] T = T ) 2 M i M i 1 = T. }{{} X i 6 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

4 Quadratic Variation Symmetric Random Walks Scaled Symmetric Random Walks Let Π = { t 0, t 1, t 2,...,t m 1, t m } be a partition of [0, T ]. The quadratic variation of a stochastic process Y up to time T is defined to be [Y, Y ] T = plim Π 0 m ) 2 Yti Y ti 1. Π max {t i t i 1 } is the maximum step size of the partition. i plim X n = X lim P X n X <ɛ)=1. n n 7 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance A Scaled Symmetric Random Walk Symmetric Random Walks Scaled Symmetric Random Walks Fix n > 0. Define where α t = nt [nt]. W n) t) = 1 n αt M [nt] α t ) M[nt] ) Note that W n) t) = 1 n M nt = 1 n nt X i if nt Z. 8 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

5 Exercise Symmetric Random Walks Scaled Symmetric Random Walks Suppose nt i Z for all i. Show the following: 1 The r.v. W n) t 1 ) W n) t 0 ), W n) t 2 ) W n) t 1 ),..., W n) t n ) W n) t n 1 ) are independent. 2 For 0 i < j m, ) E W n) t j ) W n) t i ) =0, ) Var W n) t j ) W n) t i ) = t j t i, ) E W n) t j ) F ti = W n) t i ). 9 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Quadratic Variation Symmetric Random Walks Scaled Symmetric Random Walks For t 0 such that nt Z, nt [W n), W n) ] t = W n) ) i n W n) ) i 1 n }{{} = nt = t. 1 n X i 2 1 n X i ) 2 10 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

6 Limiting Distribution Symmetric Random Walks Scaled Symmetric Random Walks Fix t 0. As n approaches infinity, the distribution of the scaled random walk W n) t) evaluated at time t converges to the normal distribution with mean 0 and variance t, i.e. W n) t) D N 0, t) as n. 11 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Proof Symmetric Random Walks Scaled Symmetric Random Walks Suppose nt Z. ) ) E e uwn) t) = E e u n M nt = = nt ) E e u n X i 1 2 e u n e u n ) nt. = E e u nt ) n X i Let f n u) =nt ln 1 2 e u n e u n ). 12 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

7 Symmetric Random Walks Scaled Symmetric Random Walks Since ln lim f nu) =t lim n n 1 2 e u n e u n ) 1 n = 1 2 u2 t, the m.g.f. of W n) t), ) E e uwn) t) e 1 2 u2 t as n. 13 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Differentiability Symmetric Random Walks Scaled Symmetric Random Walks The scaled random walk W n) has some natural time step; is linear between these time step; and is approximately normal. Note that W ) = lim n W n) is not differentiable. W ) t + h) W ) t) h d = Z h as h / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

8 Definition of Let Ω, F, P ) be a probability space. A process W is a Brownian motion if: 1 For each fixed ω Ω, the path W t) fort 0 is continuous that satisfies W 0) = 0. 2 For any 0 = t 0 < t 1 < t 2 < < t m, the increments W t 1 ) W t 0 ), W t 2 ) W t 1 ),...,W t m ) W t m 1 ) are independent; 3 and each increment is normally distributed with E W t i ) W t i 1 )) = 0, Var W t i ) W t i 1 )) = t i t i / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Covariance Let W be a Brownian motion. Suppose s < t. Then E W s)w t)) = E W s) W t) W s) ) + W 2 s) ) = E W s)) E W t) W s) )) + E W 2 s) ) = s. Therefore, Cov W s), W t)) = s t. 16 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

9 Distribution Let W Π) = W t 1 ), W t 2 ),...,Wt m ) ).ThenW Π) is a multi-variate normal r.v. and t 1 t 1 t 1... t 1 Var W Π)) = ) t 1 t 2 t 2... t 2 t i t j = t 1 t 2 t 3... t 3, t 1 t 2 t 3... t m E e u W Π)) = e 1 2 u Σu = e 1 m 2 m j=i u j) 2 t i t i 1 )). 17 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Filtration Let Ω, F, P ) be a probability space on which a Brownian motion W is defined. A filtration for the Brownian motion is a collection of σ-field { Ft) } t 0 satisfying 1 Information accumulation) For 0 s < t, Fs) Ft). 2 Adaptivity) For each t 0, W t) isft)-measurable. 3 For 0 s < t, the increment W t) W s) is independent of Fs). 18 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

10 Exercise Brownian motion W is a martingale, i.e. for 0 s t. EW t) Fs)) = W s) 19 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance First-Order Variation The first order variation of a function f up to time T is defined by FV T f ) = lim Π 0 m f ti ) f t i 1 ) where Π = { t 0, t 1, t 2,...,t m 1, t m } is a partition of [0, T ]. 20 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

11 Quadratic Variation The quadratic variation of a function f up to time T is defined by [f, f ]T ) = lim Π 0 m f ti ) f t i 1 ) ) 2 where Π = { t 0, t 1, t 2,...,t m 1, t m } is a partition of [0, T ]. 21 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Differentiable Functions Functions with continuous derivatives have zero quadratic variation. m f ti ) f t i 1 ) ) 2 m = f ξ i )t i t i 1 ) ) 2 m Π f ξ i ) ) 2 ti t i 1 ) }{{} T 0 f ξ)) 2 dξ 0 as Π / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

12 Let W be a Brownian motion. For all T 0, [W, W ]T ) = lim Π 0 m W ti ) W t i 1 ) ) 2 = T a.s. where Π = { t 0, t 1, t 2,...,t m 1, t m } is a partition of [0, T ]. 23 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Proof Let Q Π) = m W ti ) W t i 1 ) ) 2.Then Var E Q Π)) = Q Π)) = m W E ti ) W t i 1 ) ) ) 2 = T } {{ } t i t i 1 m W Var ti ) W t i 1 ) ) ) 2 2 Π T } {{ } ) 2 2 t i t i 1 0 as Π 0. lim Π 0 QΠ) = E Q Π)) = T a.s. 24 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

13 Quadratic Variation Since [W, W ]t) =t for all t, [W, W ]t 2 ) [W, W ]t 1 )=t 2 t 1. Informally, dw t) dw t) =dt. 25 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Cross Variation The cross variation of W T )andt is zero, i.e. for all T 0, Informally, lim Π 0 m W ti ) W t i 1 ) ) ) t i t i 1 =0 a.s. dw t) dt =0. 26 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

14 Proof m W ti ) W t i 1 ) ) ) t i t i 1 W T ) Π }{{} Π 0 as Π / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Remarks In addition, lim Π 0 n ) 2 ti t i 1 T Π 0 as Π 0. Informally, dt dt =0. 28 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

15 Example: Geometric Let W be a Brownian motion and S0) be a positive constant. For t 0, define St) =S0) e μ 1 2 σ2 )t+σw t). Let 0 T 1 < T 2 and Π = { } t 0, t 1, t 2,...,t m 1, t m be a partition of the interval [T 1, T 2 ]. Then ) Sti ) ln = μ 1 ) ti ) St i 1 ) 2 σ2 t i 1 + σ W ti ) W t i 1 ) ). 29 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance m )) 2 Sti ) ln St i 1 ) = μ 1 ) 2 2 σ2 m ti t i 1 ) 2 +2σ μ 1 ) m ) 2 σ2 ti t i 1 W ti ) W t i 1 ) ) m + σ 2 W ti ) W t i 1 ) ) 2 σ 2 T 2 T 1 ) as Π / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

16 Volatility of Geometric σ 2 1 T 2 T 1 m Sti ) ln St i 1 ) )) 2. } {{ } Realized volatility 31 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Let W be a Brownian motion and m be a constant. The first passage time to level m is τ m =min { t 0 W t) =m }. If W never reaches the level m, setτ m =. 32 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

17 W t 33 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance W t m Τ m 34 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

18 W t W t Τ m m Τ m 35 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Exercise Let W be a Brownian motion and, for t 0, define the exponential process Zt) = e 1 2 σ2 t+σw t). 1 Show that the process {Zt)} t 0 is a martingale. 2 Show that the stopped process {Zt τ m )} t 0 is a martingale. 36 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

19 Solution 1 For 0 s t, show that EZt) Fs)) = e 1 2 σ2 t+σw s) E e σw t) W s))) = Zs). 2 Let 0 s t. Consider a partition Π = { t 0, t 1,...,t n } of [0, t] such that it contains all those values s τ m and t τ m can take. First show that and Zt τ m ) Zs τ m )= n 1 s τm<t i t τ m Zti ) Zt i 1 ) ) { s τm < t i t τ m } Fti 1 ) for i =1, 2,...,n. 37 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance 2 Then show that E 1 A Zt τm ) Zs τ m ) )) n = E 1 A 1 s τm<t i t τ m Zti ) Zt i 1 ) )) = n E 1 A 1 s τm<t i t τ m EZt i ) Zt i 1 ) Ft i 1 ))) =0, for any A Fs). 38 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

20 Distribution The first passage time of Brownian motion to level m R is finite almost surely and the Laplace transform of its distribution is for all α>0. E e ατm) = e m 2α 39 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Proof Let σ>0andm > 0. Consider the martingale Then and Zt τ m )= e 1 2 σ2 t τ m)+σw t τ m). 0 Zt τ m ) e σm lim t Zt τ m)= 1 τm< e 1 2 σ2 τ m+σm. 40 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

21 Since E Zt τ m )) = Z0) = 1, taking t and applying the Dominated Convergence Theorem gives ) E 1 τm< e 1 2 σ2 τ m = e σm. Taking σ 0 + gives P τ m < ) =1. Hence, ) E e 1 2 σ2 τ m = e σm, for σ>0andm > / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance For m > 0, set σ = 2α to get E e ατm) = e m 2α. D For m < 0, note that τ m = τ m, thus E e ατm) = E e ατ m ) = e m 2α. For m = 0, the result follows since τ m = 0. Hence, E e ατm) = e m 2α for all α>0. 42 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

22 Remark Although τ m < a.s., E τ m )= if m / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Reflection Principle For a fixed positive level m and time t, there are two possibilities for τ m t: 1 τ m t and W t) w; 2 τ m t and W t) w; where w m. Note that for the first case each τ m t, W t) w ) path has a reflected W t) 2m w ) path. 44 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

23 W t m w Τ m t 45 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance W t 2 m w m w Τ m t 46 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

24 W t Τm t w m 2 m w 47 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance For m > 0andw m, P τ m t, W t) w) =P W t) 2m w). For m < 0andw m, P τ m t, W t) w) =P W t) 2m w). 48 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

25 Distribution of For all m 0, the random variable τ m has the distribution function and the density function F τm t) = 2 2π f τm t) = e 1 m t m m2 e 2t 2πt 3 2 z2 dz for all t / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance Proof Suppose m > 0. Then by the reflection formula In addition, Therefore, P τ m t, W t) m) =P W t) m). P τ m t, W t) m) =P W t) m). P τ m t) =2P W t) m) =2 1 m t 2π e 1 2 z2 dz. 50 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance

26 Suppose m < 0. Then by the reflection formula In addition, Therefore, P τ m t, W t) m) =P W t) m) P τ m t, W t) m) =P W t) m). t m P τ m t) =2P W t) m) =2 1 =2 e 1 2 z2 dz. m t 2π 1 2π e 1 2 z2 dz 51 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance