Lecture 11: Ito CalculusTuesday, October 23, 12 Continuous time models • We start with the model from Chapter 3 log Sj • Sum it over j: log SN log Sj log S0 = 1 =µ t+ N X µ t+ j=1 p N X p tZj tZj j=1 • Can we take the limit as N approaches infinity (delta t tends to zero)? • What are the benefits? • last sum converges to a normal random variable, so we call it lognormal! • what is more important than the distribution of S at a fixed time? • increments: log SN log SM = N X j=M +1 Tuesday, October 23, 12 µ t+ N X j=M +1 p tZj .a stochastic process • Not only are we concerned about individual Sj as a random variable.when you enter at time j and exit at time k. . SN • We must consider them as a collection of random variables • Obviously the order is important . you care about log Sj log Sk . another random variable • A collection of time indexed random variables . S2 . Is it appropriate for stock prices? Tuesday. . we ask for their distributions. But the relations between different increments can be crucial for dependence consideration • Natural first step: independent increments. . 12 . we also need to consider all possible increments log Sj log Sk • As random variables. S1 .Stock price as a process • Prices at different times: S0 . October 23. justified by CLT. we are still dealing with binomial rv’s • As the partition increases. • Statistics: the mean and the variance (of increments) should depend on the time elapsed: µ(tj tk ) and 2 (tj tk ) • Independent increments: as long as individual rv’s are independent! Tuesday.Increments • Price change over a time period • What we get from our discrete model: a sum of independent Bernoulli rv’s binomial rv • If we further divide the time period into subintervals. October 23. 12 . these binomial rv’s converge to normal rv’s (in distribution). only through X at k • This is called the Markov property! Tuesday. each consisting of two components (drift + Z) • Called a random walk. independent of all the previous X’s before k • Distribution of X at j.Random walk and Markov property • Use notation Xj = log Sj • A sum of steps. October 23. is unaffected by the X values before k • Dependence of the history up to k . X_j is the position of the walk at time j • Increments Xj Xk . 12 . given X at k. N t = T • Xj = Xtj ! Xt . 2 (t s) • increments from nonoverlapping periods are independent • The path. • increment Xt Xs is a normal random variable:N µ(t s). 12 . collection of rv’s indexed by a continuous time variable t • Properties inherited or extended: • X at t is a normal random variable. is continuous. October 23. t ! 0. but nowhere differentiable • Standard notation: Wt Tuesday. X as a function of t.From random walk to Brownian motion • Think of the limiting process as N ! 1. Just compare with Xt = p tY where Y = N (0.Definition of BM • A process Wt indexed by t for t>=0 is a Brownian motion if W0 = 0 . October 23. and for every t and s (s<t). and the random variable Wt Ws is independent of the W random variables before s. 1) • Quadratic variations and the relevance: • why is it that the variance is proportional to the time elapsed? • why is that BM paths are so ragged? • how does the stock price variance grow in time? Tuesday. • The above says much more. we have Wt Ws distributed as a normal random variable with mean 0 and variance t-s. 12 . Extending BM • Add a (time-dependent) drift • Allow local variance (for each step) to be time-dependent p • Discrete time: Xj Xj 1 = µj t + j tZj • Continuous time: dXt = µ(t) dt + (t) dWt • Stock return over (t. October 23.t+dt): dSt = µ(t) dt + (t) dWt St • This is the Black-Scholes model for stock price dSt = d log St • Attempt to solve . 12 ? .do we have St Tuesday. Ito’s lemma • assume that f(x) is continuously twice differentiable • usual differential: df = f’(x) dx • if x=x(t) is also continuously differentiable (in t): df = f’(x) x’(t) dt • now let x=X_t from a stochastic process as described in the previous slide • notice W_t is nowhere differentiable • guess: df (Xt ) = f 0 (Xt ) dXt = f 0 (Xt ) (µdt + • not quite! as we see 2 Wt+h Wt2 = (Wt+h Wt )(Wt+h + Wt ) = 2Wt (Wt+h • expect dWt2 = 2Wt dWt + dt Tuesday. 12 dWt ) ? Wt ) + (Wt+h Wt )2 . October 23. t) dWt with approximations: p Xt+h Xt = µh + hZ + e (Xt+h Xt )2 = µ2 h2 + 2 hZ 2 + 2µ h3/2 Z + · · · • Leading term (in h) after replacing Z^2 with 1: 2 h • Justifications: the difference has mean and variance: 2 Tuesday.From Taylor expansion • Assuming f(x) twice differentiable 1 00 Xt ) + f (Xt )(Xt+h 2 f (Xt+h ) = f (Xt ) + f (Xt )(Xt+h 0 • Ito process: dXt = µ(Xt . 4 2 h V ar(Z 2 1) = 3 4 2 h . t) dt + (Xt . October 23. 12 Xt )2 + · · · hE[Z 2 1] = 0. t) 2 • Theorem 5. t)) = ✓ 2 ◆ 1 ft (Xt . t) + fx (Xt . 12 dt + f 0 (Xt ) dWt 2 ◆ dt + fx (Xt . t) dWt .Ito’s lemma • Letting h ! dt • Assuming differentiability again d(f (Xt )) = ✓ 1 00 f (Xt )µ + f (Xt ) 2 0 • If we allow f to be time dependent d(f (Xt . t)µ + fxx (Xt .1 (page 110) notations dt2 = 0 dt dWt = 0 (dWt )2 = dt Tuesday. October 23. Applications • Product rule: let X_t and Y_t be Ito processes d(Xt Yt ) = Xt dYt + Yt dXt + dXt dYt • If • then dXt = µ1 dt + 1 dWt dYt = µ2 dt + 2 dWt dXt dYt = • What about d Tuesday. October 23. 12 ✓ Xt Yt ◆ 1 2 (dW ) = 2 t 1 2 dt . 12 ✓ µ ✓ µ 1 2 1 2 ◆ 2 2 ◆ T + WT T + WT . October 23.Applications in stock price modeling • Solving SDE dSt = µ dt + St dWt • Try f (St ) = log St ✓ ◆ 1 1 1 2 2 df (St ) = dSt + St dt 2 St 2 St ✓ ◆ 1 2 = µ dt + dWt 2 • Integrate in t. assuming constant mu and sigma log ST log S0 = ST = S0 exp Tuesday. CEV model • Assuming volatility is S-dependent dSt = µdt + St St • 0< • 1 dWt < 1 implies that the volatility is inverse proportional to S S1 f (S) = 1 . Ito’s lemma gives d(f (St )) = ✓ S1 µ 2 S 1 2 ◆ • No luck in explicit solution unless beta=1 Tuesday. 12 dt + dWt . October 23. expiration T • Assume S follows a geometric BM • Risk free interest rate r • At time t<T.Deriving Black-Scholes Equation • Consider the pricing of a call option C. October 23. with strike K. the price of call is a function of stock price at the time (S) • Recognizing C=C(S. t) = dt + dSt + (dS ) t 2 @t @S 2 @S ✓ ◆ 1 2 2 = Ct + µSCS + S CSS dt + SCS dWt 2 Tuesday. 12 .t) @C @C 1 @2C 2 dC(St . 0) Ct + rSCS + S CSS = rC 2 • Compare with the standard heat equation.t+dt). suggest backward in time Tuesday. 12 .Deriving Black-Scholes Equation (continued) • Forming a portfolio: one share of call + alpha shares of the stock • Change of the portfolio over (t. assuming constant alpha: d(C + ↵S) = ✓ 1 Ct + µSCS + 2 2 S 2 CSS + ↵µS ◆ dt + S (CS + ↵) dWt • If we choose ↵ = CS (delta hedging). October 23. the random component disappears.no effect of stock price fluctuation • Portfolio is iick-free. T ) = max(ST K. which implies that the portfolio is hedged . we must have d(C + ↵S) = r(C + ↵S)dt • This leads to the Black-Scholes PDE with terminal condition 1 2 2 C(ST . October 23. solutions will be smoothed in time (backward) • Set up a region in (S. C(S_max.Use of the PDE • The PDE is parabolic. 0<S< S_max • Terminal condition imposed at t=T • Solve backward in time to 0: C(S.t): 0 < t < T.K) exp(-r(T-t)) • Advantage of the PDE approach: • easy to extend to time-dependent sigma • efficient numerical methods available Tuesday.0) • Enter the observed current price S(0) in place of S • Boundary conditions: C(0.t) = 0.t) = (S_max . 12 . 12 (t) to be adjusted. no matter what happens to the market • Replication strategy: invest C(S. beta units of the bond P (t) = ↵(t)S(t) + (t)B(t) • ↵(t).0).0) in stock and the risk-less bond. when the stock price is S.0). adjusting according to the call delta. you want to end up with the value max(S_T-K. By following the delta hedge strategy. according to the strategy . October 23. Tuesday. verify at T that the total value matches the call payoff • Composition of the portfolio: alpha shares of the stock.Justification of the derivation • How do we justify this price (solution from a PDE)? • Imagine you start with C(S. 12 t) t) B(t)) ! ↵dS + dB (t)) B(t + t) .Change of value in the portfolio • Change of portfolio value in time: P (t + t) P (t) • In differential: dP = ↵(t)dS(t) + (t)dB(t) + S(t)d↵(t) + B(t)d (t) • In discrete form: ↵(t + t)S(t + t) ↵(t)S(t) =↵(t + t)S(t + t) ↵(t)S(t + = (↵(t + ↵(t)) S(t + t) t) + ↵(t) (S(t + (t + t)B(t + t) (t)B(t) = (t + t)B(t + t) (t)B(t + = ( (t + (t)) B(t + t) t) + ↵(t)S(t + t) t) + (t)B(t + t) + (t) (B(t + t) t) ↵(t)S(t) S(t)) t) (t)B(t) B(t)) • Total change in two parts: ↵(t) (S(t + (↵(t + t) t) S(t)) + (t) (B(t + ↵(t)) S(t + t) + ( (t + ! Sd↵ + Bd + d↵dS + d dB Tuesday. October 23. holding shares fixed over time period • Second part: adjusting the number of shares. bond price. October 23. all at the end of the time period • Self-financing strategy: making sure the second part is zero • This corresponds to rebalancing in such a way that no money is taken out of the portfolio. 12 .Self-financing strategy • First part in the last slide: change in stock price. and no money is injected into the portfolio either • Such is the name of the strategy: self-financing • Consequence of this trading strategy: dP = ↵dS + dB Tuesday. 0) @C • Following ↵ = . dB = rBdt. and a beta such that it is a self-financing strategy @S • Want to show P(T) = C(S(T).T). t)) = dP dC @C @C @C dS + dB dt dS = @S @t @S @C 1 @2C 2 2 = rBdt dt S dt @t 2 @S 2 • We use • Result: Tuesday. 12 dS @C = µdt + dW.Replicating the call • Begin with a portfolio P = ↵(0)S(0) + (0)B(0) = C(S(0). October 23. P = S+ B S @S d(P C) = r (P SCS ) dt r (C 1 @2C 2 (dS) 2 @S 2 . no matter what S(T) ends up with • Consider the differential d (P (S. and the BS equation SCS ) dt = r(P C)dt . t) C(S. t). the rebalancing needs to observe the following condition: there can only be transfer of money within the stock and bond accounts Tuesday. t) = C(S. given alpha is a smooth function of S and an initial portfolio value P(0). • Implication on the hedging practice: by the end of the trading adjustment period. such that P = ↵S + B is a self-financing portfolio with initial value P(0). 12 . October 23.3: • A unique beta exists. the call is replicated! • Need to check the self-financing condition • Theorem 5.Matching at T • Solving the ODE: P (t) • We have C(t) = (P (0) C(0)) ert = 0 P (S. for 0 < t T . variable coefficients • A series of changes of variables introduced to reduce to the heat equation Z S = e • First. we arrive at a constant-coefficient equation ✓ 1 2 @C + r @t 2 ◆ • Change of time variable @C @⌧ • C=e r⌧ ✓ r 2 2@ ⌧ =T t 1 2 @C @Z 2 C = rC @Z 2 2@ 2 1 2 2@ C + rC = 0 2 @Z D @D @⌧ Tuesday. . 12 1 2 ◆ @C 1 + @Z 2 ✓ r 1 2 2 ◆ @D @Z 2 D =0 @Z 2 .Solving the PDE • Linear PDE. October 23. 12 . October 23.Heat Equation • Eliminate the first-order term: ✓ y=Z+ r 1 2 2 ◆ ⌧ • Standard heat equation @D 1 = @⌧ 2 2 @ D 2 @y 2 • Initial condition is also likewise transformed • Solution transformed into the original variables • Black-Scholes formula reproduced Tuesday. one share at t will grow to exp(d(T-t)) shares at T • Buying exp(-d(T-t)) shares is equivalent to one futures contract: • Price of a futures contract: S(t)e d(T t) • or delivery contract price X(t) = S(t)e delivered at T Tuesday. October 23.foreign currency as the underlying and it grows at its rf rate • This model assumes reinvestment • If the dividend rate is d.Dividend-paying stock • The previous model assumes no dividend paying stocks • Many stocks do pay dividends • FX products . 12 Ke d(T t) r(T t) . the price at t to have one share . Tuesday. 12 .it costs money to hold commodities (d=-q). this is the cost of carry.Call option on X • An option on X with expiration T must have the same value as an option on S • But the delivery contract pays no dividend (X is its price) • Process for X: dXt = (µ + d) dt + dWt Xt • Drift does not matter! • Call price: C(S. d 2 = d1 t) N (d2 ) p T t • Applies to commodity options . October 23. t) = XN (d1 ) = Se • with d(T log(S/K) + r d + 12 p d1 = T t Ke r(T t) N (d1 ) 2 (T t) t) N (d2 ) Ke r(T .