Lecture: Continuous Time Finance Lecturer: o. Univ. Prof. Dr. phil. Helmut Strasser

Transcription

1 Lecture: Continuous Time Finance Lecturer: o. Univ. Prof. Dr. phil. Helmut Strasser Part 1: Introduction Chapter 1: Review of discrete time finance Part 2: Stochastic analysis Chapter 2: Stochastic processes Chapter 3: Stochastic calculus Part 3: Continuous time finance Chapter 4: Pricing and hedging Chapter 5: Pricing measures QFin - ContTimeFinance Slide 1 - Title c Helmut Strasser, May 1, 2010

2 Introduction - Review of discrete time finance Chapter 1: Review of discrete time finance Section 1.1: Single period models Section 1.2: Multiperiod models Section 1.3: From discrete to continuous time QFin - ContTimeFinance Slide 2 - Contents c Helmut Strasser, May 1, 2010

3 Introduction - Review of discrete time finance - Single period models Chapter 1: Review of discrete time finance Section 1.1: Single period models Basic concepts Risk neutral pricing Syllabus: binomial model - lognormal model - portfolio - no arbitrage property - claim - arbitrage free price - replicating portfolio - market completeness - risk neutral model - fundamental theorem - risk neutral pricing QFin - ContTimeFinance Slide 3 - Contents c Helmut Strasser, May 1, 2010

4 Introduction - Review of discrete time finance - Single period models - Basic concepts Single period model Basic probability space (Ω, F, P ) A market with d + 1 assets: M = (S 0, S 1,..., S d ), period [t, T ]. Spot prices: St 0, St 1,..., St d Terminal values: ST 0, S1 T,..., Sd T (constants) (random variables) The simplest case: A market with two assets M = (B, S) Spot prices: B t, S t Terminal values: B T, S T Example: Binomial model r(t t) B bank account with fixed interest rate r: B t = 1, B T = e S risky asset: P (S T = S t u) = p or P (S T = S t d) = 1 p with d < u and 0 < p < 1. QFin - ContTimeFinance Slide 4 - Single period model c Helmut Strasser, May 1, 2010

5 Introduction - Review of discrete time finance - Single period models - Basic concepts Stock price of MSFT msft : to QFin - ContTimeFinance Slide 5 - Diagram c Helmut Strasser, May 1, 2010

6 Introduction - Review of discrete time finance - Single period models - Basic concepts Log-returns of MSFT msft : to QFin - ContTimeFinance Slide 6 - Diagram c Helmut Strasser, May 1, 2010

7 Introduction - Review of discrete time finance - Single period models - Basic concepts Histogram of the log-returns of MSFT QFin - ContTimeFinance Slide 7 - Diagram c Helmut Strasser, May 1, 2010

8 Introduction - Review of discrete time finance - Single period models - Basic concepts Normal plot of the log-returns of MSFT QFin - ContTimeFinance Slide 8 - Diagram c Helmut Strasser, May 1, 2010

9 Introduction - Review of discrete time finance - Single period models - Basic concepts Cumulated squared volatility of the log-returns of MSFT msft : to QFin - ContTimeFinance Slide 9 - Diagram c Helmut Strasser, May 1, 2010

10 Introduction - Review of discrete time finance - Single period models - Basic concepts Stock price of Dow Jones ^dji : to QFin - ContTimeFinance Slide 10 - Diagram c Helmut Strasser, May 1, 2010

11 Introduction - Review of discrete time finance - Single period models - Basic concepts Log-returns of Dow Jones ^dji : to QFin - ContTimeFinance Slide 11 - Diagram c Helmut Strasser, May 1, 2010

12 Introduction - Review of discrete time finance - Single period models - Basic concepts y=getstocks("^dji",from=c(1,1,2004),to=c(31,12,2004)) print(y) pplot(y) pplot(returns(y)) y=getstocks(c("msft","nok"),from=c(1,1,2004),to=c(31,12,2004)) pplot(y) pplot(returns(y)) QFin - ContTimeFinance Slide 12 - Calc: Retrieving empirical stock prices c Helmut Strasser, May 1, 2010

13 Introduction - Review of discrete time finance - Single period models - Basic concepts y=getstocks(c("ko"),from=c(1,1,2000),to=c(31,12,2001)) Scatter(y) Scatter(Returns(y)) summary(returns(y)) Hist(Returns(y)) Normalplot(Returns(y)) Boxplot(Returns(y)) Shape(as.vector(Returns(y))) QFin - ContTimeFinance Slide 13 - Calc: Statistics of empirical stock prices c Helmut Strasser, May 1, 2010

14 Introduction - Review of discrete time finance - Single period models - Basic concepts Example: Lognormal model r(t t) B bank account with fixed interest rate: B t = 1, B T = e Assume that the log-returns of the risky asset are normally distributed: log S T S t N(a(T t), σ 2 (T t)) Volatility: Standard deviation σ of the log-returns (per time unit). ( Average asset price: E(S T S t ) = S t exp a(t t) + σ2 (T t) ) 2 Usual parametrization: µ = a + σ 2 /2 growth rate of average prices = S t e µ(t t) (A1) Lognormal model: where Z N(0, 1). ) S T = S t exp [(µ σ2 (T t) + σ ] T t Z 2 QFin - ContTimeFinance Slide 14 - Lognormal model c Helmut Strasser, May 1, 2010

15 Introduction - Review of discrete time finance - Single period models - Basic concepts Lognormal density: µ = 0, σ = QFin - ContTimeFinance Slide 15 - Diagram c Helmut Strasser, May 1, 2010

16 Introduction - Review of discrete time finance - Single period models - Basic concepts Lognormal density: µ = 0.04, σ = QFin - ContTimeFinance Slide 16 - Diagram c Helmut Strasser, May 1, 2010

17 Introduction - Review of discrete time finance - Single period models - Basic concepts Lognormal density: µ = σ2 2, σ = QFin - ContTimeFinance Slide 17 - Diagram c Helmut Strasser, May 1, 2010

18 Introduction - Review of discrete time finance - Single period models - Basic concepts ff=function(meanlog=0,sdlog=1) { fun=function(x) dlnorm(x,meanlog,sdlog) } plot(ff(),0,5);grid() plot(ff(meanlog=0.04,sdlog=0.5),0,5);grid() sd=0.5 plot(ff(meanlog=-sd^2/2,sdlog=sd),0,5);grid() x=rlnorm(500,meanlog=-sd^2/2,sdlog=sd) Scatter(x) Hist(x,breaks=20) plot(density(x),xlim=c(0,5)) QFin - ContTimeFinance Slide 18 - Calc: Lognormal distribution c Helmut Strasser, May 1, 2010

19 Introduction - Review of discrete time finance - Single period models - Basic concepts X=rpBS(rate=0.04,vol=0.2) pplot(x(1)) pplot(x(20)) x=rowmeans(x(5000)) plot(attr(x(),"time"),x,type="l") QFin - ContTimeFinance Slide 19 - Calc: Lognormal model c Helmut Strasser, May 1, 2010

20 Introduction - Review of discrete time finance - Single period models - Basic concepts Portfolios Portfolio: H = (H 0, H 1,..., H d ) (number of units in the portfolio) Value: V t = d i=0 Hi St i : initial value, price of the portfolio V T = d i=0 Hi ST i : terminal value, payoff of the portfolio Special case: Two assets Portfolio: H = (H B, H S ) (number of units in the portfolio) Value: V t = H B B t + H S S t : initial value, price of the portfolio V T = H B B T + H S S T : terminal value, payoff of the portfolio QFin - ContTimeFinance Slide 20 - Portfolios c Helmut Strasser, May 1, 2010

21 Introduction - Review of discrete time finance - Single period models - Basic concepts No Arbitrage property Definition: The market M has the no-arbitrage property (NA) if for every portfolio value V V t 0 and V T 0 V t = V T = 0 Example: Binomial model (A2) NA holds iff d < e r(t t) < u. Consequences: Let M be a market with NA. Then: (A3) There is no free lunch (NFL): V t = 0 and V T 0 V T = 0 whenever V V. (A4) Prices are uniquely determined, i.e. V T = V T V t = V t whenever V, V V (This means: If NA holds then pricing makes sense.) (A5) There is at most one admissible portfolio value V V which is riskless. QFin - ContTimeFinance Slide 21 - No arbitrage property c Helmut Strasser, May 1, 2010

22 Introduction - Review of discrete time finance - Single period models - Basic concepts (A6) Multinomial case: Characterize the NA-property! Bet and Win: A simple example (A7) A bet is offered on a soccer game with teams A and B (remis excluded). A company offers bets with quotes (a, b). Characterize the quotes where arbitrage (a riskless profit) is possible! (A8) Assume that two companies offer bets on the same game, but with different quotes. Discuss the NA property! (A9) Extend your findings to bets with k alternatives (e.g. horse races) and m companies offering quotes! (A10) A theoretical problem: Let (B, S) be a single period market. Assume that there is a numeraire (a strictly positive portfolio N V). Show that NA is equivalent to NFL! QFin - ContTimeFinance Slide 22 - Problems c Helmut Strasser, May 1, 2010

23 Introduction - Review of discrete time finance - Single period models - Basic concepts Claims Definition: A claim is a payoff at time T depending on the history of the market. Forwards: Period [t, T ], market model: B = (B t, B T ) = (1, e r(t t) ), S = (S t, S T ). C T = S T A: This claim can be replicated by the portfolio V = S Ae r(t t) B r(t t) Price: V t = S t Ae V t = 0 implies A = S t e r(t t) : forward price Options: Call C T = (S T K) +, Put P T = (K S T ) +, (K strike price). Binary Call C b T = 1 (S T K), Binary Put P b T = 1 (S T <K) QFin - ContTimeFinance Slide 23 - Claims c Helmut Strasser, May 1, 2010

24 Introduction - Review of discrete time finance - Single period models - Basic concepts Call prices: Call price: T= 1, vol= 0.5, rate= 0.05 call price spot price QFin - ContTimeFinance Slide 24 - Diagram c Helmut Strasser, May 1, 2010

25 Introduction - Review of discrete time finance - Single period models - Basic concepts Call prices: Call price: T= 0.1, vol= 1, rate= 0.05 call price spot price QFin - ContTimeFinance Slide 25 - Diagram c Helmut Strasser, May 1, 2010

26 Introduction - Review of discrete time finance - Single period models - Basic concepts Call prices: Call price: T= 0.01, vol= 1, rate= 0.05 call price spot price QFin - ContTimeFinance Slide 26 - Diagram c Helmut Strasser, May 1, 2010

27 Introduction - Review of discrete time finance - Single period models - Basic concepts Put prices: Put price: T= 1, vol= 1, rate= 0.05 put price spot price QFin - ContTimeFinance Slide 27 - Diagram c Helmut Strasser, May 1, 2010

28 Introduction - Review of discrete time finance - Single period models - Basic concepts Put prices: Put price: T= 1, vol= 1, rate= 0.05 put price spot price QFin - ContTimeFinance Slide 28 - Diagram c Helmut Strasser, May 1, 2010

29 Introduction - Review of discrete time finance - Single period models - Basic concepts Put prices: Put price: T= 0.01, vol= 1, rate= 0.05 put price spot price QFin - ContTimeFinance Slide 29 - Diagram c Helmut Strasser, May 1, 2010

30 price Introduction - Review of discrete time finance - Single period models - Basic concepts Call price surface vol spot QFin - ContTimeFinance Slide 30 - Diagram c Helmut Strasser, May 1, 2010

31 Introduction - Review of discrete time finance - Single period models - Basic concepts call.price.diagram(rate=0.05) put.price.diagram(rate=0.05) pf=function(x,y) Call.BS(spot=x,vol=y)$Value Persp("pf(x,y)",xlim=c(0,2),ylim=c(0,1),xlab="spot", ylab="vol",zlab="price") QFin - ContTimeFinance Slide 31 - Calc: Diagrams c Helmut Strasser, May 1, 2010

32 Introduction - Review of discrete time finance - Single period models - Basic concepts Pricing problem Definition: Let M = (S 0, S 1,..., S d ) be a an arbitrage-free market and let C T be a claim. A number π is an arbitrage-free price of C T if the market (S 0, S 1,..., S d, (π, C T )) is arbitrage-free. Vanilla claims: Claims with liquid prices. The claims are traded at an exchange, and market efficiency provides arbitrage-free prices (hopefully). Exotic claims: Claims which are not traded at an exchange but sold over the counter. Pricing problem: How to find arbitrage-free prices for exotic claims? QFin - ContTimeFinance Slide 32 - Pricing problem c Helmut Strasser, May 1, 2010

33 Introduction - Review of discrete time finance - Single period models - Basic concepts Put-Call parity Let C and P be a call and a put for the same strike price K with arbitrage free prices. Put-Call parity: C T P T = (S T K) + (K S T ) + r(t t) = S T K C t P t = S t Ke Implications: r(t t) C t S t Ke which Early exercise of a call is of no value. P t Ke r(t t) S t is > S t K which is < K S t Early exercise of a put can be valuable. QFin - ContTimeFinance Slide 33 - Put-Call parity c Helmut Strasser, May 1, 2010

34 Introduction - Review of discrete time finance - Single period models - Basic concepts Pricing by replication: Definition: A claim is attainable if there is a portfolio whose payoff is equal to the claim (a replicating portfolio). Pricing rule for attainable claims: Find a replicating portfolio (H i ), i.e. such that the portfolio payoff satisfies V T = C T. Then take V t as the price of the claim C T at time t. If the market satisfies NA, then this rule provides an arbitrage-free price, and this price is uniquely determined. Question: When is it possible to replicate a claim? QFin - ContTimeFinance Slide 34 - Replication c Helmut Strasser, May 1, 2010

35 Introduction - Review of discrete time finance - Single period models - Basic concepts Example: Binomial model. Let C T := { C u T whenever S T = S u T = S tu C d T whenever S T = S d T = S td Then a replicating portfolio necessarily satisfies: We see: C T H S S T = H B e r is riskless! H B e r + H S S u T = Cu T H B e r + H S S d T = Cd T C H S S T has only a single outcome: C u T HS S u T = Cd T HS S d T This implies H S = Cu T Cd T S u T Sd T =:, H B = e r (C u T S u T) The number of units of S in the portfolio equals the Delta of the claim. QFin - ContTimeFinance Slide 35 - Binomial case c Helmut Strasser, May 1, 2010

36 Introduction - Review of discrete time finance - Single period models - Basic concepts (A11) Problem: A stock price is currently 20$, and it is known that at the end of three months the stock price will be either 18$ or 22$. We are interested in pricing a European call option to buy the stock for 21$ at the end of three months. Suppose that the risk-free rate is 12% per annum. QFin - ContTimeFinance Slide 36 - Problem c Helmut Strasser, May 1, 2010

37 Introduction - Review of discrete time finance - Single period models - Basic concepts In a binomial model every claim can be replicated. The single period binomial model is complete. Definition: A market model is complete if every claim can be replicated by a portfolio value. (A12) Problem: Discuss completeness in the multinomial case! QFin - ContTimeFinance Slide 37 - Market completeness c Helmut Strasser, May 1, 2010

38 Introduction - Review of discrete time finance - Single period models - Basic concepts Pricing based on liquid options Assume that the market provides prices for calls C 1, C 2,..., C m with strikes K 1 < K 2 <... < K m. Let C T = f(s T ) be a claim whose payoff f(x) is continuous and piecewise linear payoff function with edges at K 1 < K 2 <... < K m. This claim is replicable: Let k 1, k 2,..., k m+1 be the slopes of the payoff function f(x). Then m f(x) = f(0) + k 1 x + (k i+1 k i )(x K i ) + Therefore C T = f(s T ) = f(0) + k 1 S T + m i=1 (k i+1 k i )C i T and C t = f(0)e r(t t) + k 1 S t + m i=1 (k i+1 k i )C i t i=1 QFin - ContTimeFinance Slide 38 - Using liquid options c Helmut Strasser, May 1, 2010

39 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Introduction - Review of discrete time finance - Single period models - Risk neutral pricing QFin - ContTimeFinance Slide 39 - Title c Helmut Strasser, May 1, 2010

40 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing M = (B, S): a market, B a bank account. P, P : probability models such that the expectation E(S T ) exists. Definition: A probability model P is called risk neutral if the expectation of S T equals the forward price of S t : E (S T S t ) = e r(t t) S t. P : risk neutral probability measure P := {P P : E (ST i ) = er(t t) St i, i = 1, 2,..., d} Risk neutrality of P means: The growth rate of expected prices equals the riskless rate. Important: Typical statistical probability models P need not be risk neutral! QFin - ContTimeFinance Slide 40 - Risk neutral models c Helmut Strasser, May 1, 2010

41 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Risk neutrality: Explanation Utility maximization: An economic agent with utility function u(x) (strictly concave, increasing and differentiable) has to choose a portfolio consisting of a riskless asset c and a risky asset X (probability P ). Expected utility: f(λ) := E(u((1 λ)x + λc)), λ R. Let λ [0, 1] the utility-optimal distribution: f(λ ) = max f(λ). Theorem: The agent takes risks (λ 1) iff E(X) c. (A13) The risk neutral case corresponds to λ = 1, i.e. E(X) = c. Question: Assume that the riskless asset returns r and the risky asset returns (S π)/π. When will the agent take risks? (A14) QFin - ContTimeFinance Slide 41 - Risk neutrality c Helmut Strasser, May 1, 2010

42 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Example: Binomial model. Let Q(S T = S t u) = q and Q(S T = S t d) = 1 q. Then Q is risk neutral iff q = er(t t) d. (A15) u d Note: For the binomial case there exists a risk neutral model iff the NA-property holds. (A16) (A17) Problem: Discuss risk neutral models for the multinomial case! (A18) Problem: Discuss uniqueness of risk neutral models both for the binomial and for the multinomial case! QFin - ContTimeFinance Slide 42 - Examples c Helmut Strasser, May 1, 2010

43 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Pricing rule: Let P be a risk neutral probability model. Then the prices (initial values) of portfolios equal the expectations under P of the discounted payoffs (terminal values). Proof: Risk neutrality means ( E (S T ) = e r(t t) S t E ST ) = S e r(t t) t This carries over to all attainable portfolio values: E ( VT ) e r(t t) ( H = E B B T + H S S ) T e r(t t) = H B B t + H S S t = V t In other words: Let C T be an attainable claim. Then the unique arbitrage-free price π(c T ) satisfies for every P P ( E CT ) ST 0 = π(c ( T) St 0 π(c T ) = St 0 E CT ) ST 0 QFin - ContTimeFinance Slide 43 - Attainable claims c Helmut Strasser, May 1, 2010

44 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Question: Can we be sure to be able finding risk neutral probability model? Fundamental theorem: For any (single period) market there exists a risk neutral model iff the NA-property holds. Proof: The proof consists of two parts. Part 1 (easy): If there is an equivalent risk neutral model then the NA-property holds. Part 2 (difficult): If the NA-property holds, then there is some equivalent risk neutral model. Remark: It is even possible to find a risk neutral model P such that dp /dp is bounded. QFin - ContTimeFinance Slide 44 - Fundamental theorem c Helmut Strasser, May 1, 2010

45 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Proof of the easy part: The proof rests on two facts. Let S 0 > 0 be a numeraire. Let S t := S t S 0 t denote the discounted prices. (A19) Replication is maintained under a change of numeraire, i.e. V T = V t + d H i (ST i St) i V T = V t + i=1 i=1 d H i (S i T S i t) (A20) : M satisfies NA iff d H i (S i T S i t) 0 i=1 d H i (S i T S i t) = 0 i=1 If there exists an equivalent risk neutral model then M satisfies NA. (A21) QFin - ContTimeFinance Slide 45 - Proof c Helmut Strasser, May 1, 2010

46 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Question: How to find prices, if the statistical model P is not risk neutral? Answer: Find some (equivalent) risk neutral probability model P! Let C T 0 be a (not necessarily attainable) claim. Theorem: If the market M = (S 0, S 1,..., S m ) satisfies NA, then the set of arbitrage-free prices of a claim C T 0 is { ( π(c T ) = St 0 E CT ) } ST 0 : P P, E (C T ) < This set is not empty. Proof: (A22) (A23) Problem: Solve binomial pricing via risk neutral pricing! QFin - ContTimeFinance Slide 46 - General claims c Helmut Strasser, May 1, 2010

47 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Risk neutral lognormal model: where Z N(0, 1) (P ). µ(t t) This model satisfies: E P (S T ) = S t e ) S T = S t exp [(µ σ2 (T t) + σ ] T t Z 2 It follows: A lognormal model P is risk-neutral iff µ = r, i.e. S T = S t exp [(r )(T σ2 t) + σ ] T t Z 2 where Z N(0, 1) (P ). Risk neutral pricing: If C T is some claim, we may calculate and call it the price of the claim C T at time t. C t = E (C T ) er(t t) QFin - ContTimeFinance Slide 47 - Risk neutrality c Helmut Strasser, May 1, 2010

48 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Calculation strategy Let C T = f(s T ), i.e. assume that the payoff depends on the terminal value only. ( S ) T Writing C T = f S t we have E (C T S t ) = h(s t ) where S t h(x) := E (f [ x S T S t ]) [ = E (f x exp The main calculation work consists in evaluating ((r σ2 2 ) )]) (T t) + σ (T t) Z u(x, a, b) := E(f(x exp(a + bz))) where Z N(0, 1) Then the result is [ E (C T ) = h(s t ) = u S t, (r σ2 2 ) (T t), σ ] T t QFin - ContTimeFinance Slide 48 - Calculation strategy c Helmut Strasser, May 1, 2010

49 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Price formula for a European call: C T = (S T K) + Let F t := S t e r(t t) and σ t = σ T t. Then (A24) where C t e r(t t) = F t N(d 1 ) KN(d 2 ) d 1 = log(f t/k) + σ 2 t /2 σ t, d 2 = log(f t/k) σ 2 t /2 σ t Price formula: C t = S t N(d 1 ) Ke r(t t) N(d 2 ) QFin - ContTimeFinance Slide 49 - Call prices c Helmut Strasser, May 1, 2010

50 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing (A25) Problem: Draw the graph of call price as a function of the spot price. (A26) Problem: Describe the behaviour of call price under variation of volatility and of maturity. What is the influence of the riskless rate? (A27) Problem: Give a rough estimate of the call price if the call is deep in the money or deep out of the money. (A28) Problem: Assume that the riskless rate is zero and that volatility is known. How can we estimate the call price of a call at the money? QFin - ContTimeFinance Slide 50 - Problems c Helmut Strasser, May 1, 2010

51 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing r=0.04 S=rpBS(vol=0.2,rate=r,spot=1,time=1) mean(call(1)(s(100000)))*exp(-r) Call.BS(vol=0.2,rate=r)$Value QFin - ContTimeFinance Slide 51 - Calc: MC Call Value c Helmut Strasser, May 1, 2010

52 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Linear interpolation S=rpBS(vol=0.2,rate=0.04,time=1) Call(1)(S(10)) Call(c(1.2,1,0.8))(S(10)) sp=seq(from=0.5,to=1.5,by=0.1) Call(1/sp)(S(15)) pr=sp*colmeans(sp*call(1/sp)(s(15)))*exp(-0.04) plot(sp,pr,type="l") S=rpBS(vol=0.2,rate=0.04,time=1) pr=sp*colmeans(call(1/sp)(s(10000)))*exp(-0.04) plot(sp,pr,type="l") pr0=call.bs(spot=sp,vol=0.2,rate=0.04)$value lines(sp,pr0,col="red") QFin - ContTimeFinance Slide 52 - Calc: MC Call Value c Helmut Strasser, May 1, 2010

53 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing Smoothing spline interpolation S=rpBS(vol=0.2,rate=0.04,time=1) sp=seq(from=0.5,to=1.5,length=50) pr=sp*colmeans(call(1/sp)(s(50000)))*exp(-0.04) sm=smooth.spline(sp,pr,spar=0.5) price=function(x) predict(sm,x)$y plot(price,0.5,1.5) pr0=call.bs(spot=sp,vol=0.2,rate=0.04)$value lines(sp,pr0,col="red") QFin - ContTimeFinance Slide 53 - Calc: MC Call Value c Helmut Strasser, May 1, 2010

54 Introduction - Review of discrete time finance - Single period models - Risk neutral pricing S=rpBS(vol=0.2,rate=0.04,time=1) BinCall(1)(S(10)) BinCall(c(1.2,1,0.8))(S(10)) S=rpBS(vol=0.2,rate=0.04,time=1) sp=seq(from=0.5,to=1.5,length=50) pr=colmeans(bincall(1/sp)(s(50000)))*exp(-0.04) sm=smooth.spline(sp,pr,spar=0.5) price=function(x) predict(sm,x)$y plot(price,0.5,1.5) QFin - ContTimeFinance Slide 54 - Calc: MC BinCall Value c Helmut Strasser, May 1, 2010

55 Introduction - Review of discrete time finance - Multiperiod models Chapter 1: Review of discrete time finance Section 1.2: Multiperiod models Basic concepts Binomial model Risk neutral models Syllabus: self financing portfolio - admissible value process - no arbitrage property - attainable claim - binomial tree - completeness - early exercise - numeraire - risk neutral model - martingale measure - fundamental theorem QFin - ContTimeFinance Slide 55 - Contents c Helmut Strasser, May 1, 2010

56 Introduction - Review of discrete time finance - Multiperiod models - Basic concepts Two asset case: M = (B, S) n (equidistant) periods: t 0 = t, t 1, t 2,..., t n = T (trading times) Asset prices: (B 0, S 0 ), (B 1, S 1 ),..., (B n, S n ) Trading strategy: (H B k, HS k ) number of units in the portfolio between times t k 1 and t k Value process: V = (V 0, V 1,..., V n ) Definition: A trading strategy (H B k, HS k ) is called self-financing if at every trading time t k, k = 1, 2,..., n 1 it satisfies the rebalancing condition V k = H B k B k + H S k S k = H B k+1b k + H S k+1s k The value process of a self-financing trading strategy is called an admissible value process. QFin - ContTimeFinance Slide 56 - Self-financing property, two assets c Helmut Strasser, May 1, 2010

57 Introduction - Review of discrete time finance - Multiperiod models - Basic concepts An admissible value process depends only on the value changes of the underlying assets. Proof: V k V k 1 = (Hk B B k + Hk S S k ) (Hk B B k 1 + Hk S S k 1 ) = Hk B (B k B k 1 ) + Hk S (S k S k 1 ) In other words: V n = V 0 + n Hk B (B k B k 1 ) + k=1 n Hk S (S k S k 1 ) k=1 Short notation: V is admissible iff V = H B B + H S S and dv = H B db + H S ds QFin - ContTimeFinance Slide 57 - Self-financing property, two assets c Helmut Strasser, May 1, 2010

58 Introduction - Review of discrete time finance - Multiperiod models - Basic concepts Self-financing property A market with two assets: M = (S 0, S 1,..., S d ) n periods: t 0 = t, t 1, t 2,..., t n = T (trading times) Asset prices/payoffs: (Sk 0, S1 k,..., Sd k ) : k = 0, 1, 2,..., n Trading strategy: (H 0 k, H1 k,..., Hd k ) units in the portfolio between times t k 1 and t k Value process: V = (V 0, V 1,..., V n ) Definition: A trading strategy (Hk B, HS k ) is called self-financing if at every trading time t k, k = 1, 2,..., n 1 it satisfies the rebalancing condition: d d V k := HkS i k i = i=0 i=0 H i k+1s i k The value process of a self-financing trading strategy is called an admissible value process. QFin - ContTimeFinance Slide 58 - Self-financing property c Helmut Strasser, May 1, 2010

59 Introduction - Review of discrete time finance - Multiperiod models - Basic concepts Self-financing trading Theorem: An admissible value process depends only on the value changes of the underlying assets, i.e. Proof: (A29) V n = V 0 + n k=1 d Hk(S i k i Sk 1) i i=0 Theorem: Every linear combination of admissible value processes is an admissible value process. Proof: (A30) QFin - ContTimeFinance Slide 59 - Self-financing trading c Helmut Strasser, May 1, 2010

60 Introduction - Review of discrete time finance - Multiperiod models - Basic concepts No arbitrage Let V be the set of all admissible value processes. Definition: The market M has the no-arbitrage property (NA) if for every V V V 0 0 and V n 0 V 0 = V n Theorem: A multiperiod market model satisfies the NA-property iff for each period the corresponding single period model is arbitrage-free. Proof: Literature. Note: Single period submodels of a multiperiod model are contingent, i.e. the initial values of the assets are random (depend on the past of the period). QFin - ContTimeFinance Slide 60 - No arbitrage c Helmut Strasser, May 1, 2010

61 Introduction - Review of discrete time finance - Multiperiod models - Basic concepts Pricing of claims Definition: An attainable claim is a random variable C T which can be written as the payoff of an admissible value process (is replicated by a self-financing trading strategy). Terminology: Replicating claims by self-financing trading strategies is called hedging. Obvious: If the market satisfies the NA property then for every attainable claim the one and only price which avoids arbitrage is the cost of a hedging strategy. Question: How to obtain the price of an attainable claim? QFin - ContTimeFinance Slide 61 - Pricing of claims c Helmut Strasser, May 1, 2010

62 Introduction - Review of discrete time finance - Multiperiod models - Binomial model Example: Binomial case Given S 0, S 1,..., S n, the value process of a self-financing portfolio can be obtained recursively from its terminal value V n. At every trading time t k we have: V k = H B k B k + H S k S k Therefore V k H S S k = H B k B k is riskless: Given S k 1, the right hand side is a known constant, since H B k depends only on S k 1. This implies that (given S k 1 ) V u k H S k S u k = V d k H S k S d k Hk S = V k u V k d Sk u Sd k =: k Thus, we may reconstruct the value process backwards: H B k = V k H S k S k B k, V k 1 = H B k B k 1 + H S k S k 1 QFin - ContTimeFinance Slide 62 - Binomial case, Delta c Helmut Strasser, May 1, 2010

63 Introduction - Review of discrete time finance - Multiperiod models - Binomial model Binomial tree: Stock values s k,i = u i d k i s 0 Bond values b k = e kr (Reasonable choices of u, d, r are discussed later.) Time S V 0 s 0 v 0 1 s 1,0 s 1,1 v 1,0 v 1,1 2 s 2,0 s 2,1 s 2,2 v 2, v 2,1 v 2,2 n s n,0 s n,1 s n,2... s n,n v n,0 v n,1 v n,2... v n,n If C = f(s) is a claim let v n,i := f(s n,i ) and calculate backwards: H S k := v k,i+1 v k,i s k,i+1 s k,i, H B k = v k,i H S k s k,i b k, v k 1,i = H B k b k 1 + H S k s k 1,i The initial value v 0 is the price of the claim. QFin - ContTimeFinance Slide 63 - Binomial tree algorithm c Helmut Strasser, May 1, 2010

64 Introduction - Review of discrete time finance - Multiperiod models - Binomial model Theorem: In a multiperiod binomial model every claim can be replicated. The multiperiod period binomial model is complete. QFin - ContTimeFinance Slide 64 - Market completeness c Helmut Strasser, May 1, 2010

65 Introduction - Review of discrete time finance - Multiperiod models - Binomial model Numerical example: u = up 1/ n, d = 1/u, r = rate/n t=binomtree(n=5,up=1.06,rate=0.04,claim=call(1)) lapply(t$stock,round,3) lapply(t$value,round,3) [[1]] [[1]] [1] 1 [1] [[2]] [[2]] [1] [1] [[3]] [[3]] [1] [1] [[4]] [[4]] [1] [1] [[5]] [[5]] [1] [1] [[6]] [[6]] [1] [1] QFin - ContTimeFinance Slide 65 - Calc: Binomial tree example c Helmut Strasser, May 1, 2010

66 Introduction - Review of discrete time finance - Multiperiod models - Binomial model system.time(binomtree(n=100,up=1.06,rate=0.04,claim=call(1))) system.time(binomtree(n=500,up=1.06,rate=0.04,claim=call(1))) system.time(binomtree(n=1000,up=1.06,rate=0.04,claim=call(1))) QFin - ContTimeFinance Slide 66 - Calc: Binomial tree efficiency c Helmut Strasser, May 1, 2010

67 Introduction - Review of discrete time finance - Multiperiod models - Binomial model Early exercise: European claim: The right to exercise at maturity American claim: The right to exercise before maturity (early exercise) Classroom questions: (A31) What is more valuable? A European claim or an American claim (all side conditions equal)? (A32) When is early exercise preferable? (A33) When is early exercise valuable for a call, for a put? QFin - ContTimeFinance Slide 67 - Early exercise c Helmut Strasser, May 1, 2010

68 Introduction - Review of discrete time finance - Multiperiod models - Binomial model Early exercise in a binomial tree: Let v k,i be the values of the American claim. Beginning with v n,i = f(s n,i ) calculate backwards: but H S k := v k,i+1 v k,i s k,i+1 s k,i, H B k = v k,i H S k s k,i b k v k 1,i = max(f(s k 1,i ), H B k b k 1 + H S k s k 1,i ) The initial value v 0 is the price of the American claim. QFin - ContTimeFinance Slide 68 - Binomial algorithm with early exercise c Helmut Strasser, May 1, 2010

69 Introduction - Review of discrete time finance - Multiperiod models - Binomial model t=binomtree(n=30,up=1.06,rate=0.04,claim=call(1)) t=binomtree(n=30,up=1.06,rate=0.04,claim=call(1),mode="am") t=binomtree(n=30,up=1.06,rate=0.04,claim=put(1)) t=binomtree(n=30,up=1.06,rate=0.04,claim=put(1),mode="am") QFin - ContTimeFinance Slide 69 - Calc: Early exercise example c Helmut Strasser, May 1, 2010

70 Introduction - Review of discrete time finance - Multiperiod models - Risk neutral models Are risk neutral probability models useful for pricing claims in the multiperiod case? Change of Numeraire: Use B k = e kr n Discounting: S k := S k B k, V k := V k B k, B k := B k B k = 1 = er(t k t) as numeraire Lemma: An admissible value process depends only on the discounted value changes of the underlying assets, i.e. Proof: (A34) V n = V 0 + n k=1 i=1 d Hk(S i i k S i k 1) QFin - ContTimeFinance Slide 70 - Discounting c Helmut Strasser, May 1, 2010

71 Introduction - Review of discrete time finance - Multiperiod models - Risk neutral models Notation: F k := F(S 0, S 1,..., S k ) past of time k. Definition: A probability measure P is called risk neutral or a martingale measure if E (S i k+1 F k ) = S i k, k = 0, 1, 2,..., n, i = 1, 2,..., d. Theorem: Let (V k ) be the value sequence of a self-financing strategy. If V n 0 then (V k ) is a martingale under each martingale measure. Under a risk-neutral probability model the following statements are true: (A35) The expected return of a self-financing trading strategy equals the riskless rate. (A36) The initial value of an admissible value process equals the expected value of the discounted terminal value. (A37) The arbitrage-free price of an attainable claim equals the expected value of the discounted claim. QFin - ContTimeFinance Slide 71 - Risk neutrality c Helmut Strasser, May 1, 2010

72 Introduction - Review of discrete time finance - Multiperiod models - Risk neutral models Proof of the theorem: Since (H B k, HS k ) is self-financing we have V n = V 0 + n Hk S (S k S k 1 ) k=1 Since Q is a risk-neutral probability model the discounted prices satisfy the martingale property E Q (S k F k 1 ) = S k 1 By the redundancy rule we obtain E Q (H S k (S k S k 1 ) F k 1 ) = H S k E Q (S k S k 1 F k 1 ) = 0 By the expectation rule it follows that E Q (H S k (S k S k 1 )) = 0 (A38) Fill in the details of the arguments! QFin - ContTimeFinance Slide 72 - Proof c Helmut Strasser, May 1, 2010

73 Introduction - Review of discrete time finance - Multiperiod models - Risk neutral models Fundamental theorem: For any (multiperiod) market model there exists a martingale measure P P iff the NA-property holds. Proof: The proof consists of two parts. Part 1 (easy): If there exists an equivalent martingale measure then the NA-property holds. Part 2 (difficult): If the NA-property holds, then there exist an equivalent martingale measure. (A39) Prove part 1 of the Fundamental theorem. QFin - ContTimeFinance Slide 73 - Risk neutral models c Helmut Strasser, May 1, 2010

74 Introduction - Review of discrete time finance - From discrete to continuous time Chapter 1: Review of discrete time finance Section 1.3: From discrete to continuous time Syllabus: calibration of binomial trees - Cox Ross Rubinstein approximation - Black Scholes model QFin - ContTimeFinance Slide 74 - Contents c Helmut Strasser, May 1, 2010

75 Introduction - Review of discrete time finance - From discrete to continuous time Binomial model as discretization: How to calibrate a binomial model to given statistical properties? Suppose that the multiperiod binomial model arises through discretization of a time interval [t, T ] such that µ(t t) E(S T S t ) = S t e growth V (log S T /S t ) = σ 2 (T t) rate of expected prices volatility How to choose u, d and p when n periods are used? (large n) QFin - ContTimeFinance Slide 75 - Discretization c Helmut Strasser, May 1, 2010

76 Introduction - Review of discrete time finance - From discrete to continuous time Let S n be the terminal asset value of a binomial market model. (A40) Problem: Show that E(S n /S 0 ) = (pu + (1 p)d) n. (A41) Problem: Show that E(S n /S 0 ) = e µ(t t) implies p = eµ(t t)/n d u d Let u = e a, d = 1/u = e a and L n := log S n S 0. (A42) Problem: Show that V (L n ) = 4a 2 np(1 p). (A43) Problem: (L n ) n N is a random walk and has a binomial distribution (up to a constant factor). QFin - ContTimeFinance Slide 76 - Calibration c Helmut Strasser, May 1, 2010

77 Introduction - Review of discrete time finance - From discrete to continuous time The case of large n: (n ) Since a 2 n const, we should have a c/ n. This implies (by Taylor expansion e x 1 + x + x 2 /2) p n ( µ(t t) 2c c 4 ), E(L n ) µ(t t) c2 2, V (L n) c 2 Choosing c 2 = σ 2 (T t) (i.e. a = σ (T t)/n) gives E(L n ) µ(t t) σ2 (T t), V (L n ) σ 2 (T t) 2 Note: This is approximately the lognormal model, since L n is approximately normally distributed by the CLT. QFin - ContTimeFinance Slide 77 - Asymptotics c Helmut Strasser, May 1, 2010

78 Introduction - Review of discrete time finance - From discrete to continuous time Solution: u = 1/d = e σ (T t)/n, p = e µ(t t)/n d u d Interpretation: Volatility determines the size of the jumps in the binomial model. Expected returns can be varied by the choice of probabilities without changing volatility or the jumps. The set of available paths of the binomial model depends only on the volatility but not on the expected returns. Hedging requires only the set of available paths but no probabilities. For the calibration of a model for pricing reasons volatility only has to be taken into account. QFin - ContTimeFinance Slide 78 - Interpretation c Helmut Strasser, May 1, 2010

79 Introduction - Review of discrete time finance - From discrete to continuous time Call.BS(vol=0.2,strike=1,rate=0.04,spot=1,maturity=1)$Value CRR(n=100,vol=0.2,rate=0.04,spot=1,maturity=1,claim=Call(1)) CRR(n=1000,vol=0.2,rate=0.04,spot=1,maturity=1,claim=Call(1)) Pricing under a risk neutral lognormal model and binomial hedging render the same results. QFin - ContTimeFinance Slide 79 - Calc: Cox-Ross-Rubinstein c Helmut Strasser, May 1, 2010

80 Introduction - Review of discrete time finance - From discrete to continuous time Numerical example: Monte Carlo simulation S=rpBS(vol=0.2,rate=0.04,time=1,spot=1) mean(call(1)(s( )))*exp(-0.04) CRR(n=1000,vol=0.2,rate=0.04,spot=1,maturity=1,claim=Call(1)) Call.BS(vol=0.2,rate=0.04,maturity=1,spot=1,strike=1)$Value QFin - ContTimeFinance Slide 80 - Calc: Cox-Ross-Rubinstein c Helmut Strasser, May 1, 2010

81 Introduction - Review of discrete time finance - From discrete to continuous time From discrete to continuous time: Binomial model S k = S 0 exp(l k ) (T t)k Elapsed time t k = : n E(L k ) µt k σ2 2 t k, V (L k ) σ 2 t k L k = E(L k ) + σ ( Lk E(L k ) ) }{{ σ } =: Z k (µ σ2 2 ) t k + σz k (Z k ) k N is a random walk with V (Z k ) = t k. The increments Z k+h Z k are approximately normally distributed and centered, are independent of the past F k, have variances V (Z k+h Z k ) = h(t t)/n equal to elapsed time. QFin - ContTimeFinance Slide 81 - Path structure c Helmut Strasser, May 1, 2010

82 Introduction - Review of discrete time finance - From discrete to continuous time The Black-Scholes model For t [0, T ] let ] S t = S 0 exp [(µ )t σ2 + σw t 2 where the increments W t W s are normally distributed and centered, are independent of the past F s, have variances V (W t W s ) = t s equal to elapsed time. A market model M = (B, S) where B t = e rt and where S t satisfies the conditions stated above is called a Black-Scholes model. For each fixed time t the asset prices S t follow a lognormal model. The log-returns have independent and stationary increments. QFin - ContTimeFinance Slide 82 - Black-Scholes model c Helmut Strasser, May 1, 2010

83 Stochastic analysis - Stochastic processes Chapter 2: Stochastic processes Section 2.1: Basic concepts Section 2.2: Wiener process Section 2.3: Variation of paths QFin - ContTimeFinance Slide 83 - Contents c Helmut Strasser, May 1, 2010

84 Stochastic analysis - Stochastic processes - Basic concepts Chapter 2: Stochastic processes Section 2.1: Basic concepts Syllabus: stochastic process - path - past - increment - martingale - continuity - jumps QFin - ContTimeFinance Slide 84 - Contents c Helmut Strasser, May 1, 2010

85 Stochastic analysis - Stochastic processes - Basic concepts Stochastic processes Definition: A stochastic process is a family (X t ) t [0,T ] of random variables defined on a time interval [0, T ]. Path: A path of a process (X t ) t [0,T ] is a particular realization (x t ) considered as a function t x t of time. Past: The past of a process (X t ) t [0,T ] at time s is the information set F s := F(X u : u s) corresponding to all paths up to time s. Increments: The random variable X t X s, s < t, is called the increment of the process (X t ) t [0,T ] on the subinterval [s, t]. Martingale: A process (X t ) is a martingale if E(X t F s ) = X s whenever s < t. QFin - ContTimeFinance Slide 85 - Stochastic process c Helmut Strasser, May 1, 2010

86 Stochastic analysis - Stochastic processes - Basic concepts Continuity and jumps: Let (x t ) t [0,T ] be a particular path of a process (X t ) t [0,T ]. If at a particular time point t the difference x t := x t x t is zero then the path is continuous at t. Otherwise it has a jump at t with jump height x t. A process is called continuous if all paths are continuous everywhere. There are no jumps. QFin - ContTimeFinance Slide 86 - Path properties c Helmut Strasser, May 1, 2010

87 Stochastic analysis - Stochastic processes - Wiener process Chapter 2: Stochastic processes Section 2.2: Wiener process Syllabus: Wiener process - martingale properties - generalized Brownian motion - drift - diffusion - geometric Brownian motion - Black Scholes model - risk neutrality QFin - ContTimeFinance Slide 87 - Contents c Helmut Strasser, May 1, 2010

88 Stochastic analysis - Stochastic processes - Wiener process Wiener process Definition: A stochastic process (W t ) t [0,T ] is a Wiener process (Brownian motion process) if 1. it is continuous, 2. the increments satisfy W t W s N(0, t s), 3. the increments are independent of the past. Mathematical probability theory: There are Wiener processes. (A44) Show that the Wiener process is a martingale. (A45) Find E(W 2 t F s ). (A46) Find A t such that (W 2 t A t ) is a martingale. (A47) Find lim t W t t. QFin - ContTimeFinance Slide 88 - Wiener process c Helmut Strasser, May 1, 2010

89 Stochastic analysis - Stochastic processes - Wiener process A path of a Wiener process QFin - ContTimeFinance Slide 89 - Diagram c Helmut Strasser, May 1, 2010

90 Stochastic analysis - Stochastic processes - Wiener process 20 paths of a Wiener process QFin - ContTimeFinance Slide 90 - Diagram c Helmut Strasser, May 1, 2010

91 Stochastic analysis - Stochastic processes - Wiener process pplot(rpwiener()(1)) pplot(rpwiener()(20)) QFin - ContTimeFinance Slide 91 - Calc: Wiener process c Helmut Strasser, May 1, 2010

92 Stochastic analysis - Stochastic processes - Wiener process Generalized Brownian motion: X t = µt + σw t Drift: The drift of a continuous process is the average forward velocity ( lim E Xt+h X ) t F t := µ(t) h 0 h (A48) Show that the drift of a generalized Brownian motion is constant and equal to µ. What about the Wiener process? Diffusion: The diffusion coefficient of a continuous process is the average forward quadratic increment rate ( lim E (Xt+h X t ) 2 ) F t := σ 2 (t) h 0 h (A49) Show that the diffusion coefficient of a generalized Brownian motion is constant and equal to σ 2. What about the Wiener process? QFin - ContTimeFinance Slide 92 - Drift and Diffusion c Helmut Strasser, May 1, 2010

93 Stochastic analysis - Stochastic processes - Wiener process A generalized Brownian motion with µ = 1 and σ 2 = QFin - ContTimeFinance Slide 93 - Diagram c Helmut Strasser, May 1, 2010

94 Stochastic analysis - Stochastic processes - Wiener process X=rpWiener(mean=1,cov=0.1) pplot(x(1)) QFin - ContTimeFinance Slide 94 - Calc: Generalized Brownian Motion c Helmut Strasser, May 1, 2010

95 Stochastic analysis - Stochastic processes - Wiener process Black-Scholes Model ] Let (W t ) be a Wiener process and S t = S 0 exp [(µ )t σ2 + σw t. 2 The logarithms of a Black-Scholes model have drift µ σ 2 /2 and diffusion coefficient σ 2. (A50) Find E(S t ). (A51) Find E(S t F s ), s < t. Definition: A Black Scholes market model M = (B, S) is a risk neutral model if B t = e rt and µ = r. (A52) Let (B, S) be a risk neutral Black Scholes model. Show that (S t ) is a martingale. QFin - ContTimeFinance Slide 95 - Black Scholes model c Helmut Strasser, May 1, 2010

96 Stochastic analysis - Stochastic processes - Wiener process pplot(rpbs()(1)) pplot(rpbs()(20)) pplot(rpbs(vol=0.2,rate=0.08)(20)) QFin - ContTimeFinance Slide 96 - Calc: Geometric Brownian Motion c Helmut Strasser, May 1, 2010

97 Stochastic analysis - Stochastic processes - Variation of paths Chapter 2: Stochastic processes Section 2.3: Variation of paths Syllabus: size of increments - microscopic order of size - Riemannian subdivisions - total variation - quadratic variation - finite variation - variation of smooth functions - variation of Wiener paths QFin - ContTimeFinance Slide 97 - Contents c Helmut Strasser, May 1, 2010

98 Stochastic analysis - Stochastic processes - Variation of paths Size of increments: V (W t W s ) = t s V ( Wt W s t s ) = 1 It follows: As (t s) 0 the difference W t W s shrinks of the order t s. This is remarkable: As (t s) 0 the differences (t s) are much smaller than t s. f(t) f(s) For smooth functions f(t) the ratios remain stable as (t s) 0 t s (actually tend to f (s)). For the Wiener process the ratios W t W s t s explode as (t s) 0. QFin - ContTimeFinance Slide 98 - Size of increments c Helmut Strasser, May 1, 2010

99 Stochastic analysis - Stochastic processes - Variation of paths Subdivisions: Subdivision of an interval [a, b]: a = t 0 < t 1 < t 2 <... < t n = b Sequence of subdivisions: t 10 < t 11 t 20 < t 21 < t 22 t 30 < t 31 < t 32 < t t n0 < t n1 < t n2 <... < t n,n 1 < t nn... Riemannian sequence of subdivisions: max i t ni t n,i 1 0 as n QFin - ContTimeFinance Slide 99 - Subdivisions c Helmut Strasser, May 1, 2010

100 Stochastic analysis - Stochastic processes - Variation of paths Variation of paths: Let f(t) be any function defined on an interval [a, b]. Total variation (first order): Va b (f) := lim n for any Riemannian sequence of subdivisions of [a, b]. i f(t ni ) f(t n,i 1 ) Quadratic variation: QVa b (f) := lim n for any Riemannian sequence of subdivisions of [a, b]. i f(t ni ) f(t n,i 1 ) 2 QFin - ContTimeFinance Slide Variation c Helmut Strasser, May 1, 2010

101 Stochastic analysis - Stochastic processes - Variation of paths The sinus function: QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

102 Stochastic analysis - Stochastic processes - Variation of paths Total variation t V0 t (f) of the sinus function: QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

103 Stochastic analysis - Stochastic processes - Variation of paths Quadratic variation t QV t 0 (f) =: [f] t of the sinus function: QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

104 Stochastic analysis - Stochastic processes - Variation of paths A path (W t ) of a Wiener process: QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

105 Stochastic analysis - Stochastic processes - Variation of paths Total variation t V0 t (W ) of a Wiener path: QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

106 Stochastic analysis - Stochastic processes - Variation of paths Quadratic variation t QV t 0 (W ) =: [W ] t of a Wiener path: QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

107 Stochastic analysis - Stochastic processes - Variation of paths Path of W t + sin(2πt): QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

108 Stochastic analysis - Stochastic processes - Variation of paths Quadratic variation of W t + sin(2πt):: QFin - ContTimeFinance Slide Diagram c Helmut Strasser, May 1, 2010

109 Stochastic analysis - Stochastic processes - Variation of paths pplot(as.path("sin(2*pi*t)",ngrid=5000),ylim=c(-1,1)) pplot(pvariation(as.path("sin(2*pi*t)",ngrid=5000),order=1)) pplot(pbracket(as.path("sin(2*pi*t)",ngrid=5000)),ylim=c(-1,1)) W=rpWiener(ngrid=5000) pplot(w(1)) pplot(pvariation(w(1),order=1)) pplot(pbracket(w(1))) pplot(papply("x+sin(2*pi*t)",w(1))) pplot(pbracket(papply("x+sin(2*pi*t)",w(1)))) QFin - ContTimeFinance Slide Calc: Quadratic Variation c Helmut Strasser, May 1, 2010