0.1 Valuation Formula:

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1 0. Valuatio Formula: 0.. Case of Geeral Trees: q = er S S S 3 S q = er S S 4 S 5 S 4 q 3 = er S 3 S 6 S 7 S 6 Therefore, f (3) = e r [q 3 f (7) + ( q 3 ) f (6)] f () = e r [q f (5) + ( q ) f (4)] = f () = e r [q f (3) + ( q ) f ()] = e r [q q 3 f (7) + q ( q 3 ) f (6) ( q ) q f (5) + ( q )( q ) f (4)] i.e., Iitial value of a claim X = e rn (the probability of the path associated with fial state i) (payoff at state i) fial states

2 0.. Case of Recombiat Trees q = er d u d f () = e "q r f (7) + q ( q) f (5) +( q) f (4) recombied paths For the N-step: The preset value of a optio with payoff f (S T ): e rn N X k=0 µ N k q k ( q) N k f S 0 u k d N k µ N where istheumberofwaysofhavigk steps up ad N k steps dow i a total k N time-steps. e.g., A Europea call has the preset value: e rn Recall: For recombiat trees, N X k=0 µ N k q k ( q) N k S 0 u k d N k K + the risk-eutral probability is q = er d u d

3 For the biomial tree, T = N, is The preset value of a optio with payoff f (S T ) X N µ z } { N V (f) = e rn q j j ( q) N j f Su j d N j j=0 Q j Note that the umber of ways of j ups ad N j dows i N steps µ N Q j = j q j ( q) N j is the probability that the fial state is i j th state with stock price S j (T )=Su j d N j at time t = T, ad it, obviously, satisfies Therefore, i.e., it is a expectatio of f (S T ): NX Q j =. j=0 V (f) =e rt N X j= Q j f (S j (T )) V (f) =e rt E RN [f (S T )] Note that the subscript RN refers to the risk-eutral probability for the tree, which is simply µ N Q j = q j j ( q) N j for the j th state with stock price S j (T )=Su j d N j. 3

4 0. Some Simple Results:. The cotiget claim pays the stock price itself, i.e., f (S T )=S T Replicatig portfolio: share of stock; 0 bod; No eed for tradig therefore, at t = 0, the preset value is S; at t = T, f (S j )=Su j d N j at the j th state V (f) = e rt E RN [S T ] X N µ N = e rt j j=0 = e rt S [qu +( q) d] N q j ( q) N j Su j d N j [qu +( q) d ] e r =, ad T = N V (f) =e rt Se rn = S as expected. To summarize e rt E RN [S T ]=S today s value of the stock which is the geeralizatio of S ow = e r [qsu +( q) Sd] =e r E RN [S ext ] to a biomial tree.. The cotiget claim is a forward cotract with strike K : The payoff the is ad the preset value is f (S T )=S T K V (f) = e rt E RN [f (S T )] = e rt E RN [S T K] = e rt {E RN [S T ] E RN [K]} = e rt E RN [S T ] e rt E RN [K] = S e rt K 4

5 which is the familiar, old result, which we obtaied by usig the replicatig portfolio: 3. A Europea call with K À S 0 : Replicatig portfolio: ½ S Ke rt stocks; bod. (a) If S 0 u N >K>S 0 u N d, the probability associated with this path is Q N = q N Sice q<, if N is very large, the Q N very small I geeral, if K À S 0, there are exceptioally small umber of paths, therefore, the preset value of the call is (small probability) (S K) + 0 i.e., the call is early worthless. (b) If S 0 À K, the almost all states cotribute with very small exceptioal paths yieldig 0 values, therefore, The value of such a call e rt E RN [S T K] =S Ke rt = forward Logormal Price Dyamics passage to the cotiuum limit. Itroductio of the logormal model of stock price dyamics 5

6 . Its relatio with biomial trees 3. The Black-Scholes formulas Termiologies: what is a retur? It is δs S Therateofreturr j is more clearly, it is r j = δs S = δ log S e r j = e δ log S log S[(j+)] log S[j] = e i.e., Therefore, e r j = S [(j +)] S [j] S 0 = S (0), S () = S 0 e r 0 S () = S () e r = S 0 e r +r 0 S (k) = S ((k ) ) e rk = S 0 e r k + +r +r 0 Stock price moves radomly ad this radomess ca be described by the radom variable r j, log S (k) log S 0 = r k + + r + r 0.. Logormal Model of Stock Price Movemet Assumptios about the stock price movemet: Assumptio (): r j is iid Note that 6

7 . Idepedece: Iformatio before j caot be used to predict ext r j ;. Idetically Distributed: r j does ot deped o previous level of the stock price S. 3. This assumptio about market yields a "efficiet market". Assumptio (): r j is a Gaussia radom variable with mea µ ad variace σ where µ, σ are costats: µ the mea rate of retur; σ volatility... Properties:. For ay time iterval (t,t ), log S (t ) log S (t ) is a Gaussia radom variable with mea µ (t t ) ad variace σ (t t ) This is why the dyamics is called logormal, i.e., log S is a ormal process. Proof: (a) The reaso that log S (t ) log S (t ) is Gaussia is because log S (t ) log S (t ) = X i r i = sum of Gaussia radom variables. (b) " X E [log S (t ) log S (t )] = E r i i = X E [r i ] i = X i µ (Assumptio ()) = µ (t t ) 7

8 (c) Ã! X Var(log S (t ) log S (t )) = Var r i = X i i idepedece = X Var(r i ) i σ (Assumptio ()) = σ (t t ). log S T log S 0 N [µt, σ T ] therefore, log S T N log S 0 + µt, σ T If X N [log S 0 + µt, σ T ], i.e., the pdf of x is " p (x) = πσ T exp (x log S 0 µt ) σ T that is, the probability of the radom variable X havig values i [x, x + dx] is " p (x) dx = πσ T exp (x log S 0 µt ) dx σ T The, the probability of S is [S, S + ds] is " πσ T S exp (log S log S 0 µt ) ds σ T whichcabeseeasfollows: ds x =logs, S = dx p (S) ds = p (x) dx p (S) =p (x) dx ds = p (x (S)) S 8

9 Homework: E [S T ] = S 0 e µt + σ T Var(S T ) = S 0e µt +σ T ³ e σt. 3. The distributio of the rate of retur, η, which is defied via S T S 0 e ηt therefore, η = T log S T S 0 η N µµ, σ T i.e., the rate of retur is Gaussia distributed with mea µ... Empirical Evidece:. Logormal dyamics of stock price movemet is a approximatio (cf. volatility smile);. What are the causes of the volatility? Issues: σ is caused by o Arrival of ew iformatio; o Due to tradig Observatio: If it is due to the arrival of ew iformatio, the Market ope: Var ope = σ T Market closed : Var close = σ (3T ) iformatio from 3 days But Var close.0 Var ope observed Similar results hold for agricultural commodities. These idicate that the volatility is largely caused by tradig. 9

10 . Logormal Dyamics as the Limit of Multiperiod Biomial Trees. The stock dyamics as the limit of a biomial tree;. The he o-arbitrage valuatio as the limit of a biomial tree with risk-eutral probability... Dyamics of Stock Movemet u = e µ+σ, p = with (accurate to O (δ)) where d = e µ σ, p = mea = pu +( p) d = e ( µ+ σ ) var = pu +( p) d (mea) = σ. µ is the mea rate of cotiuous compoud retur over ; i.e., S () =S 0 e µ. µ + σ istherateofexpected retur of S over. i.e., E (S ()) = S 0 e (µ+ σ ). Note that i Lecture 3 we used symbol µ to represet this whole express µ + σ. This sloppiess is deliberate: to make agai the poit really, the o-arbitrary price, itheed,hasothigtodowiththedriftofthemarket Choosewhateveryou like to make life easier (here meaig makig our otatio easier). But you eed to fix oe covetio for cosistet computatio. Of course, this should ot be cofused with the fact that there is a true, ecoometric value of µ i the real world. 0

11 If the path has j times up, j times dow, the, the price of the stock h S (t) = S (0) exp j µ + σ i h +( j) µ σ io = S (0) exp µ + jσ ( j) σ o = S (0) exp µt +(j ) σ o Note that j is a radom variable the umber of up s. The process is the same as flippigacoiadj would be the umber of heads i flips of a fair coi (i.e., p =/), whichisthesameas X = the sum of idepedet radom variables Y i,i=,, of 0 or with probability p =. Therefore, E (Y i ) =, i =,, " Var(Y i ) = p µ +( p) µ 0 = 4 ad the Cetral Limit Theorem says P i= Y i mea of Y = VarY X q 4 = X N (0, ) as Hece, S (t) = S (0) exp µt +(X ) σ o = t ½ = S (0) exp µt + σ t (X ¾ ) S (0) exp µt + σ o tz as 0, where Z N (0, ) the stadard ormal distributio log S (t) log S (0) N µt, σ t whichrecoversthelogormaldyamicsofstockpricemovemet.

12 .. The Risk-Neutral Valuatio as 0 Recall the simplest biomial tree for risk-eutral valuatio is mea = pu 0 +( p) d 0 = e r Var = pu 0 +( p) d 0 mea = σ which has mea ad variace: accurate up to O (δ) We already kow that µ does ot matter ad we ca use this simplest biomial tree brach to costruct the whole tree. Note that we still have a fair coi sice q =. At t =, for the path with j times of up, we have ½ S (t) = S (0) exp j µr σ + σ +( j) µr σ σ ¾ = S (0) exp ½µr σ +(j ) σ ¾ = S (0) exp ½µr σ +(X ) σ ¾ (idetify j with the radom variable X above) = t = S (0) exp ½µr σ t + σ t (X ¾ ) S (0) exp ½µr σ t + σ ¾ tz as 0, which is the risk-eutral stock price dyamics ad is idepedet of µ (of course!). Note that, i a world of o-arbitrary pricig, we could simply thik the stock dyamics is described by the risk-eutral probability (risk-eutral measurevaluatio Recall

13 As where p (S T ) is the pdf of S T. Defie V (f) =e rt N X j=0 Q j f (S j ) Z V (f) =e rt ds T p (S T ) f (S T ) 0 x = log S T S (0), S T = S 0 e x x N µµr σ T,σ T p (S T ) ds T = p (x) dx = e (x ( r σ ) T ) σ T dx πσ T Therefore, V (f) =e rt Z f (S 0 e x ) e (x ( r σ ) T ) σ T dx () πσ T Note that. Whe f itheexpressioisthepayoff of a call or put, the formula becomes the so-called Black-Scholes formula.. For o-arbitrage pricig, we eed to use the risk-eutral measure for which S (t) =S (0) e (r σ )t+σ tz i cotrast to the "real" world measure for which S (t) =S (0) e µt+σ tz. 3

14 ..3 Properties. If f (S T )=S T, we kow its value at t =0is S (0). The formula () should also give V (f) =S 0. Let us verify this: Z V (f) =e rt (S 0 e x ) e (x ( r σ ) T ) σ T dx πσ T Usig whichcabeseeasfollows, E (e x )=e m+ a E (e x ) = = = πa πa πa Z + = e m+ a πa = e m+ a e x e (x m) a dx Z " + (x m) exp x dx a Z " + exp m + a [x (m + a )] dx a Z " + exp [x (m + a )] dx a {z } = Lettig m = µr σ T a = σ T V (f) = S 0 e rt e (r σ )T + σ T = S 0 as expected. i.e., e rt E RN (S T )=S 0. Europea call: f (S T )=(S T K) + 4

15 c V (f) Z = e rt (S 0 e x x (r K) + πσ T e ( σ )T) σ T dx N.B. S 0 e x K 0 = x K S 0 therefore, Z c = e rt (S 0 e x K) e (x ( r σ ) T ) σ T dx K S πσ 0 T This ca further be expressed as c = S 0 N (d ) Ke rt N (d ) where N (x) = d, = Z x π e y dy " σ T log S 0 e ( r± σ )T K cumulative distributio fuctio 3. How to valuate a Europea put? Use c p = S 0 Ke rt. 5

Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge

Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity