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# Dynamic equity price pdf

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http://awesomefinance.weebly.com/

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### Dynamic equity price pdf

1. 1. 8/28/14 Dynamic Equity Models Learning Objec>ves ¨ Simula>on ¤ Daily, monthly, annual sta>s>cal rela>onships ¨ Lognormal probability density ¨ Stochas>c differen>al equa>on ¨ Con>nuous >me price process ¨ Exact solu>on ¨ Price and return probabili>es in con>nuous >me ¨ Probability basics for op>on deriva>ves 2 More Simula>on 3 Perform a stock price simula>on for which current stock price, S0 = \$40.00, the expected monthly con>nuously compounded mean rate of return, u, is 1%, and the expected standard devia>on, s, is 5%. Perform the simula>on with daily >me increments for one year. Use floa>ng point >me, annualized, μ and σ, sta>s>cs. Run the simula>on 10,000 >mes. u 12 12.000% μ = ⋅ = s 12 17.321% σ = ⋅ = .004 years t 1 Δ = = 252 T = 1.000 years μ ⋅Δ t + z ⋅ σ ⋅ Δ t .12 .004 z .17321 .004 S = S e +Δ ⋅ t t t S = S e + ⋅ t .004 t ⋅ + ⋅ ⋅ Simula>on: 4 \$60 \$55 \$50 \$45 \$40 \$35 \$30 \$25 \$20 \$15 \$10 \$5 \$0 μ ⋅Δ t + z ⋅ σ ⋅ Δ t .12 .004 z .17321 .004 S = S e +Δ ⋅ t t t S = S e + ⋅ t .004 t ⋅ + ⋅ ⋅ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Stock Price Time [years] Dynamic Equity Models 1
2. 2. 8/28/14 Simula>on: 5 From Simulation Daily Mean rate: u 0.04859% Standard deviation: s 1.09460% -­‐6% -­‐5% -­‐4% -­‐3% -­‐2% -­‐1% 0% 1% 2% 3% 4% 5% 6% Natural Log Daily Return Rate Simula>on: 6 From simula>on M[ST] \$ 45.09 E[ST] \$ 45.91 Min[ST] \$ 23.95 Max[ST] \$ 93.91 From input M[S ] S e μ T = ⋅ T 0 ⋅ \$40.00 e = ⋅ \$45.10 = E[S ] S e * μ T = ⋅ T 0 ⋅ \$40.00 e = ⋅ \$45.78 .12 ⋅ 1.0 .135 ⋅ 1.0 The median price is the 5,000th in an ordered list of 10,000 simulated prices at T=1.0 years. The expected price is the average of the 10,000 prices. \$20 \$25 \$30 \$35 \$40 \$45 \$50 \$55 \$60 \$65 \$70 \$75 \$80 \$85 \$90 \$95 Stock Price At 1 Year = Lognormal PDF 7 The lognormal pdf is • Asymmetric • Mode, median, and mean not equal • Never nega>ve • Over >me the mode, median, and mean driZ further apart • Over >me the distribu>on skews more posi>vely In the standard price theory Simple rates, future value factors, and asset prices are distributed lognormal Return Rate and Future Value Factor PDFs 8 [ ] E[v] u M v = = v ~N(u,s2 ) ~N( , s2 ) μ [ ] = [μ] M μ E [ v ] u E [ e v ] e u* Me e = = e v ~ e N ( u,s2 ) Me12⋅[ v ] = eμ E [ e12⋅v ] eμ * = e12⋅v ~ eN(μ,σ2 ) Dynamic Equity Models 2
3. 3. 8/28/14 Exact Solu>on The differen>al equa>on for dln(S) is dln(S) = μ⋅ dt + σ ⋅ dw The solu>on with ini>al condi>on is ln(S ) = ln(S ) + μ ⋅ t + σ ⋅ dw t 0 ln(S ) = ln(S ) + μ ⋅ t + z ⋅ σ ⋅ t t 0 t μ t z σ t ⎞ ⋅ ⋅ + ⋅ = ⎟⎟⎠ ~N[μ t, t] ln S S t 0 ⎛ ⎜⎜⎝ ln S S 2 t 0 ⎞ ⋅ σ ⋅ ⎟⎟⎠ ⎛ ⎜⎜⎝ 9 At >me t the natural log of price ln(St) is distributed normally as ln(S ) ~Nln(S ) μ t, σ t t 0 Therefore [ + ⋅ ⋅ ] [ * ] N ln(S ) + μ ⋅ t, σ ⋅ t [ * ] 0 N μ t, σ t S ~ e t S ~ S e t 0 ⋅ [ ] [ ] t MS = e E S e t t t μ⋅ * μ ⋅ ⋅ ⋅ = Simula>on 10 ( ) [ ] [ ] [ ] 2 v N(u,s ) f 1 r e ~ e 2 2 ≡ + = k u k ⋅ s k 2 th ⋅ + E f e k moment for f u s 2 2 st = + E f e 1 moment for f 2 = 2 2 u 2 s nd E f e ⋅ + ⋅ 2 moment for f u = M[f] = e Median for f Var[f] E[f ] (E[f]) Variance for f 2 2 = − u u s 2 2 * * = + u = ln(1 + a) u* a = e − 1 u = ln(1 + g) u = − g e 1 d2 = Var[r] = Var[f] = E[f2 ] − (E[f])2 Monthly Statistics Specified Rate of return: u 1.0% Standard deviation, s 5.0% Annual frequency, m 12 Computed Variance, s2 0.00250 Expected rate of return, u* 1.12500% Expected first moment of f 1.01131 Expected second moment of f 1.02532 Simple mean rate, a 1.13135% Geometric rate, g 1.00502% Simple standard deviation, d 5.05973% Simula>on 11 k k k 2 2 σ ⋅ [ ] ⋅μ+ σ 2 [ ] 2 [ ] [ ] [ 2 ] ( [ ])2 E f = e E f = e μ+ 2 2 2 2 E f e 2 = ⋅μ+ ⋅σ Var f = E f − E f 2 σ 2 μ = μ + ln(1 ) μ = + α * e 1 * * μ α = − ln(1 ) μ = + γ e μ 1 γ = − 2 Var[ ] Var[f] E[f2 ] (E[f])2 δ = α = = − Monthly Statistics Specified Rate of return: u 1.0% Annual Statistics Computed μ 12.00000% Standard deviation, s 5.0% σ 17.32051% Annual frequency, m 12 Computed Computed Variance, s2 0.00250 σ2 0.03000 Expected rate of return, u* 1.12500% μ* 13.50000% Expected first moment of f 1.01131 1.14454 Expected second moment of f 1.02532 1.34986 Simple mean rate, a 1.13135% α 14.45368% Geometric rate, g 1.00502% γ 12.74969% Simple standard deviation, d 5.05973% δ 19.97357% Simula>on 12 Monthly Statistics Annual Statistics Daily Statistics Computed Specified Computed Rate of return: u 1.0% μ 12.00000% μ Δt 0.04762% Standard deviation, s 5.0% σ 17.32051% σ √Δt 1.09109% Annual frequency, m 12 m 252 Computed Computed Computed Variance, s2 0.00250 σ2 0.03000 σ2 t 0.00012 Expected rate of return, u* 1.12500% μ* 13.50000% μ∗ Δt 0.05357% Expected first moment of f 1.01131 1.14454 1.00054 Expected second moment of f 1.02532 1.34986 1.00119 Simple mean rate, a 1.13135% α 14.45368% 0.05359% Geometric rate, g 1.00502% γ 12.74969% 0.04763% Simple standard deviation, d 5.05973% δ 19.97357% 1.09171% Dynamic Equity Models 3
4. 4. 8/28/14 Daily Sta>s>cs 13 m ( ) E[S ] \$40 1 a \$45.78 T = ⋅ + = a .05357% * * u m u 252 = E[S ] = S ⋅ e ⋅ = \$40 ⋅ e ⋅ = \$45.78 T 0 * u .05356% um u 252 = M[S ] = S ⋅ e ⋅ = 40 ⋅ e ⋅ = \$45.10 T 0 u .04762% 252 ( ) = M[S ] \$40 1 g \$45.10 T = ⋅ + = g = .04763% Price as a Stochas>c Diff Eqn 14 Difference eqn for price as geometric Brownian mo>on with posi>ve expected rate of return S S St t t = ⋅ Δ = − +Δ ΔS * = ⋅ + ⋅ Δw z Δt μ Δt σ Δw S Transform to a differen>al eqn as Δt -­‐> dt with the goal to solve the eqn for price, S dS S μ* dt S σ dw = ⋅ ⋅ + ⋅ ⋅ dw= z ⋅ dt μ: con>nuously compounded natural log mean rate of return μ*: con>nuously compounded simple mean or expected rate of return To understand stochas>c differen>al, dS, introduce F which is a func>on of stochas>c process, S. S is dependent on Weiner process, w. F = f(S) Stochas>c Differen>al, dF 15 Write dF as a Taylor series expansion 2 dF F F 2 dS higher order terms S dt F t dS 1 S + ⋅ 2 2 + ∂ ∂ ∂ + ∂ ∂ = ∂ Subs>tute dS into dF F 1 ∂ F ⋅ ⋅ + ⋅ ⋅ * (μ S dt σ S dw) * 2 2 2 S ⋅ ⋅ + ⋅ ⋅ + ⋅ 2 (μ S dt σ S dw) F ∂ + S dt t dF ∂ ∂ ∂ = ∂ Ignore dt2 and dw·∙dt terms and subs>tute dw2 = dt which will be explained on the next slide. 2 dF F 2 2 σ S ⎞ dt F ∂ + ⋅ ⎟⎟⎠ ∂ ∂ * ⋅ ⋅ σ S dw S F S 1 + ⋅ ∂ 2 μ S F S t 2 ∂ ⎜⎜⎝⎛ ⋅ ⋅ ∂ ∂ ⋅ ⋅ + ∂ = Stochas>c Differen>al, dF 16 Determine: E[dW], E[dW2], VAR[dW], VAR[dW2] to resolve dw2 = dt E[dw]= E[z ⋅ dt]= dt ⋅E[z]= 0 E[dw2 ] E[(z dt)2 ] dt E[z2 ] dt = ⋅ = ⋅ = = − [( ) ] ( [ ]) [ 2 ] ( [ ]) 2 [ ] dt E[z ] dt 1 dt VAR(dw) E dw E dw 2 E z dt 0 = ⋅ − 2 = ⋅ = ⋅ = 2 2 2 2 2 VAR (dw ) = E dw − E dw [ ] ( ) [ ] dt 3 dt 0 4 2 2 E z dt dt = ⋅ − 2 4 2 dt E z dt = ⋅ − 2 2 = ⋅ − = dw ∼ N(0,dt) dw2 ∼ N(dt,0) Stochas>c Determinis>c Dynamic Equity Models 4
5. 5. 8/28/14 Probability Distribu>ons Related to dw and dw2 17 Z distribu>on -­‐4 -­‐3 -­‐2 -­‐1 0 1 2 3 4 Z2 distribu>on 0 1 2 3 4 5 6 7 8 9 10 Z4 distribu>on 0 5 10 15 20 25 Solve For Price . 18 2 Price differen>al eqn dF F 2 2 μ S F S ⎞ ∂ + ⋅ ⎟⎟⎠ ∂ ∂ * ⋅ ⋅ ∂ ⋅ ⋅ + ∂ This differen>al equa>on cannot be solved analy>cally, but can be solved under a change of variable, S. Ln(S) can be solved for ⎛ ⎜⎜⎝ ∂ 1 + ⋅ ∂ t ln(S) σ S dt F F S 1 2 2 ⋅ ⋅ ∂ μ * S ∂ lnS ⋅ + S ∂ μ S 0 1 S t + ⋅ ∂ ( 1 2 F =ln(S) = = ⎛ ⋅ ⋅ + + − ⋅ S 2 dt σ dw * μ σ 2 2 ⎜⎝ ⎛ = * − ⎞ ⋅ + ⋅ ⎟⎟⎠ ⎜⎜⎝ μ dt σ dw σ S dw S 2 2 ⎞ ∂ + ⋅ ⎟⎟⎠ σ S dt 1 σ S dw S 2 ∂ ln(S) S 1 2 ∂ ∂ 2 2 2 ⎞ )σ S dt σ S dw ln(S) S ⎜⎜⎝⎛ = dln(S) = ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⎟⎠ ⋅ ⋅ ∂ Solu>on For Price 19 μ t z σ t S S e * ⋅Δ + ⋅ ⋅ Δ = +Δ ⋅ μ ⋅ t + z ⋅ σ ⋅ t [ ] t t S S e t ⋅ = S ~ S e t [ ] t μ σ 2 2 * N μ ⋅ t , σ ⋅ t ⋅ E S = S ⋅ e = S ⋅ e 0 * μ t 0 0 t 0 t ⎞ ⋅ ⎟⎟ ⎠ ⎛ ⎜⎜ ⎝ + ⋅ ln(S ) = ln(S ) + μ ⋅Δ t + z ⋅ σ ⋅ Δ t t +Δ t t ln(S ) = ln(S ) + μ ⋅ t + z ⋅ σ ⋅ t t 0 μ t z σ t ⎞ ⋅ ⋅ + ⋅ = ⎟⎟⎠ [ ] t μ σ 2 2 ~Nμ t, t ln S S ⎛ ⎜⎜⎝ ln S S ⎞ ⋅ σ ⋅ ⎟⎟⎠ ⎛ ⎜⎜⎝ M[S ] = S ⋅ e = S ⋅ e 0 μ t t 0 t 0 t 0 2 * ⎞ ⋅ ⎟⎟ ⎠ ⎛ ⎜⎜ ⎝ − ⋅ Log and Expecta>on Operators 20 Note nonlinearity of expecta>on and natural log Start with natural log of price, Start with price expecta>on, then take expected value then take natural log ⎛ [ ] μ t z σ t ⎞ ⋅ ⋅ + ⋅ = ⎟⎟⎠ μ t ln S S t 0 ⎜⎜⎝ ⎡ E ln S S t 0 t ⎤ ⋅ = ⎥⎦ ⎢⎣ ⎞ ⎟⎟⎠ ⎛ ⎜⎜⎝ E[ln(S )] = ln(S ) + μ ⋅ t t 0 E S S e * μ t = ⋅ t 0 * ⎤ t μ t e E S S 0 = ⎥⎦ ⎡ ⎢⎣ ⋅ ⋅ ln(E[S ]) ln(S ) μ * t t = + ⋅ 0 ( [ ]) * [ ] ln E S + μ ⋅ t = E ln(S ) + μ ⋅ t t t ( [ ]) [ ] ( * ) ln E S − E ln(S ) = μ -­‐μ ⋅ t t t ln(E[S ]) E[ln(S )] 2 σ ⋅ t 2 > t t = μ μ σ 2 2 * = + μ μ σ 2 2 * − = Dynamic Equity Models 5
6. 6. 8/28/14 Simula>on: Probability of Median and Mean Price 21 ln S MED T S 0 σ⋅ ⎛ ⎜⎜⎝ 0.0011 ln \$45.09 ⎞ ⋅ μ − ⎟⎟⎠ T T = = − = [ ] % 4995 . 0 Sˆ z 0 Pr S .12 1.0 \$40.00 ⎞ ⋅ − ⎟⎠ .12 1 < = T MED ⋅ ⎛ ⎜⎝ ln S EXP T S 0 σ⋅ ⎛ ⎜⎜⎝ = 0.1022 ln \$45.91 ⎞ ⋅ μ − ⎟⎟⎠ T T = = [ ] % 071 . 54 Sˆ z 0 Pr S .12 1.0 \$40.00 ⎞ ⋅ − ⎟⎠ .12 1 ≤ = T EXP ⋅ ⎛ ⎜⎝ Simula>on: Probability of Min and Max Price 22 ln S min T S 0 σ⋅ ⎞ ⋅ μ − ⎟⎟⎠ ⎛ ⎜⎜⎝ 3.6547 T T = = − = [ ] % 013 . Sˆ z 0 Pr S .12 1.0 ln \$23.95 \$40.00 ⎞ ⋅ − ⎟⎠ .12 1 ≤ = T MIN ⋅ ⎛ ⎜⎝ ln S MAX T S 0 σ⋅ ⎛ ⎜⎜⎝ = 4.2347 ⎞ ⋅ μ − ⎟⎟⎠ T T = = [ ] % 001 . Sˆ z 0 Pr S .12 1.0 ln \$93.91 \$40.00 ⎞ ⋅ − ⎟⎠ .12 1 ≤ = T MAX ⋅ ⎛ ⎜⎝ Probability of a Price Decline 23 Using the IBM equity price sta>s>cs of μ=8% and the probability of the drop in IBM price during the week ending October 10, 2008? IBM stock opened Monday October 6th at \$101.21, 10th at \$87.75, S0. January 1962 to September 2008. μ T ln S S T ⎛ ⎜⎜⎝ z 0 ⎞ ⋅ − ⎟⎟⎠ σ ⋅ T 4.16064 Recall that the IBM return sta>s>cs were computed from .08 1 ln 87.75 101.21 ⎞ ⋅ − ⎟⎠ 0.25 1 52 52 0 = = − ⋅ ⎛ ⎜⎝ = σ = 25% (Topic 9) , what was ST, and closed Friday October ) z ( N ~ Pr S S T ≤ 0 = 0 = − = [ ] % 00159 . ) 16064 . 4 ( N ~ That weekly decline was expected once in 1,212 years [ ] ( ) 0 Pr S ≤ S = T 0 z N ~ Probability of Not Exceeding a Cri>cal Value 24 An investor owns 100 shares of an equity with a current price per share of \$40.00. The equity has an expected rate of return μ*=16% and annual standard devia>on σ = 20%. What is the probability that the investor’s \$4,000, S0, will grow to no more than \$6,000, K, aZer 5 years? 14.0% 2 2 16.0% 20% * = − = − = 2 μ μ σ 2 ⎞ ⋅ μ − ⎟⎟⎠ T ln K S 0 σ⋅ ⎛ ⎜⎜⎝ 0.65860 T z 0 = = − ) z ( N ~ Pr S K .14 5.0 ln \$6,000 \$4,000 ⎞ ⋅ − ⎟⎠ .2 ⋅ 5.0 ⎛ ⎜⎝ = [ ≤ ] = = ( − 0.65860 ) = 25 . 51 % ~ N T 0 [ ] ( ) 0 Pr S ≤K = [ ] T z N ~ Pr S >K = T z N ~ ( ) 2 Dynamic Equity Models 6
7. 7. 8/28/14 Probability of a Loss of Value 25 What is the probability that the investor will have a loss aZer 5 years? ( S0 = K = \$4,000 ) μ T ln K S 0 ⎞ ⋅ − ⎟⎟⎠ σ ⋅ T ⎛ ⎜⎜⎝ 1.56525 z 0 = = − ) (z N ~ Pr S K .14 5.0 ln \$4,000 \$4,000 ⎞ ⋅ − ⎟⎠ .2 ⋅ 5.0 ⎛ ⎜⎝ = [ ≤ ] = = ( − 1.56525) = ~ 5.88% N T 0 The probability of a loss is 5.88% Pr S ≤K = [ ] [ ] ( ) 0 T z N ~ Pr S >K = T z N ~ ( ) 2 Probability of Exceeding a Cri>cal Value 26 An investor owns 100 shares of an equity with a current price per share of \$40.00. The equity has an expected rate of return μ*=16% and annual standard devia>on σ = 20%. What is the probability that the investor’s \$4,000, S0, will grow to more than \$6,000, K, aZer 5 years? μ T ln K S ⎛ ⎜⎜⎝ = [ ] Z 0 0 = − ⎞ ⋅ − ⎟⎟⎠ σ ⋅ T 0.65860 .14 5.0 ln \$6,000 \$4,000 ⎞ ⋅ − ⎟⎠ .2 ⋅ 5.0 ⎛ ⎜⎝ = ) Z ( N ~ ) Z ( N ~ Pr S K 1 T > = − 0 = − 0 = 2 = [ ] % 49 . 74 ) Z ( N ~ The probability that the value of the shares exceeds \$6,000 is 74.49% ( ) .14 5.0 ln \$4,000 \$6,000 ⎞ ⋅ + ⎟⎠ .2 5.0 ⎛ ⎜⎝ 0.65860 ⎞ 0 * 2 μ .5 σ T ⋅ ⋅ − + ⎟⎠ σ T ln S K Z 2 = = ⋅ ⋅ ⎛ ⎜⎝ ≡ Pr S ≤K = [ ] T z N ~ ( ) 0 Pr S >K = T z N ~ ( ) 2 Simple Binary Op>on 27 A security, C, is offered as follows: If an equity, S, currently priced at \$40, S0, exceeds \$45, \$K, aZer one year (T=1.0), then the buyer of this security, C, will receive \$K, if the equity, S, is less than or equal to K, then the buyer will receive nothing. The annual standard devia>on of the equity, σ, is 20% and the annual expected risk free rate of return, r*, is 6%. If ST > K, then CT = K If ST ≤ K, then CT = 0 d N ~ C e E C e K [ ] ( ) * * r T − ⋅ − ⋅ = ⋅ = ⋅ ⋅ T r T .38892 -­‐ N ~ e \$45 ( ) .06 1 − ⋅ = ⋅ ⋅ .06 2 e \$45 .34867 \$14.78 0 − = ⋅ ⋅ = [ ] [ ] E C = K ⋅ Pr S > K T T d N ~ K ( ) 2 ( ) = ⋅ ⎞ 0 * 2 r .5 σ T ⋅ ⋅ − + ⎟⎠ σ ⋅ T ( ) .38892 2 .06 .5 .2 1 ⎞ ⋅ ⋅ − + ⎟⎠ .2 1 ln S K ⎛ ⎜⎝ ln 40 45 d 2 = − ⋅ ⎛ ⎜⎝ = = The fair value of this security known as a “cash or nothing call op>on” is \$14.78 [ ] ( ) 0 Pr S ≤K = [ ] T d N ~ Pr S >K = T d N ~ ( ) 2 Confidence Intervals 28 What are the upper and lower bounds on a future stock price for which one is 95% (=1-­‐α) confident? St+ and St-­‐ are the upper and lower bounds at >me T = 0.5 years S S e * μ T 1.95996 σ T 0 ⋅ + ⋅ ⋅ ⋅ \$40.00 e \$57.17 + T = = = S S e .16 0.5 1.95996 0.2 0.5 * ⋅ + ⋅ ⋅ ⋅ μ T 1.95996 σ T 0 ⋅ − ⋅ ⋅ ⋅ \$40.00 e \$32.84 .16 0.5 1.95996 0.2 0.5 − T = = = ⋅ − ⋅ ⋅ ⋅ Confidence Level (1-­‐α) α α/2 -­‐Z +Z 90% 10% 5.00% -­‐1.64485 1.64485 95% 5% 2.50% -­‐1.95996 1.95996 99% 1% 0.50% -­‐2.57583 2.57583 ( − 1 . 95996 ~ ) N Dynamic Equity Models 7
8. 8. 8/28/14 Value at Risk (VaR) 29 What is the maximum loss that an investor would expect over some >me period t ? For example, what is the maximum loss expected with 95% confidence from owning an equity over a 10 day period? The equity has μ*= 16%, σ = 20%, and S0 = \$40.00. Unlike the confidence interval, which uses a two tailed confidence , VaR is a one-­‐tail interval. Confidence Level (1-­‐α) α -­‐Z 90% 10% -­‐1.28155 95% 5% -­‐1.64485 99% 1% -­‐2.32635 S S e * μ T 1.64485 σ T − ⋅ 0 = ⋅ − ⋅ ⋅ \$40.00 e \$37.70 1.64485 0.2 10 252 .16 10 ⋅ − ⋅ ⋅ 252 T = = ⋅ ( − 1 . 64485 ~ ) N Value at Risk (VaR) 30 The minimum 95% confident price is \$37.67, thus the 95% maximum expected loss is \$3.63 or value at risk, VaR And commonly approximated for short >me periods as follows VaR = \$40.00 − \$34.34 = \$5.66 VaR is computed directly as follows ( ) VaR S 1 e * μ T z σ T = ⋅ − 0 ⎛ ⋅ + ⋅ ⋅ \$40.00 1 e = ⋅ − \$2.30 1.64485 0.2 10 252 .16 10 ⋅ − ⋅ ⋅ 252 = ⎞ ⎟⎟ ⎠ ⎜⎜ ⎝ VaR = S ⋅ 1 − e 0 μ* T z σ T ⎛ ⎜⎝⎛ ⋅ + ⋅ ⋅ \$40.00 1 e = ⋅ − \$2.54 ⎟⎠⎞ 1.64485 0.2 10 252 = ⎞ ⎟⎟ ⎠ ⎜⎜ ⎝ − ⋅ ⋅ Expected Value Exceeding Cri>cal Value 31 ~ The same N problem as last slide, but now -­‐ what is the expected value of the equity posi>on given that the cri>cal value, K, has been exceeded? [ ] [ ] ( z ) 1 z N ~ ~( ) N ~ N 2 ( z ) ( z ) 2 * > = ⋅ μ T 1 E S |S K E S T T T S e ⋅ = ⋅ ⋅ 0 The deriva>on details are not included in this course. ( ) ⎞ 0 * 2 μ .5 σ T ⋅ ⋅ + + ⎟⎠ σ ⋅ T ( ) σ T ⎞ 0 * 2 μ .5 σ T ln S K ⎜⎝⎛ ln S K z z 1 2 ⋅ ⋅ − + ⎟⎠ ⋅ ⎛ ⎜⎝ = = 1.10581 0.65860 .18 5.0 ln \$4,000 \$6,000 ⎞ ⋅ + ⎟⎠ .2 5.0 .14 5.0 ⋅ ⎛ ⎜⎝ ln \$4,000 \$6,000 ⎞ ⋅ + ⎟⎠ .2 5.0 z z 1 2 = = ⋅ ⎛ ⎜⎝ = = E S |S \$6,000 \$4,000 e.16 5 .086560 [ ] .074492 > = ⋅ ⋅ \$8902.16 .086560 = ⋅ \$10,344 .074492 T T = Example: Price Distribu>on at >me T (5Yrs) 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 N [ ln(S ) μ T , σ T ] N[4.388879 E[ST|ST>K]=\$103.44 S ~ e 0 + ⋅ ⋅ , .447214] T ~ e Mode[ST]=\$65.95 K=\$60 Median[ST]=\$80.55 E[ST]=\$89.02 S0 \$0 \$20 \$40 \$60 \$80 \$100 \$120 \$140 \$160 \$180 \$200 32 Dynamic Equity Models 8
9. 9. 8/28/14 Another Simple Binary Op>on 33 A security, C, is offered as follows: If an equity currently priced at \$40, S0, exceeds \$45, K, aZer exactly one year (T=1.0), then the buyer of this security will receive the price of the equity, ST, if the equity, S, is less than or equal to K, then the buyer will receive nothing. If ST > K, then CT = ST If ST ≤ K, then CT = 0 [ ] [ ] [ ] E C = Pr S > K ⋅ E S |S > K T T T T d N ~ ( ) [ ] ( ) 1 d N ~ = ⋅ ⋅ ( ) ( ) 1 S E d N ~ 2 T S e ~ N r * T d ( ) ⎞ 0 * 2 r .5 σ T ⋅ ⋅ + + ⎟⎠ σ ⋅ T ( ) .18892 2 .06 .5 .2 1 ⎞ ⋅ ⋅ + + ⎟⎠ .2 1 ln S K ⎛ ⎜⎝ ln 40 45 d 1 = − ⋅ ⎛ ⎜⎝ = = ⋅ The fair value of this security known as a “asset = ⋅ ⋅ 0 2 [ ] ( ) ( ) * r T C e E C T = ⋅ − ⋅ .18892 -­‐ N ~ 40 \$ d N ~ S = ⋅ = ⋅ 0 1 \$40 .42509 \$17.00 0 = ⋅ = or nothing call op>on” is \$14.78 [ ] ( ) 0 Pr S ≤K = [ ] T d N ~ Pr S >K = T d N ~ ( ) 2 Comparing the Two Binary Op>ons ¨ cash or nothing call op>on ¨ asset or nothing call op>on 0.05 0.04 0.03 0.02 0.01 0 S K S K T > T ≤ E[S |S K] T T > K \$10 \$20 \$30 \$40 \$50 \$60 \$70 \$80 \$90 34 [ ] [ ] [ ] E C = Pr S > K ⋅ E S |S > K T T T T d N ~ ( ) [ ] ( ) 1 d N ~ = ⋅ ⋅ ( ) ( ) S E d N ~ 2 T S e ~ N r * ⋅ T d = ⋅ ⋅ 0 − r * ⋅ T [ ] T ( ) 0 1 C e E C 0 1 2 = ⋅ d N ~ S = ⋅ [ ] [ ] E C = K ⋅ Pr S > K T T d N ~ K ( ) 2 = ⋅ [ ] K = E K|S > K d N ~ C e K ( ) 2 * r T 0 T − ⋅ = ⋅ ⋅ ≤ = [ ] [ ] ( ) ( ) 0 2 Pr S K T ~ N d = -­‐ ~ d N Pr S >K = T d N ~ ( ) 2 Essen>al Concepts 35 Appendix: Probability and Expecta>on Summary 36 [ ] ( ) ( ) ( ) Pr S K T > = z = ~ N ~ -­‐ N ~ z = 1 N − z 2 0 0 z N ~ [ ] [ ] ( ) 1 z N ~ ( ) 2 E S |S K E S > = ⋅ T T T Risk Neutral d N ~ Pr S K N [ ] ( ) [ ] [ ] ( d ) 1 ~ d N ~ ( ) 2 > = T 2 E S |S K E S > = ⋅ T T T [ ] ( ) ( ) Pr S K T ≤ = ~ N z = -­‐ ~ z N 0 2 z N ~ 1 ( ) 2 = − Risk Neutral [ ] ( ) 2 Pr S K T d -­‐ N ~ ≤ = [ ] E[ ] Pr Risk neutral probability Risk neutral expecta>on Dynamic Equity Models 9
10. 10. 8/28/14 Appendix: 1 Tail Confidence 37 90% 95% 99% Confidence 95% confident that return rate lies above the shaded area Appendix: 2 Tail Confidence 38 90% 95% 99% Confidence 95% confident that return rate lies between the shaded areas Dynamic Equity Models 10