: Forward index level derived from put-call parity. : First strike price below the forward index level, F

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1 Appendix A The Carr Madan Implied Volatility CM IMV The following is the CBOE formula for the VIX index, denoted as CM IMV. CM = τ CM IMV = CM i i i e rτ Qτ, i τ F τ F i : Time to expiration : Forward index level derived from put-call parity : First strike price below the forward index level, F : Strike price of the i th out-of-the-money option; a call if i >, and a put if i < ; both put and call if i = i r Qτ, i = i+ i : Risk-free spot rate of interest : The ask price/bid price for the option with strike i In the VIX index calculation, Q i is the midpoint of the bid-ask spread for each option with strike price i. The forward index level is: F = strike price + e rτ Call price P ut price where the strike price selected is that for which the absolute difference between the call and put prices is the smallest. In our paper, we use the strike price that is closest to the spot price to calculate the forward index level, sometimes called the effective forward price. The original formula proposed by Carr and Madan doesn t include the term involving F. Appendix B Appendix B- Expected Profit and Variance Portfolio Expected Profit Given Y : Assume y = ln Y Nµ Y, Y and Z N,, then X = Y Z + µ Y, Y = exp Y Z + µ Y The calculation of E Y IY y and V ar Y IY y is detailed as follows: E Y IY y =E exp Y Z + µ Y Iexp Y Z + µ Y y =E exp Y Z + µ Y IZ lny µ Y Y = exp Y u + µ Y exp u z π du; Let z = lny µ Y Y = exp Y + µ Y exp u Y du z π = exp Y + µ Y N Y z 9

2 E Y IY y =E exp Y Z + µ Y Iexp Y Z + µ Y y = exp Y u + µ Y exp u z π du; Let z = lny µ Y Y = expy + µ Y exp u Y du z π = exp Y + µ Y N Y z V ar Y IY y =E Y IY y {E Y IY y} = exp Y + µ Y { exp Y N Y z N Y z } Let y, that is z, we have E Y = exp Y + µ Y V ar Y = exp Y + µ Y exp Y Let C τ = P Y P k +. The expectation and variance of the portfolio value can be explicitly calculated below. let z = lnk µ Y, is the holded stocks for hedging, P k is the strike price, and B = p c Y ap is the amount of money market account at very beginning. E W a =E a P τ C τ + Be rτ =E a P Y P Y P k + + Be rτ = a P exp Y + µ Y P exp Y + µ Y N Y z k N z + Be rτ. If we express the mean and volatility in annualized term, we can show the formula for C τ is the same as Black- Scholes-Merton pricing model. Let µ Y = µτ, Y = τ and = P k. Given the expectation operator is the risk neutral probability, we have e τ +µτ = e rτ. Then the following equation for C τ is exactly the same as Black-Scholes-Merton Formula after discounted by risk-free rate. P exp Y + µ Y N Y z k N z = P e rτ N = P e rτ N = P e rτ N = P e rτ N Y lnk+µ Y Y P k N lnk+µ Y Y Y / lnk+lneµ Y + Y / Y ln P e µ Y + Y / + P k Y / Y Y / Y P k N Y / lnk+lneµ Y + N ln P e µ Y + Y / P k Y / Y ln P +rτ+ Y τ/ Y τ N ln P +rτ Y τ/ Y τ The variance of the portfolio is 3

3 V ar W a =V ar a P τ C τ + B a e rτ and B a = C a ap =V ar a P Y + V ar P Y P k + a covp Y, P Y P k + = a P V ary + P V ar Y k + ap covy, Y k+ V ar W b =V ar b P τ + C τ + B b e rτ and B b = C b bp =V ar b P Y + V ar P Y P k + + b covp Y, P Y P k + = b P V ary + P V ar Y k + + b P covy, Y k+ We need to calculate V ar Y k + and covy, Y k +. V ar Y + =V ar Y k IY k Cov Y, Y k + = Cov Y, Y k IY k = E Y Y k IY k exp =V ar Y IY k k IY k =V ar Y IY k + k V ar IY k k cov Y IY k, IY k = exp Y + µ Y exp Y N Y z N Y z +k NzN z k exp Y + µ Y N Y znz Y + µ Y = E Y IY k k EY IY k exp = exp Y + µ Y NY z k + exp +k exp Y + µ Y N z exp Y Y + µ Y + µ Y Y + µ Y exp N Y z k N z exp Y + µ Y Y + µ Y N Y z k N z N Y z Plugging V ar Y k + and covy, Y k + into V ar W a, and rearranging the terms, we obtain V ar W a = V ar W b = P C + A N Y z + A N Y z + A 3 N z where C = a exp Y + µ Y exp Y A = exp Y + µ Y a A = exp Y + µ Y exp + µ Y Y N Y z + k Nz a k + exp Y + µ Y A 3 = k k Nz a exp Y + µ Y Portfolio Mean and Variance given a Bernoulli type Volatility. 3

4 We assume a Bernoulli type random volatility, which is a probability with independence assumption 4 for different annual volatility levels, L and H. In our math derivation, we let L = L and H = H. We also use L, H in subscripts to denote the volatilities L, H. Traders assign the subjective probability φ and φ to volatility L and H, respectively. Expected Profit given a Bernoulli type Volatility For a written and a long call option position, after traders apply delta hedging, the final wealth of W a and W b are W a = a P T C T + Ca a P e rτ W b = b P T + C T + C b b P e rτ where C a and C b are selling price and buying price respectively, C T is final option payoff C T = MaxP T,, and are the hedging positions. C T and is cash-outflow for a written call, while C a and is cash inflow for a written call. Conversely, C T and Cb are cash-inflow and cash-outflow respectively for a long position. A positive negative means a long short position of the stock; therefore, if a is positive, a P T is the amount of cash-inflow from selling stocks at P T and a P is the cash-outflow to buy stocks at P. Given assumption 4, EP T = P E P { T = P φe L P T + φe H P } T = P { φe rτ + φe rτ } = P e rτ. P P P And the expected final wealth for selling a call option and purchasing a call options are, EW a =φe H { C a e rτ C T + ap T P e rτ } + φe L { C a e rτ C T + ap T P e rτ } =C ae rτ φe H {C T } φe L {C T } = C ae rτ E {C T } EW b =φe H { C b e rτ + C T + b P T P e rτ } + φe L { C b e rτ + C T + b P T P e rτ } =E {C T } C b erτ Using Appendix B- formula, we can derive the option price formula which is the same as Black-Schole-Merton Model. The expected prices under different volatility levels are, E H C τ =e rτ P Nz H + H τ e rτ Nz H and z H = ln P + r H τ H τ E L C τ =e rτ P Nz L + L τ e rτ Nz L and z L = ln P + r L τ L τ P where z H = ln E H P T J.Therefore, the expected final wealth is EW a = C ae rτ e rτ { P φ NzH + H τ + φ Nz L + L τ e rτ φ Nz H + φ Nz L } Variance given a Bernoulli type Volatility V arw a = φv ar H { C a e rτ C T + ap T P e rτ } + φv ar L { C a e rτ C T + ap T P e rτ } + φ φe L E H 3

5 where E L = E L { C a e rτ C T + ap T P e rτ } and E H = { C ae rτ C T + ap T P e rτ }. To calculate V ar H { C a e rτ C T + ap T P e rτ }, we can rewrite V ar H { C a e rτ C T + ap T P e rτ } as V ar H { C a e rτ C T + ap T P e rτ } = av ar H {P T } + V ar H {C T } a Cov H P T, C T ln Pτ follows N r P H τ, H τ. Then, the variance of P τ is { } V ar H {P T } = E H P T EH P T = P e erτ H τ. The variance of C τ is V ar H C τ =V ar H P τ IP τ =V ar H P τ IP τ IP τ =V ar H P τ IP τ + V ar H IP τ Cov H P τ IP τ, IP τ =P {e erτ H τ N H τ + z H N H } τ + z H + Nz H N z H P e rτ NH τ + z H N z H The co-variance of P τ and C τ is Cov H P τ, C τ =Cov H P T, P T IP T =E H { PT IP T } E H {P T IP T } E H P T E H PT + =P erτ+ H τ N H τ + z H P e rτ + P erτ N H τ + z H + P e rτ Nz H Therefore, { } V ar H C a e rτ C T + a P T P e rτ = C + A NH τ + z H + A NH τ + z H + A 3 Nz H C = P a erτ e H τ A = P erτ+ H τ a A = P erτ NH τ + z H P e rτ N z H + ap e rτ + P e rτ A 3 = N z H ap e rτ The derivation for V ar L is the same. And the unconditional Variance for a bernoulli type random volatility is: V arw a = φv ar H { C T + a P T } + φv ar L { C T + a P T } + φ φe L C T E H C T For the general log-normal random variable lnx Nµ,, we have the following general results: The First and Second Moment of X. EX = e µ+ and EX = e µ+. The Restricted First and Second Moment of X. E X IX x = EX Φ + lnx + µ and E X IX x = EX Φ + lnx + µ. 33

6 V arw b = φv ar H {C T + b P T } + φv ar L {C T + b P T } + φ φe L C T E H C T The derivation of V ar W b is the same as we did for V ar W a. Variance are the same, because a = b, the proof of which is shown next. Appendix B- The Optimal Delta The portfolio value at the expiration date W a and W b can have two volatility realizations L, H with subjective probability φ and φ respectively. The variance of W a and W b can be written as V arw a = φv ar L W a + φv ar H W a + φφ E L W a E H W a = φ { av ar L P T + V ar L C T acov L P T, C T } +φ { a V ar HP T + V ar H C T acov H P T, C T } + φφ E L W a E H W a V arw b = φv ar L W b + φv ar H W b + φφ E L W b E H W b = φ { b V ar LP T + V ar L C T + b Cov L P T, C T } +φ { b V ar HP T + V ar H C T + b Cov H P T, C T } + φφ E L W b E H W b The necessary conditions to minimize V ar W a and V ar W b are = V arwa = φ { av ar L P T Cov L P T, P T + } + φ { av ar H P T Cov H P T, P T + } + φφ { ae L P T E L PT + ae H P T + E H PT +} E L P T E H P T = V arw b = φ { b V ar L P T + Cov L P T, P T + } + φ { b V ar H P T + Cov H P T, P T + } + φφ { b E L P T + E L PT + b E H P T E H PT +} E L P T E H P T Therefore, the optimal a and b are a = ECovP T +,P T V EV arp T V = φcov LP T,P T + +φcov HP T,P T + + φφe L P T E H P T {E LP T + E HP T + } φv ar L P T +φv ar H P T + φφe L P T E H P T b = ECovP T +,P T V EV arp T V = φcov LP T,P T + +φcov HP T,P T + + φφe L P T E H P T {E LP T + E HP T + } φv ar L P T +φv ar H P T + φφe L P T E H P T Appendix B-3 Comparative Statics for τ Let fa,, L, H, τ; φ = EW a Q V arwa c =. We show the following propositions. γ 34

7 When time to maturity decreases, the ask volatility increases for ATM options, i.e. A = A < Assume the ask volatility corresponds to the ask price C a in the risk neutral probability measure, i.e. We have C a = P NA τ + z A e rτ Nz A and z A = ln P + r A τ A τ Now, we derive the equations for A and. To calculate EWa, we first calculate erτ C a. 3 e rτ C a A = erτ C a A >. = EWa Q V arw a γ = nz A A τ + rp e rτ Nz A + A τ Since C H and C L are similar to erτ C a EW a with H, L replacing a respectively, we can finally write as EW a = τ nz AA φ nz H H φ nz L L +rp e rτ Nz A + A τ φ Nz H + H τ φ Nz L + L τ To calculate V arwa, we first calculate V ar H and the details follow. V ar H = C + A NH τ + z H + A NH τ + z H + A 3 Nz H NH τ+z +A H NH τ+z + A H Nz + A H 3 We have 3 We have the basic results regarding the normal density function. nz A + A τ = π e A τ+z A = π e A τ z A A τz A = P e rτ nz A nz A + A τ = e A τ+z A = e A τ z A A τz A = π π P e rτ A τ nz A 35

8 C = P r + H e rτ+ H τ re rτ A NH τ + z H = P r + H e rτ+ H τ NH τ + z H NH τ+z A H = H τ + z H nz H A NH τ + z H = P e rτ H τ + z H nz H NH τ + z H rp erτ NH τ + z H +P e rτ z H nz H NH τ + z H rp e rτ N z H NH τ + z H +r P e rτ NH τ + z H + 4r P erτ NH τ + z H A NH τ+z H = { P e rτ NH τ + z H N z H + + P e rτ} H τ + z H nz H A 3 Nz H = z H nz H r P e rτ Nz H A 3 Nz H = N z H P e rτ z H nz H Adding up all the parts and rearranging, we finally have 4 V ar H = D + B NH τ + z H + B NH τ + z H + B 3 Nz H + Nz H P e rτ NH τ + z H + P e rτ H τ nz H D = P r + H e rτ+ H τ re rτ B = P r + H e rτ+ H τ B = rp erτ NH τ + z H rp e rτ N z H + r P e rτ + 4r P erτ B 3 = r P e rτ To calculate V arwa, we also need to calculate E H E L. E H E L = E H {C T } E L {C T } E H {C T } E L {C T } = E H {C T } E L {C T } τ nz HH nz L L + rp e rτ Nz H + H τ Nz L + L τ We discuss the limiting behavior of For ATM call option, i.e. = P e rτ, we have for ATM, OTM and ITM call options in turn. nz H = nz L = nz A π and Nz H = Nz L = Nz A, when τ 4 ϑv ar L ϑτ can be written in the similar format with L replacing H. 36

9 Then we have the limiting behavior EWa V arwa 5 and as follows, EW a = A φh φl + o πτ τ V arw a = φp H a a + π + φp L a a + π + φ φ H L + o 4π EW As τ, a approaches positive infinity 6, while approaches maturity date, which implies that A <. For OTM call option, i.e. > P e rτ, we have V arwa is a constant. Therefore, > as time nz H τ = nz L τ = nz A τ and Nz H = Nz L = Nz A, when τ Then, the limiting behavior of EWa and V arwa is as follows, 5 The last term of ϑv ar H ϑτ EW a =o V arw a =φp a H + φp a L + o is messy, and needs special attention. The detailed derivation, assuming τ, follows. N H τ N H H τ τ nz H Nz H P e rτ NH τ + z H + P e rτ H τ nz H = = H N H τ N H τ H τ As time τ goes to, we have Therefore, n H τ N lim H τ N H τ τ H = n = and lim n τ π τ H τ = n = π lim Nz H P e rτ NH τ + z H + P e rτ H τ nz H = τ π H The limiting behavior of ϑe H E L ϑτ is also tricky, and the derivation is ϑe H E L = E H {C τ } E L {C τ } ϑτ τ nz HH nz L L + o = P e N rτ H τ N L τ τ nz HH nz L L + o As time τ goes to, we have = P e rτ H L N H τ N L τ H L τ ϑe H E L lim = H L + o τ ϑτ 4π n + o 6 To compensate for the hedging uncertainty and transaction cost, we have EW a >. For ATM call option, EW a = N A τ φ N H τ φ N L τ > N A τ φ N H τ φ N L τ > N + A τn φ N + H τn φ N + L τn + o τ > A φh φl + o > A λh λl > as τ 37

10 As τ, EWa V arwa becomes, while becomes a constant positive number. Therefore, < as time approaches maturity date, which implies that A >. 3 For ITM call option, i.e. < P e rτ, we have nz H τ = nz L τ = nz A τ and Nz H = Nz L = Nz A, when τ Then, the limiting behavior EWa and V arwa is As τ, EWa EW a =o V arw a =φp a H + φp a L + o becomes, while maturity date, which implies that A >. V arwa Appendix B-4 Second Derivative of τ Based on B-S model, we have A becomes a positive constant. Therefore, < as time approaches = / A = f A f A A A = τnz A > f A = τnz A τ z A z A In Proposition, we have already discussed the limiting behavior of and now we continue on discussing the limiting behavior of f For ATM call option, we have for ATM,OTM and ITM call options, for the three different cases. f A EW a = 4 πτ 3 A φh φl + o τ 3 V arw a = O f A = nz A + o τ τ As τ, EW a V arwa becomes negative, tends to a positive constant, so f is negative. Also, as τ, becomes positive. Therefore, we can conclude that as time approaches maturity date, the ask volatility will increase at an increasing rate. For OTM call option, we have EW a = O nz A τ 5 V arw a = O and f A = O nz A τ 3 As τ, EW a becomes, V arw a tends to a positive constant, so f is a negative constant. The order of A is Onz A τ, while the order of f A is O nz A, which implies that f dominates. Therefore, as τ 3 A A time approaches the maturity date, the ask volatility will decrease at a decreasing rate. 3 For ITM call option, we have the same conclusion as for the OTM call option. The insight lies in noting that for OTM and ITM call option, the normal density function of z is an infinitesimal in any order of τ, i.e. τ, for all m R. Appendix B-5 The Volatility Level Effect nz τ m as 38

11 Consider A = A In Proposition, we proved that A = τnz A >. Hence, we only need to determine the sign of = EW a Q V arw a. γ For EWa, we have EW a = nz A A τ φ nz H H τ φ nz L L τ For V arwa, we first calculate V ar H using the Chain Rule. V ar H = C + A NH τ + z H + A NH τ + z H + A 3 Nz H + A NH τ + z H + A NH τ + z H + A 3 Nz H For every part of V ar H, we have C = P ah τe rτ+ H τ A N H τ + z H = ap H τe rτ+ H τ NH τ + z H A NH τ+z H = a H τ + z H nzh A NH τ + z H = P e rτ z H H τ nz H NH τ + z H A NH τ+z H = { P e rτ NH τ + z H N z H + a + ap e rτ} H τ + z H nzh A 3 Nz H = z H nz H Nz H A 3 Nz H = N z H ap e rτ z H nz H Adding up all the parts and rearranging, we finally have 7 V ar H = P H τe rτ+ H τ + ap H τe rτ+ H τ NH τ + z H + Nz H P e rτ NH τ + z H a + ap e rτ H τnz H To calculate V arwa, we also need to calculate E H E L and the details follow. E H E L = E H {C τ } E L {C τ } E H {C } E L {C τ } = E H {C τ } E L {C τ } nz H H τ nz L L τ 7 ϑv ar L ϑ can be written in the similar format with L replacing H. 39

12 We will discuss the limiting behavior of for ATM, OTM and ITM call options in turn. For ATM call option, i.e. = P e rτ, we have nz H = nz L = nz A π and Nz H = Nz L = Nz A, when τ Then we have the limiting behavior EWa and V arwa as follows, EW a = τ π A φh φl + o τ O τ and V arw a = Oτ As τ, EWa that A <. dominates V arwa For OTM call option, i.e. > P e rτ, we have. Therefore, > as time approaches maturity date, which implies EW a = Onz τ and V arw a = Oτ Since nz is an infinitesimal of oτ n for any n when z, V arwa dominates EWa, which implies that A < as τ. Therefore, > as time approaches maturity date. 3 For ITM call option, i.e. > P e rτ, we use the similar reasoning as in OTM, and we reach the same conclusion. Appendix C Appendix C- The optimal dynamic hedging strategy We first compute the variance over two periods. According to the law of total variance, the total variance for a two-periods model is V ar W a, = E V ar W a, + V ar E W a, V ar W a, = E V ar W a, + V ar E W a, = E V ar E W a, + V ar E W a, = E V ar C, + P + V ar C, + P and V ar W a, = where V ar {E W a, } = V ar { e r τ E P + + e r τ E P } = V ar { C, + P } V ar {E W a, } = V ar E P + + P = V ar C, + P Therefore, V ar W a, = E {V ar C, + P + V ar C, + P } We let C, denote the equilibrium call price at time where C, e r τ = E P +. In addition, C, = P +. Now we derive the optimal hedging strategy. We follow the methodology in Basak and Chabakauri and apply dynamic programming to the value function J t, which is defined as 4

13 J = V ar tw a,. The law of total variance yields a recursive representation for the value function. J t = min t { EtJ t+ τ + V ar t E t+ τ P T + + tp t+ τ }. where t is the stock holding and τ is time interval. We first check optimization for period. { } J = min E J + V ar E P + + P = min {E J + V ar C, + P }. By F.O.C, we get optimal as, We continue to get optimal for period. = Cov C,,P V ar P. The solution is J = min E J + V ar E P + + P = min E J + V ar C, + P. = Cov C,,P V ar P. The general solution for multiple-periods model is also provided by Basak and Chabakauri. To get analytical solution for, we advance to compute covariance. Given in each period we have two possible realizations µ H τ, H τ and µ L τ, L τ with a bernoulli random arrival rate, law of total covariance yields Cov C,, P = φcov,l C,, P + φcov,h C,, P +Cov φe,l C, + φe,h C,, φe,l P + φe,h P. Given E P V = P e r τ is constant by assumption 4, we simplify the covariance as Cov C,, P = φcov,l C,, P + φcov,h C,, P, Cov C,, P = φcov,l C,, P + φcov,h C,, P. Appendix C- The optimal dynamic hedging strategy Here we compute C, and C,. C, = E P + { } = e r τ φe P + X = x, V = H + φe P + X = x, V = L = φbsm x,h + φbsm x,l, where x, H, L are the realizations of X and V. ln P BSM x,h = P N +r τ+ H τ e r τ N H τ ln P BSM x,l = P N +r τ+ L τ e r τ N L τ ln P +r τ H τ H τ ln P +r τ L τ L τ. 4

14 C, = e r τ E P + { } = e r τ φ E E P + X, V = H V = H + φφe E P + X, V = H V = L { } +e r τ φ φe E P + X, V = L V = H + φ E E P + X, V = L V = L = φ BSM H,H + φ φbsm H,L + φ BSM L,L The following is the derivation for BSME H,H, BSME H,L and BSM L,L. Here in Appendix C, for the notation convenience, we let τ =. E E P + X, V = H V = H = E E P IP > X, V = H V = H E E IP > X, V = H V = H Let f X,H the normal density function. The first term can be derived as E E P IP > X, V = H V = H = X,H P e X,H +X,H PrX,H > ln P X,H X,H f X,HdX,H X,H = u H + Hε = X,H P e X,H +u H + H ε Prε > ln P X,H u H H X,H f X,HdX,H. let Z = ln P X,H u H H = P X e X,H +u H,H ε >Z e H ε e ε dε f π X,HdX,H = P X e X,H +u H + H e ε H dε,h Z f π X,HdX,H = P X e X,H +u H + H N ln P X,H u H,H H f X,H dx,h H = X,H P e u H +ε +u H + H Note N X,H ln P u H H H = P e u+ H ε N X,H ln P u H H H N ln P +u H + H H + ε = Prε > ln P X,H u H H. H apply theorem : Nm + sε π e ε g dε = N = P e u H + H N ln P +u H + H. H e ε H dε π X,H u H e H d πh H ε m s +g s +. Then we derive the second term. E E IP > X, V = H V = H = X,H PrX,H > ln P X,H X,H f X,HdX,H = Prε X > ln X P,H u H X,H,H f X H,HdX,H = N ln +X P,H +u H X u H e H dx X,H H πh,h N ln P +u H e ε dε π = ε = N H + ε ln P +u H H 4

15 Given u H + H = r, { } e r E E P + X, V = H V = H = P N ln P +r+ H H e r N ln P +r H H = BSM H,H Using the same procedure, we can derive { } e r E E P + X, V = L V = L = P N ln P +r+ L L e r N ln P +r L L = BSM L,L and { } e r E E P + X, V = H V = L = P N ln P +r+ H +r+ L H + L e r N ln P +r H +r L H + L = BSM L,H and { } e r E E P + X, V = L V = H = P N ln P +r+ H +r+ L H + L e r N ln P +r H +r L H + L = BSM H,L Appendix C-3 Proof of Proposition 5 T Let Y = j=t+ τ X j, P T = P e Y with each X j a Bernoulli-type random normal distribution. n n Therefore, we know C t,t = i= φ i φ n i E t PT + V ary = i H τ + n i L τ i n n { } C t,t = i= φ i φ n i P t N d e nr τ N d i d = ln P t +i r τ+ H τ +n i r τ+ L τ i H τ+n i L τ d = ln P t +i r τ H τ +n i r τ L τ, n = i T H τ+n il τ τ, Next, Cov tc t+,t, P t+ = E t CovtC t+,t, P t+ V t+ + Cov t EtC t+,t V t+, E tp t+ V t+ E tp t+ V t+ is constant by our assumption. = E t CovtC t+,t, P t+ V t+ = φ { E tc t+,t P t+ V t+ = H E tc t+,t V t+ = H EP t+ V t+ = H } + φ { E tc t+,t P t+ V t+ = L E tc t+,t V t+ = L EP t+ V t+ = L } 43

16 The detail of first term of covariance is E tc t+,t P t+ V t+ = H { n n { } } = E t i= φ i φ n i + i N d P t+ e n r τ N d V t+ = H, where n = T t+ τ. And we advance to express the terms as τ { } E t P t+ N d P t+ e n r τ N d V t+ = H ln P t eu H τ+ H τεt+ N = E t P te n r τ e u H τ+ H τεt+ N +i r τ+ H τ +n i r τ+ L τ i H τ+n i L τ ln +i r τ H τ +n i +u H τ r τ L τ + ε t+ +u H τ + ε t+ i H τ+n i L τ V t+ = H Nm + sε e ε g dε = N π = P t er τ+ H τ N ln P te n r τ N +i r τ+ H τ ln m s +g s + +n i r τ+ L τ i+ H τ+n i L τ +i r τ H τ +n i r τ L τ i+ H τ+n i L τ +r τ+ 3 H τ +r τ+ H τ. Let r + = r τ + H τ, r H + = r τ + L τ, r L H = r τ H τ and r = r τ L τ L { } E t P t+ N d P t+ e n r τ N d V t+ = H = e r+ ln P t H N +i r+ H +n i r + L + r + H + H τ i+ H τ+n i L τ P te n r τ ln P t N +i r H +n i r r L + + H. i+ H τ+n i L τ. Similarly, details inside E tc t+,t P t+ V t+ = L are 44

17 { } E t P t+ N d P t+ e n r τ N d V t+ = L ln P t eu L τ+ L τεt+ N = E t P te n r τ e u L τ+ L τεt+ N +i r τ+ H τ ln +n i r τ+ L τ +u L τ + ε t+ i H τ+n i L τ +i r τ H τ +n i r τ L τ +u L τ + ε t+ i H τ+n il τ V t+ = L = P t er τ+ L τ N ln P te n r τ N = P t e r+ L N +i r τ+ H τ ln +n i r τ+ L τ i H τ+n i+ L τ +i r τ H τ ln P t +i r+ H +n i r + L + r + L + L τ P te n r τ N i H τ+n i+ L τ +n i r τ L τ i H τ+n i+ L τ ln P t +i r H +n i r r L + + L i H τ+n i+ L τ. +r τ+ 3 L τ +r τ+ L τ The second term E tc t+,t V t+ = H EP t+ V t+ = H and E tc t+,t V = H EP t+ V t+ = L can be expressed separately as and E tc t+,t V t+ = H EP t+ V t+ = H n = n { } i= φ i φ n i E t + N d e n r τ N d V t+ = H e u H τ+ H τ i {{ } } E t + N d e n r τ N d V t+ = H e u H τ+ H τ P te u H τ+ H τεt+ N = P te r τ E t e n r τ N ln ln +i r τ+ H τ +i r τ H τ +n i r τ+ L τ +u H τ + ε t+ i H τ+n i L τ +n i r τ L τ +u H τ + ε t+ i H τ+n i L τ V t+ = H = P t er τ N ln +i+ P te n r τ N = P t er τ N r τ+ H τ +n i i+ H τ+n i +i+ L τ r τ H τ ln ln P t +i+ r+ H +n i r + L P te n r τ N Similarly i+ H τ+n i L τ r τ+ L τ +n i i+ H τ+n i L τ ln P t +i+ r H +n i r L i+ H τ+n i L τ r τ L τ 45

18 and E tc t+,t V = L EP t+ V = L n = n { } i= φ i φ n i E t + N d e n r τ N d V t+ = L e u L τ+ L τ i { } E t + N d e n r τ N d V t+ = L e u L τ+ L τ P te u L τ+ L τεt+ N = P te r τ E t e n r τ N ln ln +i r τ+ H τ +i r τ H τ +n i r τ+ L τ i H τ+n il τ +n i r τ L τ +u L τ +u L τ + ε t+ + ε t+ i H τ+n il τ V t+ = L = P t er τ N +i ln r τ+ H τ +n i+ r τ+ L τ i H τ+n i+ L τ P te n r τ N ln +i r τ H τ ln P t +i r+ H +n i+ r + L +n i+ i H τ+n i+ L τ = er τ N i H τ+n i+ L τ P te n r τ N ln P t +i r H +n i+ r L. i H τ+n i+ L τ Combining, r τ L τ φe tc t+,t V = H EP t+ V = H + φe tc t+,t V = L EP t+ V = L P n = + n + te r τ N i= φ i φ n + i P t i e n r τ N The analytical solution for covariance is ln P t +i+ r+ H +n i+ r + L i H τ+n i+ L τ ln P t +i+ r H +n i+ r L i H τ+n i+ L τ 46

19 Cov tc t+,t, P t+ n = φ φ i φ n i {P } t i er+ H N d,h e n r τ N d,h n + φ φ i φ n i {P } t i er+ L N d,l e n r τ N d,l n + n + i= φ i φ n + i {P } t i er τ N d e n r τ N d. and d,h = ln P t +i r+ H +n i r + L + r + H + H τ i+ H τ+n i L τ d,h = ln P r t +i r H +n i r L + + H i+ H τ+n i L τ d,l = ln P t +i r+ H +n i r + L + r + L + L τ i H τ+n i+ L τ d,l = ln P t +i r H +n i r + r + L L i H τ+n i+ L τ d = ln P t +i+ r+ H +n i+ r + L i H τ+n i+ L τ d = ln P t +i+ r H +n i+ r L i H τ+n i+ L τ 47