MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING

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1 MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING G. BERKOLAIKO Summary Deriving BSM equation using self-financing portfolio and Ito calculus. Boundary conditions for European call price. The Greeks: Delta, Gamma, Theta, Rho and Vega; checking that Black-Scholes formula is a solution to the PDE. Hedging procedure, examples and simulations. Estimating volatility: historic volatility, implied volatility. 1. Black Scholes Merton equation 1.1. Ito differential of the option value. We have seen Ito chain rule: let f = f(t, g, g = g(t, w, then the differential of f t = f ( t, g(t, W t with respect to time is (1.1 df t = f t dt f 2 g (dg t 2 + f 2 g dg t. Let V (t, S t be the value of a financial derivative which depends on time and the price of the underlying S t. Assume the price of the underlying follows geometric Brownian motion, (1.2 ds t = µs t dt + σs t dw t. To use Ito chain rule (1.1 with f = V and g = S, we need (ds t 2. Squaring equation (1.2 and discarding terms smaller than dt (note that (dt 2 dtdw t (dw t 2 = dt, we get (ds t 2 = (σs t dw t 2 = σ 2 St 2 dt. Ito chain rule then gives (1.3 dv t = ( V t σ2 St 2 2 V S 2 dt + V S ds t Self-financing portfolio. Consider of a portfolio Π = {a stock, b cash}. Here b is the amount of cash denominated in time t = dollars. We will denote the value, at time t, of a single time- dollar by β t. Of course β t = e rt and the value of the cash portion of the portfolio is bβ t = be rt. A portfolio is called self-financing if the change of its value at any time is entirely due to the change in value of the underlying assets. In other words, there is no inflow or outflow of capital from the portfolio, but the exchange of cash and stock is allowed. As an example, consider an Individual Retirement Account to which you contributed your maximal annual amount. You cannot take money out (until retirement, you cannot contribute more funds (until next year, but you can sell your stock holdings and the cash from the sale will still remain inside the account; it will attract interest and can be used to purchase stock at a different time. Because we allow exchange, the amount of stock and cash can change; the decision to exchange may be taken depending on the stock price and passage of time, thus a = a(t, S t and b = b(t, S t. We allow negative values of a (shorting the stock and b (borrowing cash. Of course, this is not allowed in one s retirement account. 1

2 2 G. BERKOLAIKO The value of the portfolio at time t is (1.4 Π t = a(t, S t S t + b(t, S t β t. Mathematically, the self-financing condition is written as (1.5 dπ t = a(t, S t ds t + b(t, S t dβ t Self-financing replicating portfolio. We want to design a self-financing portfolio which will replicate the value of a given financial derivative. This means that Π t = V t and, in particular the change in value at every t is identical, dπ t = dv t. Equating the differentials, we get (1.6 a(t, S t ds t + b(t, S t dβ t = ( V t σ2 St 2 2 V S 2 dt + V S ds t. We would like to remove the risk associated with stock price moves, ds t. If we chose a(t, S t = V, S all terms containing ds t will cancel. Note that we can reasonably do that since we chose how much stock to put into our replicating portfolio (this amount of stock we called delta. The corresponding amount of cash is determined from (1.4, ( V b(t, S t dβ t = b(t, S t rβ t dt = r(π t a(t, S t S t dt = r V t S t dt. S Equation (1.6 now reads ( ( V V r V t S t dt = S t σ2 St 2 2 V dt, S 2 or, moving everything to one side and canceling dt, V (1.7 t σ2 S 2 2 V V + rs rv =. S2 S This is the celebrated Black-Scholes-Merton partial differential equation. Remarkably, it is not stochastic, the random term was removed by delta-hedging. Since we never used any explicit information about the nature of the financial derivative, we expect it to be satisfied by any financial derivative that may be replicated by a self-financing portfolio. To find the particular solution of the equation that gives the value of, for example, call option, we need to specify the appropriate boundary conditions BSM equation and boundary conditions for a Call Option. In the special case when V (t, S is the value C(t, S of a call option, the BSM equation reads (simply substituting C for V C (1.8 t σ2 S 2 2 C C + rs rc =. S2 S One condition is readily available: it is the final payoff condition (1.9 C(T, S T = max(s T E,. We also remember the estimates we derived earlier, S t Ee r(t t C t S t and C t Squeeze lemma now implies that as S, the value of the call also goes to. On the other side, if S, the value of the call grows like S. To put it mathematically, (1.1 C(t, = and lim S C(t, S S = 1.

3 MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING 3 2. Checking the Black-Scholes formula satisfies the BSM equation; the Greeks While it is possible to solve the BSM equation (1.8 by relating it to heat equation and using the fundamental solution derived in any PDE textbook, we will approach the problem from the other end. We will verify that the formula we previously derived for the value of the call option satisfies the equation and the boundary conditions. To remind, the formula is (2.1 C(t, S, E = SN(d 1 Ee r(t t N(d 2, where ( ln(s/e + r ± σ2 (T t 2 (2.2 d 1,2 = σ. T t Exercise 2.1. Using the formula for the call and the Put-Call Parity, derive the formula for the put option, (2.3 P (t, S, E = Ee r(t t N( d 2 SN( d 1, where d 1 and d 2 are the same as above Boundary conditions. We start with the boundary conditions as they are simpler. We cannot substitute t = T directly as it will result in division by zero in (2.2, we need to take the limit t T (the minus means we approach time T from below, i.e. t < T. We have ln(s/e + lim d 1,2 = lim t T t T ( r ± σ2 2 σ T t ( ln(s/e = lim t T σ T t + lim t T (T t r ± σ2 2 T t σ ln(s/e = lim t T σ T t. Since the denominator is going to, the limit is infinite. But it is important to understand whether it is or +. This is determined by the sign of ln(s/e, since everything else is positive. { (2.4 lim d if S < E, 1,2 = t T + if S > E. Thus we conclude that lim C(t, S, E = t T = { SN( EN( if S < E, SN( EN( { if S < E, S E if S > E. = max(s E,, if S > E where we used that N( = and N( = 1. Similarly, when S, the dominant term in d 1,2 is ln(s +, giving and C(t, S, E lim S S lim d 1,2 = + S + = lim S SN( Ee r(t t N( S = 1.

4 4 G. BERKOLAIKO and Finally, when S +, the dominant term in d 1,2 is ln(s, giving lim d 1,2 = S + lim C(t, S, E = SN( S + Ee r(t t N( = Calculating derivatives. To substitute C into equation (1.8, we need to compute C C. These derivatives are so important financially that they got special names. t Derivative C is called Delta, denoted by capital Greek letter. We differentiate S (2.5 C(t, S, E = SN(d 1 Ee r(t t N(d 2, with respect to S, applying product rule and chain rule to get C S = N(d 1 + SN (d 1 d 1 S Ee r(t t N (d 2 d 2 S. Explicit computation shows that d 1 S = d 2 S = 1 Sσ T t., 2 C, S S 2 On the other hand, from the definition of N(x we have N (x = 1 2π e x2 /2 and a longer computation shows that ( (2.6 SN (d 1 Ee r(t t N (d 2 = SN (d 2 Therefore we get e d 2 1 d2 2 2 Ee r(t t S =. (2.7 = C S = N(d 1 + ( SN (d 1 Ee r(t t N (d 2 d S = N(d 1. Differentiating with respect to S, we obtain Γ (Gamma, (2.8 Γ = 2 C S = 2 S N(d 1 = N (d 1 d 1 S = N (d 1 Sσ T t. The time-derivative of C is called Theta, denoted by Θ. A calculation involving the use of (2.6 again shows that (2.9 Θ = C t = Sσ 2 T t N (d 1 ree r(t t N(d 2. For reference, the Black Scholes Merton equation (1.8 can be written as (2.1 Θ σ2 S 2 Γ + rs rc =. Substituting (2.5, (2.7, (2.8 and (2.9 into it we see that all terms cancel! Other useful Greeks include Vega C C and Rho, σ r (2.11 Vega = C σ = SN (d 1 T t, (2.12 ρ = C r = (T tee r(t t N(d 2. Vega isn t actually a Greek letter, just sounds nice. Less used higher derivatives are also given names, often rather inventive: there is Vomma (also known as Volga or Weezu, Zomma and Ultima; there is Color and Charm (possibly because many quants joined finance after physics graduate school and have fond memories of Standard Model of particle physics.

5 MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING 5 3. Hedging procedure Suppose you need to price a European-style call option using Black-Scholes formula (2.1-(2.2. Of the parameters entering the problem all but one are known: prevailing interest rate r, spot price of the underlying S, parameters of the option itself strike E and time to expiration T t. The only parameter not readily available is the volatility σ. Since Vega = C >, the price of the call σ (and the put is monotone increasing with volatility. Trading options is often said to be trading volatility. It is important to have a good estimate of the volatility and we will discuss some methods for getting it. Note that it is future volatility that enters the formula, so it is impossible to get it right every time. For now we will assume we have a value of σ. Then pricing an option is a matter of using the formula; any scientific calculator has the function erf(x which can be used to compute N(x. How do we hedge against the price swings of the underlying? We follow the same procedure we developed working with trees, using the value of Delta we computed in (2.7. Suppose we priced and sold the Call Option at time t = and denote by t i the moments of time we will be adjusting our hedge, = t < t 1 < t 2 <... < t n 1 < t n = T. Denote by t i the interval between the moments of adjustment, t i = t i t i 1 (usually, in our examples, we will use constant t. Denote by A i the number of shares of stock we hold immediately after adjustment at time t = t i and D i our corresponding cash holdings, denominated in time t = t i dollars. After the initial sale and hedge, we have (3.1 A = (S, T D = C(S, T A S. Here we indicated dependence on only those parameters that will be changing through the hedging procedure. The values of r, E and σ enter the calculations but are assumed to be constant. After the first period t 1 we want to adjust the hedge. Our cash holdings attracted some interest. We recalculate Delta, this is our new target hedge. We purchase (or sell additional shares as dictated by the new Delta, and the cost of these additional shares is reflected in our cash holdings, (3.2 A 1 = (S t1, T t 1 D 1 = D e r t 1 (A 1 A S t1. This process is repeated until expiration, (3.3 A i = (S ti, T t i D i = D i 1 e r t i (A i A i 1 S ti. At the expiration time t n = T we close our positions and (hopefully have the right amount of money to make the payoff Example of the hedging procedure: real-life data. On Monday , Facebook shares closed at the price S = We would like to price a call option with strike E = 135 and expiration in 9 weeks (on , then sell one 1-share lot of options and simulate hedging procedure using real prices. We will use the value 1 σ = Using interest rate r =.1, T = 9/52 and E = 135 and spot price S = the option value computed with (2.1 is C = 1.39 with = This was the value of the VIX index that day; the VIX index measures the implied volatility of S&P5 index options, it is usually significantly smaller that volatility of individual stocks such as FB.

6 6 G. BERKOLAIKO Time Actions Position t = /52, S = Sell 1 calls at 1.39 =.34 Buy 34 shares 1 calls, 34 shares, cash t = 1/52, S = =.46 Buy 12 shares 1 calls, 46 shares, cash t = 2/52, S = =.43 Sell 3 shares 1 calls, 43 shares, cash t = 3/52, S = =.62 Buy 19 shares 1 calls, 62 shares, cash t = 4/52, S = =.71 Buy 9 shares 1 calls, 71 shares, cash t = 5/52, S = =.86 Buy 15 shares 1 calls, 86 shares, cash t = 6/52, S = =.91 Buy 5 shares 1 calls, 91 shares, cash t = 7/52, S = =.96 Buy 5 shares 1 calls, 96 shares, cash t = 8/52, S = = 1. Buy 4 shares 1 calls, 1 shares, cash t = 9/52, S = Bring forward 1 calls, 1 shares, cash Physical settlement: deliver 1 shares, receive E = 135 per share Final balance: cash Table 1. Hedging process with Black-Scholes formula and actual stock prices for the example is Section 3.1. Because we sold a 1-share lot, we receive 139 dollars and buy 34 shares 2 to hedge, A = 34, D = = First consider what happens if we do not re-adjust the hedge for the duration of the option s life. On , the closing price of FB is S T = We have to make the payoff of ( and close our hedging position by selling 34 shares at the increased price. Our total balance Final Balance = e 9r/52 ( = This isn t great, but still better that not doing any hedging at all (check that the balance in that case would be The following are the weekly 3 closing prices of Facebook (FB from to S = 132.6, 134.5, , , , 139.6, , 14.32, , S 9 = In real stock markets one may not buy or sell a fractional number of shares. In fact, this is the reason why options are sold in 1-share lots: to enable hedging without huge round-off errors. 3 The easiest place to get those prices is Yahoo Finance but, as of time of writing, weekly prices option has problems with data alignment (in particular, it reports prices for Sunday, when the markets are closed. To get the prices below, ask for daily prices and write down prices on Mondays (except for when markets were closed; the price below is for the following Tuesday.

7 MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING 7 11 Hedging a short 95. $13.96 Stock price Delta.5 Portfolio Value At expiration, we have $11.82 to pay out $11.32 Option Replicating Time Figure 1. A sample simulated price path, corresponding Delta and the value of the replicating portfolio compared with the price of the option being replicated. After one week, on , we recalculate =.46 and therefore A 1 = 46, D 1 = e r/ = Here we used t = 1/52. After another week, the new Delta is =.43 and therefore A 2 = 43, D 2 = e r/ = The entire hedging procedure is summarized in Table 1. Note that at expiration we used physical settlement : the writer delivers actual shares (which he has by virtue of hedging and receives from the holder the strike price for every share. Check the cash settlement (writer pays S E per share, sells hedging shares for S on the market results in identical final balance Example of the hedging procedure: simulated data. To test the performance of hedging more thoroughly we can turn to simulated data. In previous parts of this course we learned how to simulate a price path given parameters µ (drift and σ (volatility of the geometric Brownian motion model for the stock prices. Taking, as an example, µ =.1, σ =.25, r =.3, T = 1(year, S = 1 and strike E = 95, we generate a sample path with 5 points (essentially, one point per week and display it in the top panel of Fig. 1. While simulating hedging, we compute at each time point and display it in the middle panel. In the bottom panel we display the value of the hedging portfolio we create (black and the value of the option we are trying to replicate (red. In other words, the red curve is C(t, S t, 95 calculated from Black-Scholes formula, equation (2.1. The black curve is the value of our hedging portfolio (cash and shares combined, (3.4 D i + A i S ti,

8 8 G. BERKOLAIKO 25 Less frequent hedging 25 More frequent hedging Wrong vol used at pricing Payoff Replicating final balance Payoff corrected for misprice Wrong vol used during hedging Figure 2. Results of 1 simulation runs in different conditions. corresponding to the time point t i. See equation (3.3 for the recursive calculation of D i (A i, the number of asset shares we hold, is just the value of Delta. We see that the replicating value is shadowing the underlying option value very closely, resulting in the final balance being close to the necessary payoff. To see how far the replicating value typically lands from the target payoff, we repeat the same experiment 1 times. In Fig. 2 each run is represented by one point (blue. We now plot only the final replicating portfolio value, equation (3.4 with i = n against the final price S n = S T in this particular run. We also plot in red the payoff diagram this is the target value for the replicating portfolio to arrive to at this particular value of S T. In all figures the parameters are µ =.1, σ =.25, r =.3, T = 1(year, S = 1, E = 95, unless stated otherwise. In the top left figure we re-hedge at 5 points distributed uniformly through the year. In the top right we re-hedge at 5 points; note that the results lie closer to the target value! Bottom left is back to 5 re-hedging points but the option was initially mispriced by using a smaller value of volatility, σ =.1. The error in the original price simply brings the whole plot down. Bottom right is using correct σ for valuing the option originally, but is using incorrect σ =.1 for calculating at every step. 4. Estimating volatility The most important part of pricing an option is estimating the (future! volatility of the underlying. The most natural way to estimate is to assume that the past volatility will persist into the future.

9 MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING 9 We defined volatility as (normalized variance of the log-returns of the underlying. The formulas were (4.1 σ = 1 ( St+ t S t var or σ = 1 ( var ln (S t+ t /S t. t S t t When t is small, the two formulas will give similar results. In more detail, given historical (weekly/daily/hourly prices S M,..., S 1, S, we calculate historical log-returns ( S i R i = log, i =,..., M 1. S i 1 The average return is and the estimated variance is R = 1 M M 1 i= R i, var = 1 M 1 (R i R 2. M 1 i= Often the average return is assumed to be negligible compared to volatility (it is of order t compared to U which is of order t leading to simplified formulas for the historical volatility σ hist = 1 1 M (R i t M 2. In practice, the volatility does not stay constant. One of the easiest way to correct for it is to give bigger weight to more recent data, using, for example, the weighted estimate (4.2 σ hist,ω = 1 M i= (R i 2 ω i t M, i= ωi where < ω 1 (the value of ω =.95 is a good start. The weighted estimate is convenient if you want to update σ (to be used in Delta calculations as the new market data comes in, i= σ 2 new = ωσ 2 old + (1 ω R2 new t. We can also calculate σ after the option has expired using (for example, in equation (4.1 the prices that came to pass during the lifetime of the option. This is known as realized volatility. Of course, realized volatility isn t known at the time of pricing of an option. In the example studied in Table 1, the realized volatility was.689. This is smaller that the value of σ =.1137 we used to price the option. In effect, we overpriced the option and this is why we received a significant profit after hedging. Finally, one can look at prices charged by other option traders and ask what is the value of σ that would produce this price in the Black-Scholes formula. This is called the implied volatility, it is often reported alongside option price and can be computed by (numerically solving for σ in equation (2.1. Implied volatility of a specific set of options (options on SPX, the S&P5 Index is published by Chicago Board Options Exchange. As it is a measure of expectations of future price wobbles, VIX is colorfully termed the fear index by the media.

10 1 G. BERKOLAIKO 5. Informal derivation of Black Scholes Merton equation and hedging errors The motivation of this section is two-fold. One is to provide a slightly different derivation of Black Scholes Merton equation. The other is to identify the primary source of hedging errors. We start with the latter, which is motivated by the following observation: consider the example of real-data hedging in Section 3.1 (see Table 1. Had we priced our option according to volatility actually realized by these prices (σ =.689, call price.54 and performed hedging using this volatility, our final balance would have been an unnerving But we are using the volatility actually shown by the prices, where is this negative balance coming from? Let us re-work the hedging process with volatility σ =.689 and provide more details. The price for a single option is.54. With the new value of volatility, the Deltas and cash balances will change and the hedging process is summarized in Table 2. The new column on the right shows the total value of our position, including the value of short options (option prices reported at every time step as C, shares and cash. Note that within each week the total value does not change (the portfolio is self-financing. But from one week to the next the total value jumps, sometimes dramatically. It starts from and ends up at which is also our final balance. The biggest jumps occur when our Delta prescribes the largest hedge adjustments. What happens to option price from one week to the next? Both time t and the underlying S changes. Assuming the option price is smooth in those variables, we can use Taylor expansion C := C(t + t, S + S C(t, S = C C t + t S S C 2 S 2 ( S We included the term ( S 2 because the stock prices typically jump proportionally to t; in our asset price model, S := S t+ t S t = σs ty +... In our stochastic calculations we used Ito rule of thumb to substitute (ds 2 = σ 2 S 2 dt. This is equivalent to substituting Y 2 with its expectation EY 2 = 1 in and obtaining ( S 2 = (σs ty = σ 2 S 2 Y 2 t +... σ 2 S 2 t. (5.1 C C C t + t S S C 2 S 2 σ2 S 2 t. Equation (5.1 is of course the finite-differences version of the Ito differential (1.3. Our cash deposits D and the value of the stock we hold also change from week to week. Together, they form our replicating portfolio Π whose value increment is Π := Π(t + t, S + S Π(t, S = De r t + S t+ t (D + S t = D(e r t 1 + S. Here is the amount of stock we hold (per option which we can set to be equal to C S. Assuming our replicating portfolio has been doing its job (i.e. replicating: Π(t, S = C(t, S, the amount of cash we hold can be calculated as D = Π(t, S S = C C S S. Expanding e r t = 1 + r t +, we get Π r ( C C S S t + C S S.

11 MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING 11 Time Actions Position Total Value t = /52 Sell 1 calls S = Buy 24 shares C =.54, =.24 1 calls, 24 shares, cash. t = 1/52 1 calls, 24 shares, cash 9.73 S = Buy 18 shares C = 1.11, =.42 1 calls, 42 shares, cash 9.73 t = 2/52 1 calls, 42 shares, cash 1.36 S = Sell 4 shares C =.87, =.38 1 calls, 38 shares, cash 1.36 t = 3/52 1 calls, 38 shares, cash 32.6 S = Buy 31 shares C = 2.2, =.69 1 calls, 69 shares, cash 32.6 t = 4/52 1 calls, 69 shares, cash 3.14 S = Buy 12 shares C = 2.85, =.81 1 calls, 81 shares, cash 3.14 t = 5/52 1 calls, 81 shares, cash S = Buy 15 shares C = 4.74, =.96 1 calls, 96 shares, cash t = 6/52 1 calls, 96 shares, cash S = Buy 3 shares C = 5.3, =.99 1 calls, 99 shares, cash t = 7/52 1 calls, 99 shares, cash S = Buy 1 share C = 5.37, = 1. 1 calls, 1 shares, cash t = 8/52 1 calls, 1 shares, cash S = Do nothing C = 7.31, = 1. 1 calls, 1 shares, cash t = 9/52 Bring forward 1 calls, 1 shares, cash S = Settle C = 6.4 Final balance: Table 2. Hedging steps with realized volatility We set S = C which is necessary if the replicating portfolio is to continue replicating, cancel the S terms and obtain C (5.2 t t + 1 ( 2 C 2 S 2 σ2 S 2 t = r C C S S t. Dividing by t we recover the Black Scholes Merton equation. This derivation is largely parallel to the stochastic calculus derivation, but we can now assign clearer meaning to equation s terms. Equation (5.2 can be interpreted as follows: change in option value due to passage of time plus the volatility-induced proceeds from hedging must balance the interest expenses one incurs while hedging. Even more importantly, the finite-differences derivation we just performed can be tested against real-life data. The derivation can be informally summarized as (5.3 C C T aylor C Ito Π T aylor Π,

12 C and its approximations, in $ 12 G. BERKOLAIKO C C Taylor C Ito Taylor Week Figure 3. Weekly increments in call price and their approximations for weekly data from Table 2. where we start with C on one side and Π on the other and get down to C Ito Π T aylor which is exactly equation (5.2. For reference, the individual terms are C = C(t + t, S + S C(t, S, C T aylor = C C t + t C Ito = C C t + t Π T aylor = S S C 2 S 2 ( S2, 2 C S 2 σ2 S 2 t, S S ( C C S S Π = D(e r t 1 + S. t + C S S, All of these terms we can compute at each step of the hedging in Table 2 and compare to each other. To compute derivatives C, 2 C and C we use the expressions for, Γ and Θ we computed S S 2 t in equations (2.7, (2.8 and (2.9. The result of the calculation for each week is shown in Figure 3. We note that of all the steps in the chain of approximations (5.3 the worst performer is the approximation C T aylor C Ito. This is because at this step we swapped a random variable Y 2 for its mean EY 2 = 1. This is only correct in the integral sense or once averaged over several small time steps (and allowing the Law of Large Numbers to kick in. This is the point where frequent hedging adjustment plays an important role. The error generated by this approximation is easy to understand, C T aylor C Ito = 1 2 C ( ( S 2 σ 2 S 2 t. 2 S 2

13 Total portfolio value Stock price MATH 425 PART VI: BLACK SCHOLES MERTON EQUATION AND HEDGING Weeks Figure 4. Top: Price increments of a stock (solid line compared to average volatility prediction (dotted parabolas of equation (5.4. The background color shows the value of Γ at that point of time and stock price. Note the close to expiration time, Γ diverges around the strike price. Bottom: evolution of the total portfolio values. There is no error if (5.4 S = ±σs t. If S is small, the writer of the option is making money off the hedge; if S is large, the writer is losing money (similar to what we saw in 1-level binary tree. Importantly, the errors ( S 2 σ 2 S 2 are modulated by Γ = 2 C. This information is visually collected in Figure 4. S 2 We note that the largest drops in the total portfolio values correspond to stock prices going far outside the parabola bounds of equation (5.4 while the Γ background is large (yellow. This is what happens in week 1 and 3. In contrast, week 8 price increment exceeds the parabola bounds as much as week 1 increment, but does not result in any significant drop in total portfolio value because Γ is close to zero (blue. We also note that hedging result of week 2 almost entirely compensates for the loss of week 1. On the other hand we were very unlucky that the low volatility weeks 6 and 7 happened when Γ was too low to make any difference. Thus, hedging needs to be frequent enough to correct errors while Γ remains high: make hedge while the Gamma shines.