summary black{scholes{merton equationberko/teaching/common/math425/... · 2019. 12. 4. · math 425...

MATH 425 PART VI: BLACK–SCHOLES–MERTON EQUATION ANDHEDGING

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 ft = f

(t, g(t,Wt)

)with respect to time is

(1.1) dft =∂f

∂tdt+

1

2

∂2f

∂g2(dgt)

2 +∂f

∂gdgt.

Let V (t, St) be the value of a financial derivative which depends on time and the price of theunderlying St. Assume the price of the underlying follows geometric Brownian motion,

(1.2) dSt = µStdt+ σStdWt.

To use Ito chain rule (1.1) with f = V and g = S, we need (dSt)2. Squaring equation (1.2) and

discarding terms smaller than dt (note that (dt)2 � dtdWt � (dWt)2 = dt), we get (dSt)

2 =(σStdWt)

2 = σ2S2t dt. Ito chain rule then gives

(1.3) dVt =

(∂V

∂t+

1

2σ2S2

t

∂2V

∂S2

)dt+

∂V

∂SdSt.

1.2. Self-financing portfolio. Consider of a portfolio Π = {a stock, b cash}. Here b is theamount of cash denominated in time t = 0 dollars. We will denote the value, at time t, of asingle time-0 dollar by βt. Of course βt = ert and the value of the cash portion of the portfolio isbβt = bert.

A portfolio is called self-financing if the change of its value at any time is entirely due to thechange in value of the underlying assets. In other words, there is no inflow or outflow of capitalfrom the portfolio, but the exchange of cash and stock is allowed. As an example, consider anIndividual Retirement Account to which you contributed your maximal annual amount. Youcannot take money out (until retirement), you cannot contribute more funds (until next year), butyou can sell your stock holdings and the cash from the sale will still remain inside the account; itwill 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 exchangemay be taken depending on the stock price and passage of time, thus a = a(t, St) and b = b(t, St).We allow negative values of a (shorting the stock) and b (borrowing cash). Of course, this is notallowed in one’s retirement account.

1

2 G. BERKOLAIKO

The value of the portfolio at time t is

(1.4) Πt = a(t, St)St + b(t, St)βt.

Mathematically, the self-financing condition is written as

(1.5) dΠt = a(t, St)dSt + b(t, St)dβt.

1.3. Self-financing replicating portfolio. We want to design a self-financing portfolio whichwill replicate the value of a given financial derivative. This means that Πt = Vt and, in particularthe change in value at every t is identical, dΠt = dVt. Equating the differentials, we get

(1.6) a(t, St)dSt + b(t, St)dβt =

(∂V

∂t+

1

2σ2S2

t

∂2V

∂S2

)dt+

∂V

∂SdSt.

We would like to remove the risk associated with stock price moves, dSt. If we chose a(t, St) = ∂V∂S

,all terms containing dSt will cancel. Note that we can reasonably do that since we chose howmuch stock to put into our replicating portfolio (this amount of stock we called “delta”). Thecorresponding amount of cash is determined from (1.4),

b(t, St)dβt = b(t, St)rβtdt = r(Πt − a(t, St)St)dt = r

(Vt − St

∂V

∂S

)dt.

Equation (1.6) now reads

r

(Vt − St

∂V

∂S

)dt =

(∂V

∂t+

1

2σ2S2

t

∂2V

∂S2

)dt,

or, moving everything to one side and canceling dt,

(1.7)∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0.

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 explicitinformation about the nature of the financial derivative, we expect it to be satisfied by any financialderivative that may be replicated by a self-financing portfolio. To find the particular solution ofthe equation that gives the value of, for example, call option, we need to specify the appropriateboundary conditions.

1.4. BSM equation and boundary conditions for a Call Option. In the special case whenV (t, S) is the value C(t, S) of a call option, the BSM equation reads (simply substituting C for V )

(1.8)∂C

∂t+

1

2σ2S2∂

2C

∂S2+ rS

∂C

∂S− rC = 0.

One condition is readily available: it is the final payoff condition

(1.9) C(T, ST ) = max(ST − E, 0).

We also remember the estimates we derived earlier,

St − Ee−r(T−t) ≤ Ct ≤ St and 0 ≤ Ct

Squeeze lemma now implies that as S → 0, the value of the call also goes to 0. On the other side,if S →∞, the value of the call grows like S. To put it mathematically,

(1.10) C(t, 0) = 0 and limS→∞

C(t, S)

S= 1.

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 usingthe fundamental solution derived in any PDE textbook, we will approach the problem from theother end. We will verify that the formula we previously derived for the value of the call optionsatisfies the equation and the boundary conditions.

To remind, the formula is

(2.1) C(t, S, E) = SN(d1)− Ee−r(T−t)N(d2),

where

(2.2) d1,2 =ln(S/E) +

(r ± σ2

2

)(T − t)

σ√T − t

.

Exercise 2.1. Using the formula for the call and the Put-Call Parity, derive the formula for theput option,

(2.3) P (t, S, E) = Ee−r(T−t)N(−d2)− SN(−d1),

where d1 and d2 are the same as above.

2.1. Boundary conditions. We start with the boundary conditions as they are simpler. Wecannot substitute t = T directly as it will result in division by zero in (2.2), we need to take thelimit t→ T− (the minus means we approach time T from below, i.e. t < T ). We have

limt→T−

d1,2 = limt→T−

ln(S/E) +(r ± σ2

2

)(T − t)

σ√T − t

= limt→T−

ln(S/E)

σ√T − t

+ limt→T−

(r ± σ2

2

)√T − t

σ= lim

t→T−

ln(S/E)

σ√T − t

.

Since the denominator is going to 0, the limit is infinite. But it is important to understand whetherit is −∞ or +∞. This is determined by the sign of ln(S/E), since everything else is positive.

(2.4) limt→T−

d1,2 =

{−∞ if S < E,

+∞ if S > E.

Thus we conclude that

limt→T−

C(t, S, E) =

{SN(−∞)− EN(−∞) if S < E,

SN(∞)− EN(∞) if S > E

=

{0 if S < E,

S − E if S > E.

= max(S − E, 0),

where we used that N(−∞) = 0 and N(∞) = 1.Similarly, when S →∞, the dominant term in d1,2 is ln(S)→ +∞, giving

limS→+∞

d1,2 = +∞

and

limS→∞

C(t, S, E)

S= lim

S→∞

SN(∞)− Ee−r(T−t)N(∞)

S= 1.

4 G. BERKOLAIKO

Finally, when S → 0+, the dominant term in d1,2 is ln(S)→ −∞, giving

limS→0+

d1,2 = −∞

andlimS→0+

C(t, S, E) = SN(−∞)− Ee−r(T−t)N(−∞) = 0.

2.2. Calculating derivatives. To substitute C into equation (1.8), we need to compute ∂C∂S

, ∂2C∂S2 ,

∂C∂t

. These derivatives are so important financially that they got special names.

Derivative ∂C∂S

is called Delta, denoted by capital Greek letter ∆. We differentiate

(2.5) C(t, S, E) = SN(d1)− Ee−r(T−t)N(d2),

with respect to S, applying product rule and chain rule to get

∂C

∂S= N(d1) + SN ′(d1)

∂d1

∂S− Ee−r(T−t)N ′(d2)

∂d2

∂S.

Explicit computation shows that

∂d1

∂S=∂d2

∂S=

1

Sσ√T − t

.

On the other hand, from the definition of N(x) we have N ′(x) = 1√2πe−x

2/2 and a longer compu-

tation shows that

(2.6) SN ′(d1)− Ee−r(T−t)N ′(d2) = SN ′(d2)

(e−

d21−d2

22 − Ee−r(T−t)

S

)= 0.

Therefore we get

(2.7) ∆ =∂C

∂S= N(d1) +

(SN ′(d1)− Ee−r(T−t)N ′(d2)

) ∂d∂S

= N(d1).

Differentiating ∆ with respect to S, we obtain Γ (Gamma),

(2.8) Γ =∂2C

∂S2=

∂

∂SN(d1) = N ′(d1)

∂d1

∂S=

N ′(d1)

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 ′(d1)− rEe−r(T−t)N(d2).

For reference, the Black–Scholes–Merton equation (1.8) can be written as

(2.10) Θ +1

2σ2S2Γ + rS∆− rC = 0.

Substituting (2.5), (2.7), (2.8) and (2.9) into it we see that all terms cancel!Other useful Greeks include Vega ∂C

∂σand Rho ∂C

∂r,

Vega =∂C

∂σ= SN ′(d1)

√T − t,(2.11)

ρ =∂C

∂r= (T − t)Ee−r(T−t)N(d2).(2.12)

Vega isn’t actually a Greek letter, just sounds nice. Less used higher derivatives are also givennames, often rather inventive: there is Vomma (also known as Volga or Weezu), Zomma andUltima; there is Color and Charm (possibly because many quants joined finance after physicsgraduate school and have fond memories of Standard Model of particle physics).


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 priceof the underlying S, parameters of the option itself strike E and time to expiration T − t. Theonly parameter not readily available is the volatility σ. Since Vega = ∂C

∂σ> 0, the price of the call

(and the put) is monotone increasing with volatility. Trading options is often said to be “tradingvolatility”. It is important to have a good estimate of the volatility and we will discuss somemethods for getting it. Note that it is future volatility that enters the formula, so it is impossibleto get it right every time.

For now we will assume we have a value of σ. Then pricing an option is a matter of usingthe 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 wedeveloped working with trees, using the value of Delta we computed in (2.7).

Suppose we priced and sold the Call Option at time t = 0 and denote by ti the moments of timewe will be adjusting our hedge,

0 = t0 < t1 < t2 < . . . < tn−1 < tn = T.

Denote by ∆ti the interval between the moments of adjustment, ∆ti = ti − ti−1 (usually, in ourexamples, we will use constant ∆t). Denote by Ai the number of shares of stock we hold immediatelyafter adjustment at time t = ti and Di our corresponding cash holdings, denominated in time t = tidollars.

After the initial sale and hedge, we have

(3.1) A0 = ∆(S0, T ) D0 = C(S0, T )− A0 × S0.

Here we indicated dependence on only those parameters that will be changing through the hedgingprocedure. The values of r, E and σ enter the calculations but are assumed to be constant.

After the first period ∆t1 we want to adjust the hedge. Our cash holdings attracted someinterest. We recalculate Delta, this is our new target hedge. We purchase (or sell) additionalshares as dictated by the new Delta, and the cost of these additional shares is reflected in our cashholdings,

(3.2) A1 = ∆(St1 , T − t1) D1 = D0er∆t1 − (A1 − A0)× St1 .

This process is repeated until expiration,

(3.3) Ai = ∆(Sti , T − ti) Di = Di−1er∆ti − (Ai − Ai−1)× Sti .

At the expiration time tn = T we close our positions and (hopefully) have the right amount ofmoney to make the payoff.

3.1. Example of the hedging procedure: real-life data. On Monday 2017-02-06, Facebookshares closed at the price S0 = 132.06. We would like to price a call option with strike E = 135 andexpiration in 9 weeks (on 2017-04-10), then sell one 100-share lot of options and simulate hedgingprocedure using real prices.

We will use the value1 σ = 0.1137. Using interest rate r = 0.01, T = 9/52 and E = 135 andspot price S0 = 132.06 the option value computed with (2.1) is

C = 1.39 with ∆ = 0.3426

1This was the value of the VIX index that day; the VIX index measures the implied volatility of S&P500 indexoptions, it is usually significantly smaller that volatility of individual stocks such as FB.

6 G. BERKOLAIKO

Time Actions Positiont = 0/52, S = 132.06 Sell 100 calls at 1.39∆ = 0.34 Buy 34 shares −100 calls, 34 shares, −4350.83 cash

t = 1/52, S = 134.05∆ = 0.46 Buy 12 shares −100 calls, 46 shares, −5960.27 cash

t = 2/52, S = 133.72∆ = 0.43 Sell 3 shares −100 calls, 43 shares, −5560.25 cash







t = 9/52, S = 141.04 Bring forward −100 calls, 100 shares, −13467.31 cashPhysical settlement:deliver 100 shares,

receive E = 135 per share Final balance: 32.69 cash

Table 1. Hedging process with Black-Scholes formula and actual stock prices forthe example is Section 3.1.

Because we sold a 100-share lot, we receive 139 dollars and buy 34 shares2 to hedge,

A0 = 34, D0 = 139− 34× 132.06 = −4350.83.

First consider what happens if we do not re-adjust the hedge for the duration of the option’slife. On 2017-04-10, the closing price of FB is ST = 141.04. We have to make the payoff of(141.04 − 135) × 100 and close our hedging position by selling 34 shares at the increased price.Our total balance

Final Balance = 141.04× 34− 4350.83× e9r/52 − (141.04− 135)× 100 = −239.76.

This isn’t great, but still better that not doing any hedging at all (check that the balance in thatcase would be −602.61).

The following are the weekly3 closing prices of Facebook (FB) from 2017-02-06 to 2017-04-10.

S0 = 132.06, 134.05, 133.72, 136.41, 137.42, 139.60, 139.94, 140.32, 142.28, S9 = 141.04.

2In real stock markets one may not buy or sell a fractional number of shares. In fact, this is the reason whyoptions are sold in 100-share lots: to enable hedging without huge round-off errors.

3The easiest place to get those prices is Yahoo Finance but, as of time of writing, “weekly prices” option hasproblems with data alignment (in particular, it reports prices for Sunday, when the markets are closed). To getthe prices below, ask for daily prices and write down prices on Mondays (except for 2017-02-20 when markets wereclosed; the price below is for the following Tuesday).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 190

100

110

Sto

ck p

rice

Hedging a short 95.00 call @ $13.96

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Delta

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time

5

10

15

Port

folio

Valu

e

At expiration, we have $11.82 to pay out $11.32

Option

Replicating

Figure 1. A sample simulated price path, corresponding Delta and the value ofthe replicating portfolio compared with the price of the option being replicated.

After one week, on 2017-02-13, we recalculate ∆ = 0.46 and therefore

A1 = 46, D1 = −4350.83× er/52 − 12× 134.05 = −5960.27.

Here we used ∆t = 1/52. After another week, the new Delta is ∆ = 0.43 and therefore

A2 = 43, D2 = −4350.83× er/52 + 3× 133.72 = −5560.25.

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) andreceives from the holder the strike price for every share. Check the “cash settlement” (writer paysS − E per share, sells hedging shares for S on the market) results in identical final balance.

3.2. Example of the hedging procedure: simulated data. To test the performance of hedg-ing more thoroughly we can turn to simulated data. In previous parts of this course we learned howto simulate a price path given parameters µ (drift) and σ (volatility) of the geometric Brownianmotion model for the stock prices.

Taking, as an example, µ = 0.1, σ = 0.25, r = 0.03, T = 1(year), S0 = 100 and strike E = 95,we generate a sample path with 50 points (essentially, one point per week) and display it in thetop panel of Fig. 1. While simulating hedging, we compute ∆ at each time point and display it inthe middle panel.

In the bottom panel we display the value of the hedging portfolio we create (black) and thevalue of the option we are trying to replicate (red). In other words, the red curve is C(t, St, 95)calculated from Black-Scholes formula, equation (2.1). The black curve is the value of our hedgingportfolio (cash and shares combined),

(3.4) Di + AiSti ,

8 G. BERKOLAIKO

-15

-10

-5

0

5

10

15

20

25Less frequent hedging

-15

-10

-5

0

5

10

15

20

25More frequent hedging

80 90 100 110 120 130-15

-10

-5

0

5

10

15

20

25Wrong vol used at pricing

Payoff

Replicating final balance

Payoff corrected for misprice

80 90 100 110 120 130-15

-10

-5

0

5

10

15

20

25Wrong vol used during hedging

Figure 2. Results of 1000 simulation runs in different conditions.

corresponding to the time point ti. See equation (3.3) for the recursive calculation of Di (Ai, thenumber 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, resultingin 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 sameexperiment 1000 times. In Fig. 2 each run is represented by one point (blue). We now plot onlythe final replicating portfolio value, equation (3.4) with i = n against the final price Sn = STin this particular run. We also plot in red the payoff diagram — this is the target value for thereplicating portfolio to arrive to at this particular value of ST .

In all figures the parameters are µ = 0.1, σ = 0.25, r = 0.03, T = 1(year), S0 = 100, E = 95,unless stated otherwise. In the top left figure we re-hedge at 50 points distributed uniformlythrough the year. In the top right we re-hedge at 500 points; note that the results lie closer to thetarget value! Bottom left is back to 50 re-hedging points but the option was initially mispriced byusing a smaller value of volatility, σ = 0.1. The error in the original price simply brings the wholeplot down. Bottom right is using correct σ for valuing the option originally, but is using incorrectσ = 0.1 for calculating ∆ at every step.

4. Estimating volatility

The most important part of pricing an option is estimating the (future!) volatility of the under-lying. The most natural way to estimate is to assume that the past volatility will persist into thefuture.


We defined volatility as (normalized) variance of the log-returns of the underlying. The formulaswere

(4.1) σ =1√∆t

√var

(St+∆t − St

St

)or σ =

1√∆t

√var(

ln (St+∆t/St)).

When ∆t is small, the two formulas will give similar results.In more detail, given historical (weekly/daily/hourly) prices S−M , . . . , S−1, S0, we calculate his-

torical log-returns

R−i = log

(S−iS−i−1

), i = 0, . . . ,M − 1.

The average return is

〈R〉 =1

M

M−1∑i=0

R−i,

and the estimated variance is

v̂ar =1

M − 1

M−1∑i=0

(R−i − 〈R〉)2 .

Often the average return is assumed to be negligible compared to volatility (it is of order ∆tcompared to U which is of order

√∆t) leading to simplified formulas for the historical volatility

σhist =1√∆t

√√√√ 1

M

M∑i=0

(R−i)2.

In practice, the volatility does not stay constant. One of the easiest way to correct for it is togive bigger weight to more recent data, using, for example, the weighted estimate

(4.2) σhist,ω =1√∆t

√∑Mi=0(R−i)2ωi∑M

i=0 ωi

,

where 0 < ω ≤ 1 (the value of ω = 0.95 is a good start). The weighted estimate is convenient ifyou want to update σ (to be used in Delta calculations) as the new market data comes in,

σ2new = ωσ2

old + (1− ω)R2

new

∆t.

We can also calculate σ after the option has expired using (for example, in equation (4.1)) theprices that came to pass during the lifetime of the option. This is known as realized volatility. Ofcourse, realized volatility isn’t known at the time of pricing of an option. In the example studiedin Table 1, the realized volatility was 0.0689. This is smaller that the value of σ = 0.1137 we usedto price the option. In effect, we overpriced the option and this is why we received a significantprofit after hedging.

Finally, one can look at prices charged by other option traders and ask what is the value of σthat 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 inequation (2.1). Implied volatility of a specific set of options (options on SPX, the S&P500 Index)is published by Chicago Board Options Exchange. As it is a measure of expectations of futureprice wobbles, VIX is colorfully termed the “fear index” by the media.

10 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 ofBlack–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 exampleof real-data hedging in Section 3.1 (see Table 1). Had we priced our option according to volatilityactually realized by these prices (σ = 0.0689, call price 0.54) and performed hedging using thisvolatility, our final balance would have been an unnerving −42.22. But we are using the volatilityactually shown by the prices, where is this negative balance coming from?

Let us re-work the hedging process with volatility σ = 0.0689 and provide more details. Theprice for a single option is 0.54. With the new value of volatility, the Deltas and cash balances willchange and the hedging process is summarized in Table 2. The new column on the right shows thetotal value of our position, including the value of short options (option prices reported at everytime 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 0and ends up at −42.22 which is also our final balance. The biggest jumps occur when our Deltaprescribes the largest hedge adjustments.

What happens to option price from one week to the next? Both time t and the underlying Schanges. Assuming the option price is smooth in those variables, we can use Taylor expansion

∆C := C(t+ ∆t, S + ∆S)− C(t, S) =∂C

∂t∆t+

∂C

∂S∆S +

1

2

∂2C

∂S2(∆S)2 + . . .

We included the term (∆S)2 because the stock prices typically jump proportionally to√

∆t; in ourasset price model,

∆S := St+∆t − St = σS√

∆tY + . . .

In our stochastic calculations we used “Ito rule of thumb” to substitute (dS)2 = σ2S2dt. This isequivalent to substituting Y 2 with its expectation EY 2 = 1 in

(∆S)2 = (σS√

∆tY + . . .)2 = σ2S2Y 2∆t+ . . . ≈ σ2S2

∆t.

and obtaining

(5.1) ∆C ≈ ∂C

∂t∆t+

∂C

∂S∆S +

1

2

∂2C

∂S2σ2S2

∆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) = Der∆t + ∆ · St+∆t − (D + ∆ · St)= D(er∆t − 1) + ∆ · ∆S.

Here ∆ is the amount of stock we hold (per option) which we can set to be equal to ∂C∂S

. Assumingour replicating portfolio has been doing its job (i.e. replicating: Π(t, S) = C(t, S)), the amount ofcash we hold can be calculated as

D = Π(t, S)−∆ · S = C − ∂C

∂SS.

Expanding er∆t = 1 + r∆t+ · · · , we get

∆Π ≈ r

(C − ∂C

∂SS

)∆t+

∂C

∂S∆S.


Time Actions Position Total Valuet = 0/52 Sell 100 callsS = 132.06 Buy 24 sharesC = 0.54, ∆ = 0.24 −100 calls, 24 shares, −3115.55 cash 0.00

t = 1/52 −100 calls, 24 shares, −3116.15 cash −9.73S = 134.05 Buy 18 sharesC = 1.11, ∆ = 0.42 −100 calls, 42 shares, −5529.05 cash −9.73

t = 2/52 −100 calls, 42 shares, −5530.11 cash −1.36S = 133.72 Sell 4 sharesC = 0.87, ∆ = 0.38 −100 calls, 38 shares, −4995.23 cash −1.36





t = 7/52 −100 calls, 99 shares, −13396.70 cash −42.31S = 140.32 Buy 1 shareC = 5.37, ∆ = 1.00 −100 calls, 100 shares, −13537.02 cash −42.31

t = 8/52 −100 calls, 100 shares, −13539.62 cash −42.22S = 142.28 Do nothingC = 7.31, ∆ = 1.00 −100 calls, 100 shares, −13539.62 cash −42.22

t = 9/52 Bring forward −100 calls, 100 shares, −13544.83 cash −42.22S = 141.04 SettleC = 6.04 Final balance: −42.22

Table 2. Hedging steps with realized volatility

We set ∆S = ∆C which is necessary if the replicating portfolio is to continue replicating, cancelthe ∆S terms and obtain

(5.2)∂C

∂t∆t+

1

2

∂2C

∂S2σ2S2

∆t = r

(C − ∂C

∂SS

)∆t.

Dividing by ∆t we recover the Black–Scholes–Merton equation. This derivation is largely parallelto 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 plusthe volatility-induced proceeds from hedging must balance the interest expenses one incurs whilehedging.

Even more importantly, the finite-differences derivation we just performed can be tested againstreal-life data. The derivation can be informally summarized as

(5.3) ∆C ≈ ∆CTaylor ≈ ∆CIto ≈ ∆ΠTaylor ≈ ∆Π,

12 G. BERKOLAIKO

1 2 3 4 5 6 7 8 9

Week

-1.5

-1

-0.5

0

0.5

1

1.5

2

C a

nd it

s ap

prox

imat

ions

, in

$

CC

Taylor

CIto

Taylor

Figure 3. Weekly increments in call price and their approximations for weekly datafrom Table 2.

where we start with ∆C on one side and ∆Π on the other and get down to ∆CIto ≈ ∆ΠTaylor whichis exactly equation (5.2). For reference, the individual terms are

∆C = C(t+ ∆t, S + ∆S)− C(t, S),

∆CTaylor =∂C

∂t∆t+

∂C

∂S∆S +

1

2

∂2C

∂S2(∆S)2,

∆CIto =∂C

∂t∆t+

∂C

∂S∆S +

1

2

∂2C

∂S2σ2S2

∆t,

∆ΠTaylor =

(C − ∂C

∂SS

)∆t+

∂C

∂S∆S,

∆Π = D(er∆t − 1) + ∆ · ∆S.

All of these terms we can compute at each step of the hedging in Table 2 and compare to eachother. To compute derivatives ∂C

∂S, ∂2C∂S2 and ∂C

∂twe use the expressions for ∆, Γ and Θ we computed

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 ∆CTaylor ≈ ∆CIto. 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 severalsmall time steps (and allowing the Law of Large Numbers to kick in). This is the point wherefrequent hedging adjustment plays an important role.

The error generated by this approximation is easy to understand,

∆CTaylor − ∆CIto =1

2

∂2C

∂S2

((∆S)2 − σ2S2

∆t).


130

135

140

145

Sto

ck p

rice

0 1 2 3 4 5 6 7 8 9Weeks

-50

-40

-30

-20

-10

0

Tot

al p

ortfo

lio v

alue

Figure 4. Top: Price increments of a stock (solid line) compared to ”averagevolatility” prediction (dotted parabolas) of equation (5.4). The background colorshows the value of Γ at that point of time and stock price. Note the close to ex-piration time, Γ diverges around the strike price. Bottom: evolution of the totalportfolio 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 islosing money (similar to what we saw in 1-level binary tree). Importantly, the errors (∆S)2−σ2S2

are modulated by Γ = ∂2C∂S2 . This information is visually collected in Figure 4.

We note that the largest drops in the total portfolio values correspond to stock prices going faroutside the parabola bounds of equation (5.4) while the Γ background is large (yellow). This iswhat happens in week 1 and 3. In contrast, week 8 price increment exceeds the parabola boundsas much as week 1 increment, but does not result in any significant drop in total portfolio valuebecause 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 Γwas too low to make any difference. Thus, hedging needs to be frequent enough to correct errorswhile Γ remains high: “make hedge while the Gamma shines”.

summary black{scholes{merton equationberko/teaching/common/math425/... · 2019. 12. 4. · math 425...

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