c 2016 by Martin HaughIEOR E4706: Foundations of Financial EngineeringThe Black-Scholes ModelIn these notes we will use Itô’s Lemma and a replicating argument to derive the famous Black-Scholes formulafor European options. We will also discuss the weaknesses of the Black-Scholes model and geometric Brownianmotion, and this leads us directly to the concept of the volatility surface which we will discuss in some detail.We will also derive and study the Black-Scholes Greeks and discuss how they are used in practice to hedgeoption portfolios.1The Black-Scholes ModelWe are now able to derive the Black-Scholes PDE for a call-option on a non-dividend paying stock with strike Kand maturity T . We assume that the stock price follows a geometric Brownian motion so thatdSt µSt dt σSt dWt(1)where Wt is a standard Brownian motion. We also assume that interest rates are constant so that 1 unit ofcurrency invested in the cash account at time 0 will be worth Bt : exp(rt) at time t. We will denote byC(S, t) the value of the call option at time t. By Itô’s lemma we know that C1 2 2 2C C C σ SdWt(2)dC(S, t) µStdt σSt S t2 S 2 SLet us now consider a self-financing trading strategy where at each time t we hold xt units of the cash accountand yt units of the stock. Then Pt , the time t value of this strategy satisfiesPt xt Bt yt St .(3)We will choose xt and yt in such a way that the strategy replicates the value of the option. The self-financingassumption implies thatdPt xt dBt yt dSt(4) rxt Bt dt yt (µSt dt σSt dWt ) (rxt Bt yt µSt ) dt yt σSt dWt .(5)Note that (4) is consistent with our earlier definition of self-financing. In particular, any gains or losses on theportfolio are due entirely to gains or losses in the underlying securities, i.e. the cash-account and stock, and notdue to changes in the holdings xt and yt .Returning to our derivation, we can equate terms in (2) with the corresponding terms in (5) to obtainyt rxt Bt C S C1 2C σ2 S 2 2 . t2 S(6)(7)If we set C0 P0 , the initial value of our self-financing strategy, then it must be the case that Ct Pt for all tsince C and P have the same dynamics. This is true by construction after we equated terms in (2) with thecorresponding terms in (5). Substituting (6) and (7) into (3) we obtainrSt C C1 2C σ 2 S 2 2 rC 0, S t2 S(8)

The Black-Scholes Model2the Black-Scholes PDE. In order to solve (8) boundary conditions must also be provided. In the case of ourcall option those conditions are: C(S, T ) max(S K, 0), C(0, t) 0 for all t and C(S, t) S as S .The solution to (8) in the case of a call option isC(S, t)where d1and d2 St Φ(d1 ) e r(T t) KΦ(d2 )StK(9) (r σ 2 /2)(T t) σ T t d1 σ T t logand Φ(·) is the CDF of the standard normal distribution. One way to confirm (9) is to compute the variouspartial derivatives using (9), then substitute them into (8) and check that (8) holds. The price of a Europeanput-option can also now be easily computed from put-call parity and (9).The most interesting feature of the Black-Scholes PDE (8) is that µ does not appear1 anywhere. Note that theBlack-Scholes PDE would also hold if we had assumed that µ r. However, if µ r then investors would notdemand a premium for holding the stock. Since this would generally only hold if investors were risk-neutral, thismethod of derivatives pricing came to be known as risk-neutral pricing.1.1Martingale PricingIt can be shown2 that the Black-Scholes PDE in (8) is consistent with martingale pricing. In particular, if wedeflate by the cash account then the deflated stock price process, Yt : St /Bt , must be a Q-martingale whereQ is the EMM corresponding to taking the cash account as numeraire. It can be shown that the Q-dynamics ofSt satisfy3dSt rSt dt σSt dWtQ(10)where WtQ is a Q-Brownian motion. Note that (10) impliesST St e(r σ2/2)(T t) σ(WTQ WtQ )so that ST is log-normally distributed under Q. It is now easily confirmed that the call option price in (9) alsosatisfieshi r(T t)C(St , t) EQmax(ST K, 0)(11)t ewhich is of course consistent with martingale pricing.1.2DividendsIf we assume that the stock pays a continuous dividend yield of q, i.e. the dividend paid over the interval(t, t dt] equals qSt dt, then the dynamics of the stock price can be shown to satisfydSt (r q)St dt σSt dWtQ .(12)In this case the total gain process, i.e. the capital gain or loss from holding the security plus accumulateddividends, is a Q-martingale. The call option price is still given by (11) but now withST St e(r q σ2/2)(T t) σ (WTQ WtQ ).1 The discrete-time counterpart to this observation was when we observed that the true probabilities of up-moves and downmoves did not have an impact on option prices.2 We would need to use stochastic calculus tools that we have not discussed in these notes to show exactly why the BlackScholes call option price is consistent with martingale pricing. It can also be shown that the Black-Scholes model is completeso that there is a unique EMM corresponding to any numeraire.3 You can check using Itô’s Lemma that if S satisfies (10) then Y will indeed be a

The Black-Scholes Model3In this case the call option price is given byC(S, t)where d1and d2 e q(T t) St Φ(d1 ) e r(T t) KΦ(d2 )StK(13) (r q σ 2 /2)(T t) σ T t d1 σ T t. logExercise 1 Follow the replicating argument given above to derive the Black-Scholes PDE when the stock paysa continuous dividend yield of q.2The Volatility SurfaceThe Black-Scholes model is an elegant model but it does not perform very well in practice. For example, it iswell known that stock prices jump on occasions and do not always move in the continuous manner predicted bythe GBM motion model. Stock prices also tend to have fatter tails than those predicted by GBM. Finally, if theBlack-Scholes model were correct then we should have a flat implied volatility surface. The volatility surface is afunction of strike, K, and time-to-maturity, T , and is defined implicitlyC(S, K, T ) : BS (S, T, r, q, K, σ(K, T ))(14)where C(S, K, T ) denotes the current market price of a call option with time-to-maturity T and strike K, andBS(·) is the Black-Scholes formula for pricing a call option. In other words, σ(K, T ) is the volatility that, whensubstituted into the Black-Scholes formula, gives the market price, C(S, K, T ). Because the Black-Scholesformula is continuous and increasing in σ, there will always4 be a unique solution, σ(K, T ). If the Black-Scholesmodel were correct then the volatility surface would be flat with σ(K, T ) σ for all K and T . In practice,however, not only is the volatility surface not flat but it actually varies, often significantly, with time.Figure 1: The Volatility Surface4 Assumingthere is no arbitrage in the market-place.

The Black-Scholes Model4In Figure 1 above we see a snapshot of the5 volatility surface for the Eurostoxx 50 index on November 28th ,2007. The principal features of the volatility surface is that options with lower strikes tend to have higherimplied volatilities. For a given maturity, T , this feature is typically referred to as the volatility skew or smile.For a given strike, K, the implied volatility can be either increasing or decreasing with time-to-maturity. Ingeneral, however, σ(K, T ) tends to converge to a constant as T . For T small, however, we often observean inverted volatility surface with short-term options having much higher volatilities than longer-term options.This is particularly true in times of market stress.It is worth pointing out that different implementations6 of Black-Scholes will result in different implied volatilitysurfaces. If the implementations are correct, however, then we would expect the volatility surfaces to be verysimilar in shape. Single-stock options are generally American and in this case, put and call options will typicallygive rise to different surfaces. Note that put-call parity does not apply for American options.Clearly then the Black-Scholes model is far from accurate and market participants are well aware of this.However, the language of Black-Scholes is pervasive. Every trading desk computes the Black-Scholes impliedvolatility surface and the Greeks they compute and use are Black-Scholes Greeks.Arbitrage Constraints on the Volatility SurfaceThe shape of the implied volatility surface is constrained by the absence of arbitrage. In particular:1. We must have σ(K, T ) 0 for all strikes K and expirations T .2. At any given maturity, T , the skew cannot be too steep. Otherwise butterfly arbitrages will exist. Forexample fix a maturity, T and consider put two options with strikes K1 K2 . If there is no arbitrage thenit must be the case (why?) that P (K1 ) P (K2 ) where P (Ki ) is the price of the put option with strikeKi . However, if the skew is too steep then we would obtain (why?) P (K1 ) P (K2 ).3. Likewise the term structure of implied volatility cannot be too inverted. Otherwise calendar spreadarbitrages will exist. This is most easily seen in the case where r q 0. Then, fixing a strike K, we canlet Ct (T ) denote the time t price of a call option with strike K and maturity T . Martingale pricing impliesthat Ct (T ) Et [(ST K) ]. We have seen before that (ST K) is a Q-submartingale and nowstandard martingale results can be used to show that Ct (T ) must be non-decreasing in T . This would beviolated (why?) if the term structure of implied volatility was too inverted.In practice the implied volatility surface will not violate any of these restrictions as otherwise there would be anarbitrage in the market. These restrictions can be difficult to enforce, however, when we are “bumping” or“stressing” the volatility surface, a task that is commonly performed for risk management purposes.Why is there a Skew?For stocks and stock indices the shape of the volatility surface is always changing. There is generally a skew,however, so that for any fixed maturity, T , the implied volatility decreases with the strike, K. It is mostpronounced at shorter expirations. There are two principal explanations for the skew.1. Risk aversion which can appear as an explanation in many guises:(a) Stocks do not follow GBM with a fixed volatility. Instead they often jump and jumps to the downsidetend to be larger and more frequent than jumps to the upside.(b) As markets go down, fear sets in and volatility goes up.(c) Supply and demand. Investors like to protect their portfolio by purchasing out-of-the-money puts andso there is more demand for options with lower strikes.5 Note that by put-call parity the implied volatility σ(K, T ) for a given European call option will be also be the impliedvolatility for a European put option of the same strike and maturity. Hence we can talk about “the” implied volatility surface.6 For example different methods of handling dividends would result in different implementations.

The Black-Scholes Model52. The leverage effect which is due to the fact that the total value of company assets, i.e. debt equity,is a more natural candidate to follow GBM. If so, then equity volatility should increase as the equity valuedecreases. To see this consider the following:Let V , E and D denote the total value of a company, the company’s equity and the company’s debt,respectively. Then the fundamental accounting equations states thatV D E.(15)(Equation (15) is the basis for the classical structural models that are used to price risky debt and creditdefault swaps. Merton (1970’s) recognized that the equity value could be viewed as the value of a calloption on V with strike equal to D.)Let V , E and D be the change in values of V , E and D, respectively. ThenV V (E E) (D D) so thatV VV E ED D V V E E ED D D VEVD(16)If the equity component is substantial so that the debt is not too risky, then (16) impliesσV EσEVwhere σV and σE are the firm value and equity volatilities, respectively. We therefore haveσE VσV .E(17)Example 1 (The Leverage Effect)Suppose, for example, that V 1, E .5 and σV 20%. Then (17) implies σE 40%. Suppose σV remainsunchanged but that over time the firm loses 20% of its value. Almost all of this loss is borne by equity so thatnow (17) implies σE 53%. σE has therefore increased despite the fact that σV has remained constant.It is interesting to note that there was little or no skew in the market before the Wall street crash of 1987. So itappears to be the case that it took the market the best part of two decades before it understood that it waspricing options incorrectly.What the Volatility Surface Tells UsTo be clear, we continue to assume that the volatility surface has been constructed from European option prices.Consider a butterfly strategy centered at K where you are:1. long a call option with strike K K2. long a call with strike K K3. short 2 call options with strike KThe value of the butterfly, B0 , at time t 0, satisfiesB0 C(K K, T ) 2C(K, T ) C(K K, T ) e rT Prob(K K ST K K) K/2 e rT f (K, T ) 2 K K/2 e rT f (K, T ) ( K)2

The Black-Scholes Model6where f (K, T ) is the (risk-neutral) probability density function (PDF) of ST evaluated at K. We therefore havef (K, T ) erTC(K K, T ) 2C(K, T ) C(K K, T ).( K)2(18)Letting K 0 in (18), we obtain 2C. K 2The volatility surface therefore gives the marginal risk-neutral distribution of the stock price, ST , for any time,T . It tells us nothing about the joint distribution of the stock price at multiple times, T1 , . . . , Tn .f (K, T ) erTThis should not be surprising since the volatility surface is constructed from European option prices and thelatter only depend on the marginal distributions of ST .Example 2 (Same marginals, different joint distributions)Suppose there are two time periods, T1 and T2 , of interest and that a non-dividend paying security hasrisk-neutral distributions given byST1 e(r σ22ST2 e /2)T1 σ T1 Z1 (r σ /2)T2 σ T2 ρZ1 (19)1 ρ2 Z2 (20)where Z1 and Z2 are independent N(0, 1) random variables. Note that a value of ρ 0 can capture amomentum effect and a value of ρ 0 can capture a mean-reversion effect. We are also implicitly assumingthat S0 1.(A)(B)Suppose now that there are two securities, A and B say, with prices St and St given by (19) and (20) attimes t T1 and t T2 , and with parameters ρ ρA and ρ ρB , respectively. Note that the marginal(A)(B)distribution of St is identical to the marginal distribution of St for t {T1 , T2 }. It therefore follows thatoptions on A and B with the same strike and maturity must have the same price. A and B therefore haveidentical volatility surfaces.But now consider a knock-in put option with strike 1 and expiration T2 . In order to knock-in, the stock price attime T1 must exceed the barrier price of 1.2. The payoff function is then given byPayoff max (1 ST2 , 0) 1{ST1 1.2} .Question: Would the knock-in put option on A have the same price as the knock-in put option on B?Question: How does your answer depend on ρA and ρB ?Question: What does this say about the ability of the volatility surface to price barrier options?3The GreeksWe now turn to the sensitivities of the option prices to the various parameters. These sensitivities, or theGreeks are usually computed using the Black-Scholes formula, despite the fact that the Black-Scholes model isknown to be a poor approximation to reality. But first we return to put-call parity.Put-Call ParityConsider a European call option and a European put option, respectively, each with the same strike, K, andmaturity T . Assuming a continuous dividend yield, q, then put-call parity statese rT K Call Price e qT S Put Price.(21)

The Black-Scholes Model7This of course follows from a simple arbitrage argument and the fact that both sides of (21) equal max(ST , K)at time T . Put-call parity is useful for calculating Greeks. For example7 , it implies that Vega(Call) Vega(Put)and that Gamma(Call) Gamma(Put). It is also extremely useful for calibrating dividends andconstructing the volatility surface.The GreeksThe principal Greeks for European call options are described below. The Greeks for put options can becalculated in the same manner or via put-call parity.Definition: The delta of an option is the sensitivity of the option price to a change in the price of theunderlying security.The delta of a European call option satisfiesdelta C e qT Φ(d1 ). SThis is the usual delta corresponding to a volatility surface that is sticky-by-strike. It assumes that as theunderlying security moves, the volatility of the option does not move. If the volatility of the option did movethen the delta would have an additional term of the form vega σ(K, T )/ S. In this case we would say thatthe volatility surface was sticky-by-delta. Equity markets typically use the sticky-by-strike approach whencomputing deltas. Foreign exchange markets, on the other hand, tend to use the sticky-by-delta approach.Similar comments apply to gamma as defined below.(a) Delta for European Call and Put Options(b) Delta for Call Options as Time-To-Maturity VariesFigure 2: Delta for European OptionsBy put-call parity, we have deltaput deltacall e qT . Figure 2(a) shows the delta for a call and put option,respectively, as a function of the underlying stock price. In Figure 2(b) we show the delta for a call option as afunction of the underlying stock price for three different times-to-maturity. It was assumed r q 0. What isthe strike K? Note that the delta becomes steeper around K when time-to-maturity decreases. Note also thatdeltacall Φ(d1 ) is often loosely interpreted as the (risk-neutral) probability that the option will expire in themoney. (This risk-neutral probability is in fact Φ(d2 ).)In Figure 3 we show the delta of a call option as a function of time-to-maturity for three options of different7 Seebelow for definitions of vega and gamma.

The Black-Scholes Model8Figure 3: Delta for European Call Options as a Function of Time-To-Maturitymoney-ness. Are there any surprises here? What would the corresponding plot for put options look like?Definition: The gamma of an option is the sensitivity of the option’s delta to a change in the price of theunderlying security.The gamma of a call option satisfiesgamma 2Cφ(d1 ) e qT2 SσS Twhere φ(·) is the standard normal PDF.(a) Gamma as a Function of Stock Price(b) Gamma as a Function of Time-to-MaturityFigure 4: Gamma for European OptionsIn Figure 4(a) we show the gamma of a European option as a function of stock price for three differenttime-to-maturities. Note that by put-call parity, the gamma for European call and put options with the samestrike are equal. Gamma is always positive due to option convexity. Traders who are long gamma can makemoney by gamma scalping. Gamma scalping is the process of regularly re-balancing your options portfolio to be

The Black-Scholes Model9delta-neutral. However, you must pay for this long gamma position up front with the option premium. In Figure4(b), we display gamma as a function of time-to-maturity. Can you explain the behavior of the three curves inFigure 4(b)?Definition: The vega of an option is the sensitivity of the option price to a change in volatility.The vega of a call option satisfiesvega (a) Vega as a Function of Stock Price C e qT S T φ(d1 ). σ(b) Vega as a Function of Time-to-MaturityFigure 5: Vega for European OptionsIn Figure 5(b) we plot vega as a function of the underlying stock price. We assumed K 100 and thatr q 0. Note again that by put-call parity, the vega of a call option equals the vega of a put option with thesame strike. Why does vega increase with time-to-maturity? For a given time-to-maturity, why is vega peakednear the strike? Turning to Figure 5(b), note that the vega decreases to 0 as time-to-maturity goes to 0. This isconsistent with Figure 5(a). It is also clear from the expression for vega.Question: Is there any “inconsistency” to talk about vega when we use the Black-Scholes model?Definition: The theta of an option is the sensitivity of the option price to a negative change intime-to-maturity.The theta of a call option satisfiestheta C Tσ e qT Sφ(d1 ) qe qT SN(d1 ) rKe rT N(d2 ).2 TIn Figure 6(a) we plot theta for three call options of different times-to-maturity as a function of the underlyingstock price. We have assumed that r q 0%. Note that the call option’s theta is always negative. Can youexplain why this is the case? Why does theta become more negatively peaked as time-to-maturity decreases to0?In Figure 6(b) we again plot theta for three call options of different money-ness, but this time as a function oftime-to-maturity. Note that the ATM option has the most negative theta and this gets more negative as

The Black-Scholes Model10(a) Theta as a Function of Stock Price(b) Theta as a Function of Time-to-MaturityFigure 6: Theta for European Optionstime-to-maturity goes to 0. Can you explain why?Options Can Have Positive Theta: In Figure 7 we plot theta for three put options of different money-nessas a function of time-to-maturity. We assume here that q 0 and r 10%. Note that theta can be positive forin-the-money put options. Why? We can also obtain positive theta for call options when q is large. In typicalscenarios, however, theta for both call and put options will be negative.Figure 7: Positive Theta is PossibleDelta-Gamma-Vega Approximations to Option PricesA simple application of Taylor’s Theorem saysC(S S, σ σ) C(S, σ) S C(S, σ) C1 2C C ( S)2 2 σ S2 S σ1 S δ ( S)2 Γ σ vega2

The Black-Scholes Model11where C(S, σ) is the price of a derivative security as a function8 of the current stock price, S, and the impliedvolatility, σ. We therefore obtainΓ( S)2 vega σ2 delta P&L gamma P&L vega P&LP&L δ S When σ 0, we obtain the well-known delta-gamma approximation. This approximation is often used, forexample, in historical Value-at-Risk (VaR) calculations for portfolios that include options. We can also write 2 ΓS 2 S S vega σ P&L δSS2S ESP Return Gamma Return2 vega σwhere ESP denotes the equivalent stock position or “dollar” delta.4Delta HedgingIn the Black-Scholes model with GBM, an option can be replicated exactly by delta-hedging the option. Infact the Black-Scholes PDE we derived earlier was obtained by a delta-hedging / replication argument. The ideabehind delta-hedging is to re-balance the portfolio of the option and stock continuously so that you always havea total delta of zero after re-balancing. Of course it is not practical in to hedge continuously and so instead wehedge periodically. Periodic or discrete hedging then results in some replication error.Let Pt denote the time t value of the discrete-time self-financing strategy that attempts to replicate the optionpayoff and let C0 denote the initial value of the option. The replicating strategy is then given byP0Pti 1: C0 (22)Pti (Pti δti Sti ) r t δti Sti 1 Sti qSti t (23)where t : ti 1 ti is the length of time between re-balancing (assumed constant for all i), r is the annualrisk-free interest rate (assuming per-period compounding), q is the dividend yield and δti is the Black-Scholesdelta at time ti . This delta is a function of Sti and some assumed implied volatility, σimp say. Note that (22)and (23) respect the self-financing condition. Stock prices are simulated assuming St GBM(µ, σ) so thatSt t St e(µ σ2 /2) t σ tZwhere Z N(0, 1). Note the option implied volatility, σimp , need not equal σ which in turn need not equal therealized volatility (when we hedge periodically as opposed to continuously). This has interesting implications forthe trading P&L which we may define asP&L : PT (ST K) in the case of a short position in a call option with strike K and maturity T . Note that PT is the terminal valueof the replicating strategy in (23). Many interesting questions now arise:Question: If you sell options, what typically happens the total P&L if σ σimp ?Question: If you sell options, what typically happens the total P&L if σ σimp ?Question: If σ σimp what typically happens the total P&L as the number of re-balances increases?Some Answers to Delta-Hedging QuestionsRecall that the price of an option increases as the volatility increases. Therefore if realized volatility is higherthan expected, i.e. the level at which it was sold, we expect to lose money on average when we delta-hedge an8 Theprice may also depend on other parameters, in particular time-to-maturity, but we suppress that dependence here.

The Black-Scholes Model12option that we sold. Similarly, we expect to make money when we delta-hedge if the realized volatility is lowerthan the level at which it was sold.In general, however, the payoff from delta-hedging an option is path-dependent, i.e. it depends on the pricepath taken by the stock over the entire time interval. In fact, we can show that the payoff from continuouslydelta-hedging an option satisfiesZP&L 0TSt2 2 Vt2 S 22σimp σt2 dt(24)where Vt is the time t value of the option and σt is the realized instantaneous volatility at time t.S22The term 2t SV2t is often called the dollar gamma, as discussed earlier. It is always positive for a call or putoption, but it goes to zero as the option moves significantly into or out of the money.Returning to self-financing trading strategy of (22) and (23), note that we can choose any model we like for thesecurity price dynamics. In particular, we are not restricted to choosing geometric Brownian motion and otherdiffusion or jump-diffusion models could be used instead. It is interesting to simulate these alternative modelsand to then observe what happens to the replication error in (24) where the δti ’s are computed assuming(incorrectly) a geometric Brownian motion price dynamics.

a continuous dividend yield of q. 2 The Volatility Surface The Black-Scholes model is an elegant model but it does not perform very well in practice. For example, it is well known that stock prices jump on occasions and d