The Black Scholes Model

5 The Black Scholes Model 5.1 INTRODUCTION In the last chapter, two approaches were suggested for finding the price of an option: r Use risk neutrality to set growth rates (returns) equal to the interest rate and with these substitutions, work out the expected value of the payoff. The present value of this amount equals the fair value of the option today. r Solve the Black Scholes equation subject to the appropriate boundary conditions. Either of these methods can be used to derive the Black Scholes model for the prices of European call and put options, which is the most famous and widely used option model. It consists of a simple formula giving the value of the option as a function of a few parameters; this is called a “closed form solution”. Normally, models do not come in such a convenient form, but consist of a set of procedures which are applied in order to get a numerical value for the price of an option; these are the “numerical methods” described in Part 2 of this book. In the analysis of the previous chapters, a number of restrictive assumptions have been made. They are referred to collectively as the Black Scholes assumptions or a description of the Black Scholes world: (A) Volatility is constant. (B) Cash is borrowed or deposited at the same constant rate of interest. (C) There is no buy/sell spread or sales commission. (D) It is possible to short stock without charge. (E) Markets are continuous, so there is always a quote available. (F) Markets exist for any quantity of stock, including fractions of stock. (G) Markets are completely liquid, so we get instant execution, in any size, at the quoted price. 5.2 DERIVATION OF MODEL FROM EXPECTED VALUES (i) Risk neutrality tells us that the value at time t = 0 of a call option maturing at time T = 0is given by C 0 = e −rT E[C T ] risk neutral = e −rT E[max[(S T − X ), 0]] risk neutral For notional simplicity, the risk-neutral suffix will be dropped but it must always be remembered that we are dealing with pseudo-probabilities and pseudo-expectations. The value of the option is zero if it expires out-of-the-money (S T < X), so the expression for the value of the call option at t = 0 may be written C 0 = e −rT E [ S T − X : S T > X ] (ii) In order to obtain the expected value we need to multiply the payoff by a probability dis- tribution and integrate over S T . However, it is mathematically much simpler to transform 5 The Black Scholes Model variables: equation (3.7) states that S T = S 0 e mT+σ √ Tz T where m = µ − 1 2 σ 2 and z T is a stan- dard normal variate; in a risk-neutral world with dividends, m = (r − q) − 1 2 σ 2 . The mechan- ical details of how to evaluate the conditional expectations are given in Appendix A.1(v) The result is C 0 = e −rT  S 0 e (r−q)T N[σ √ T − Z X ] − X N[−Z X ]  where Z X = (ln(X/S 0 ) − mT)/σ √ T . This result is more usually written as C 0 = e −rT {F 0T N[d 1 ] − X N[d 2 ]} d 2 = d 1 − σ √ T d 1 =  ln(F 0T / X  + 1 2 σ 2 T  /σ √ T =  ln S 0 / X + (r − q)T + 1 2 σ 2 T  /σ √ T (5.1) where F 0T is the forward price. (iii) Note that for constant X,E[X : S T > X] = E[X | S T > X]E[S T > X] = X P[S T > X]. It is therefore sometimes stated that the factor N[d 2 ] is the probability that S T > X, i.e. that the option will be exercised. But remember that risk neutrality has led to the substitution µ → r. Therefore P[S T > X] is a pseudo-probability. The true probability that S T > X is N[d 2 ], but with r replaced by µ. (iv) General Black Scholes Formula for Put or Call: Recall the put–call parity relationship of Section 2.2(i): F 0T + P 0 e rT = X + C 0 e rT Substitute from the Black Scholes expression for C 0 : P 0 e rT = X{1 − N[d 2 ]}−F 0T {1 − N[d 1 ]} From equation (A1.4), N[d] + N[−d] = 1, so that P 0 = e −rT {X N[−d 2 ] − F 0T N[−d 1 ]} The Black Scholes formulas for European put and call options can be combined as f 0 = e −rT φ{F 0T N[φd 1 ] − X N[φd 2 ]} (5.2) where φ =+1 for a call option and φ =−1 for a put. (v) In manipulating these formulas, we often need an option price at time t.Clearly, this is obtained from equation (5.2) merely by making the substitutions f 0 → f t ; S 0 → S t ; F 0T → F tT ; T → T − t. 5.3 SOLUTIONS OF THE BLACK SCHOLES EQUATION It has been shown that the same arbitrage reasoning leads both to the risk-neutral stock price distribution (from which we derived the Black Scholes model), and to the Black Scholes equation. The two approaches should therefore lead to the same final conclusions: now comes the acid test. 52 5.4 GREEKS FOR THE BLACK SCHOLES MODEL (i) Stated formally, we seek a solution C(S 0 , T ) of the equation ∂C 0 ∂T = (r − q)S 0 ∂C 0 ∂ S 0 + 1 2 σ 2 S 2 0 ∂ 2 C 0 ∂ S 2 0 − rC 0 subject to the initial and boundary conditions r C(S 0 , 0) = max[0, S 0 − X ] r lim S 0 →0 C(S 0 , T ) → 0 r lim S 0 →∞ C(S 0 , T ) → S 0 e −qT − X e −rT (ii) Let us now make the following transformations, suggested by equation (A4.5) in the Appendix: C 0 = e −rT  / 1 2 σ 2 e −kx−k 2 T  v(x, T  ); x = ln S 0 ; T  = 1 2 σ 2 T ; k = r − q − 1 2 σ 2 σ 2 Substituting in the previous equation and doing the algebra reduces the problem to a solution of the equation ∂v ∂T  = ∂ 2 v ∂x 2 subject to initial and boundary conditions r v(x, 0) = max[0, e (k+1)x − X e kx ] r lim x→−∞ v(x, T  ) → 0 r lim x→∞ v(x, T  ) → e (k+1)x+(k+1) 2 T  − X e kx+k 2 T  (iii) The solution of this problem is demonstrated in the Appendix: using Fourier transforms in equation (A6.5) or using Green’s functions in equation (A7.8) we can write v(x, T  ) =  +∞ −∞ e kx max[0, e x − X ]  1 2 √ π T  e − (y−x) 2 4T   dy Without detailing every tedious step, the integral is performed as follows: r Get rid of the awkward “max” function in the integral, setting the lower limit of integration y = ln X. r Change the variable of integration (y−x ) 2 4T  → z. r Use the standard integral results of Appendix A.1(v). r Substitute back for x, T  and k . If the reader cares to check all this he will retrieve equation (5.1), the Black Scholes formula. 5.4 GREEKS FOR THE BLACK SCHOLES MODEL (i) Some Useful Differentials: The Black Scholes model gives specific analytical formulas for the prices of European put and call options. It is therefore possible to give formulas for the Greeks simply by differentiation. The starting point is the Black Scholes model, but before slogging away at the differentials, we note a couple of general results which much simplify the computations. 53 5 The Black Scholes Model (A) From equation (A1.2) we have ∂ N[φd] ∂θ = ∂ N[φd] ∂(φd) ∂(φd) ∂θ = φn(d) ∂d ∂θ where φ can only take values ±1 and is independent of θ , and n(d) = 1 √ 2π e − 1 2 d 2 . (B) From the definitions of d 1 and d 2 given in equation (5.1), it follows that d 2 − d 1 =−σ √ T and d 2 + d 1 = 2 σ √ T ln  S 0 e −qT X e −rT  so that d 2 2 − d 2 1 = (d 2 − d 1 )(d 2 + d 1 ) =−2ln  S 0 e −qT X e −rT  Substituting this into the explicit expression for n(d 2 ) in (A) above gives n(d 2 ) = 1 √ 2π e − 1 2 d 2 1 +ln  S 0 e −qT X e −rT  or S 0 e −qT n(d 1 ) = X e −rT n(d 2 ) (C) Differentiating the relationship d 1 − d 2 = σ √ T gives                        ∂d 1 ∂ S 0 − ∂d 2 ∂ S 0 = 0 ∂d 1 ∂r − ∂d 2 ∂r = 0 ∂d 1 ∂T − ∂d 2 ∂T = σ 2 √ T ∂d 1 ∂σ − ∂d 2 ∂σ = √ T (D) Differentiating the explicit expression for d 1 with respect to S 0 gives ∂d 1 ∂ S 0 = 1 S 0 σ √ T The Greeks can now be obtained by differentiating equation (5.2): f 0 = φS 0 e −qT N[φd 1 ] − φ X e −rT N[φd 2 ] d 1 = 1 σ √ T ln  S 0 e −qT X e −rT  + 1 2 σ √ T ; d 2 = 1 σ √ T ln  S 0 e −qT X e −rT  − 1 2 σ √ T (ii) Delta:  = ∂ f 0 ∂ S 0 = φ e −qT N[φd 1 ] + φ S 0 e −qT n(d 1 ) ∂d 1 ∂ S 0 − φ X e −rT n(d 2 ) ∂d 2 ∂ S 0 Using (B) and (C) above gives  = φ e −qT N[φd 1 ] (5.3) (iii) Gamma:  = ∂ ∂ S 0 = φ 2 e −qT n(d 1 ) ∂d 1 ∂ S 0 and ∂d 1 ∂ S 0 = 1 Sσ √ T 54 5.4 GREEKS FOR THE BLACK SCHOLES MODEL so that  = n(d 1 ) Sσ √ T e −qT Note that this is independent of φ so that  is the same for a put or a call option. (iv) Theta: The differential of the Black Scholes formula with respect to T would measure the rate of increase of the value of an option as its time to maturity increases; but theta is the rate of increase in value as time passes, i.e. as the maturity of the option decreases. Recalling the conventions described in Section 1.1(v):  = ∂ f 0 ∂t =− ∂ f 0 ∂T = φqS 0 e −qT N[φd 1 ] − φrXe −rT N[φd 2 ] −S 0 e −qT n(d 1 ) ∂d 1 ∂T + X e −rT n(d 2 ) ∂d 2 ∂T Once again using (B) and (C) above gives  = φqS 0 e −qT N[φd 1 ] − φrXe −rT N[φd 2 ] − S 0 e −qT n(d 1 ) σ 2 √ T (v) Vega:  = ∂ f 0 ∂σ = S 0 e −qT n(d 1 ) ∂d 1 ∂σ − X e −rT n(d 2 ) ∂d 2 ∂σ As before this can be simplified to  = S 0 e −qT n(d 1 ) √ T = X e −rT n(d 2 ) √ T (5.4) No! The equals sign is not a typo. A direct comparison between this and the expression for gamma gives  = S 2 σ T (vi) Rho: ρ = ∂ f 0 ∂r = S 0 e −qT n(d 1 ) ∂d 1 ∂r − X e −rT n(d 2 ) ∂d 2 ∂r + φTXe −rT N[φd 2 ] ρ = φTXe −rT N[φd 2 ] (vii) The specific functional form of the Black Scholes formula leads to a very simple expression for . The value of a call option can be written C 0 ={e −qT N[d 1 ]}S 0 −{X e −rT N[d 2 ]} = S 0 − B 0 where the first line is the Black Scholes formula and the second line represents a replicating portfolio. The model can therefore be interpreted as the recipe for replicating a call: buy e −qT N[d 1 ] units of stock and borrow cash of X e −rT N[d 2 ]. (viii) Approximate Option Values: The relative complexity of the Black Scholes model means that it is quite hard to make a quick intuitive guess at the value of an option. However, practitioners often use the formula 0.4 × σ √ T % as the price of an at-the-money-forward option, i.e. one where the strike price equals the forward price. 55 5 The Black Scholes Model If X = F 0T in equation (5.1), then we have d 1 = 1 2 σ √ T and d 2 =− 1 2 σ √ T . Ignoring the dividends, we have for the price of a call option (or for that matter a put option): C 0 = S 0 {N[+σ √ T ] − N[−σ √ T ]} The peak of a standard normal distribution is at a height of 1/ √ 2π ≈ 0.4 [see equation (A1.2)], so if 1 2 σ √ T is small we can write C 0 = S 0 × 0.4 × σ √ T For short-term, low-volatility options this works well, although the robustness of the approxi- mation is surprising, even over a wide range. The exact and approximate call option values for σ = 20%, T = 3 months are 3.99% and 4.00%; for σ = 40%, T = 4 years they are 31.08% and 32.00%. 5.5 ADAPTATION TO DIFFERENT MARKETS (i) The objective of this book is to provide the reader with a grounding in option theory, which can be applied to a variety of different markets. Most readers will be interested in one spe- cific market, and it is always easier to read material which is narrowly specific to ones own area of interest, but unfortunately this is not a practicable way to write a book. This sec- tion tries to ease the reader’s burden of adapting the material to his own specific area of interest. In much of the forgoing, the market used to develop the theory was the equity market. This was chosen since it is the most straightforward and widely understandable for newcomers to finance theory: everyone understands what the price of one share of stock means and roughly how dividends work; a futures price or convenience yield is more arcane. Where equity failed to provide an adequate example, as in the discussions of arbitrage or futures, we have turned to other markets such as foreign exchange or commodities. At the risk of some repetition, we now summarize how the theory is adapted to other markets. (ii) Equities: This is the easiest, since the theory has been developed largely with reference to this market. It is a very straightforward cash market, i.e. the commodity (stock) is purchased directly with physical delivery as soon as possible after purchase. In most established markets there are traded options on the most important stocks, although forwards and futures on single stocks have not yet become established. This begs the following question: in the absence of a forward market, can we really price an option using the arbitrage arguments of Section 1.2, which were developed for the foreign exchange market with its large forward market which can be used to execute arbitrage trades? The answer is an emphatic yes; foreign exchange was merely used as a simple illustration of the no-arbitrage principle in its various forms. The notion of a forward can be used in pricing an option, even though no formal forward market exists. The arbitrage that is actually performed if an option is mispriced is not buying spot and selling forward, but extracting the option’s fair value through delta hedging. A formal forward market is not needed to calculate the fair value of an option from the notional forward price; but the delta hedge must exist. In some markets, shorting stock is illegal or restricted to certain categories of market participant, and often stock is just not available for borrowing. This means that positive delta positions (short puts, long calls) cannot 56 5.5 ADAPTATION TO DIFFERENT MARKETS be hedged and arbitrage arguments do not apply. The “fair value” is then no more than a hypothetical construction. The “dividend” q may be different for delta hedging with long or short stock positions. If the stock is held long, q will indeed be the continuous dividend yield; but if the stock is held short, q will be the total cash that needs to be paid out on the short position, i.e. continuous dividend plus stock borrowing cost. (iii) Prices, Values and Greeks of Forwards and Futures: Before going on to discuss other markets it is worth briefly recapping on the meanings of the words “price” and “value”. In the cash markets (equities, spot FX, spot commodities) the two mean exactly the same: if a stock price is $50 its value is $50. In the case of options, usage is rather context dependent, but usually price means what someone is prepared to pay, while value is calculated from the price of the underlying commodity using a model. Confusion arises with forwards and futures contracts, but this is largely a matter of semantics: r The forward (F tT ) or futures ( tT ) price is the price at time t, at which one agrees to buy a commodity at time T in the future. It was shown in Chapter 1 that if interest rates are constant, then F tT =  tT = S t e (r−q)(T −t) . r If the forward or futures contract is entered into at the prevailing market price (is at-the- money), then its value is zero. r Suppose the contractual purchase price in a forward contract is not equal to the forward price but instead equal to X. The value of the contract is then given by equation (1.4): f tT = S t e −q(T −t) − X e −r(T −t) = e −r(T −t) (F tT − X). r It follows from the last point that the delta of a forward is  fwd = ∂ f tT /∂ S t = e −q(T −t) . r A futures contract cannot build up value since it is marked to market daily and  today −  yetsterday = δ is paid over each day. r An infinitesimal movement in the underlying price δS t will cause δ tT to be paid over at the end of a given day. From the relation  tT = S t e (r−q)(T −t) , delta is given by  fut = e (r−q)(T −t) . Futures and forward contracts both have delta close to 100% but have no gamma. They are often used in place of the underlying stock or commodity to delta hedge options, since they involve no initial cash outlay. (iv) Foreign Exchange: These are the largest and most liquid markets considered in this book. Most of the Black Scholes assumptions of Section 5.1 are fairly realistic, except for constant volatility. In addition to the spot market, foreign exchange is very actively traded between banks using over-the-counter (OTC) forward contracts; also, the important currencies have publicly traded futures markets. The theory carries over very simply from that developed for equity: the stock simply be- comes one unit of the foreign currency. The dividend throw-off is replaced by the foreign currency interest rate. This is a particularly easy substitution to make since interest rates are incurred continuously. In fact the continuous dividend yield Black Scholes model was really first developed for foreign exchange; in that context it is often known as the Garman Kohlhagen model. Delta hedging of foreign currency options is not usually carried out with physical foreign currency. It is much more convenient and less cash consuming to use forward contracts or futures. Note that the forwards or futures do not have to have the same maturity as the option being hedged; the deltas of the option and its hedge just need to match. 57 5 The Black Scholes Model (v) Stock Indices and Commodities: In theory one can invest in a stock index by buying a prescribed number of shares of each stock in the index. This is obviously too cumbersome to be practical for hedging, but a direct investment in the underlying index is often not possible since no traded instrument exists. In compensation, the futures markets on most major stock indices are very liquid and cheap to deal in, and are the normal source for delta hedges. Options on commodities may also be analyzed using the Black Scholes methodology devel- oped for equity derivatives. In this case, storage and insurance costs are treated as a negative dividend in the Black Scholes formula and in the formula relating the spot price to the futures price. In theory then, we could delta hedge a commodities option by getting the delta from the Black Scholes formula (setting dividends equal to negative storage costs); we would then hedge by buying or selling the right number of futures such that  fut balances the delta of the option. Unfortunately, this does not work well in practice. In the first place, the storage model for commodities futures prices does not describe market prices well; and second, setting futures prices equal to forward prices only really works with constant, or at least uncorrelated, interest rates [see Section 1.4.(iii)]. If we write an option and try to hedge the position in the futures market, we then run the basis risk, or risk associated with futures prices deviating from the simple models previously described. However, an alternative approach avoids this problem: instead of writing an option on the spot price of a commodity, write it on the futures price. We turn our attention to these contracts next. 5.6 OPTIONS ON FORWARDS AND FUTURES (i) Following the last section, we now examine what happens if the underlying security is itself a futures contract. For example, it was seen in the last section that a call option on a stock index could be dynamically hedged by buying or selling the appropriate number of stock index futures contracts; now we consider a call option on a stock index futures price rather than on the index itself. The analysis is very similar for forward contracts and futures contracts, so these are treated together, with any divergence in behavior pointed out as we go along. Futures contracts are of course far more important in practice, since these are traded on exchanges, while active forward markets are normally interbank (especially in foreign exchange). It is critical that the reader has a clear understanding of the concepts and notation of para- graph (iii) of the last section. now maturity of forward/ futures 0 time t T t maturity of option t (ii) The payoff of an option on a futures or forward contract is more abstract than for a simple stock. Compare the following three European call options maturing in time τ : r Options on the Underlying Stock Price: the contract is an option to buy one share of stock at a price X. Payoff = max[(S τ − X ), 0] r Options on the Forward Price: this is an option that at time τ we can enter a forward contract maturing at time T, at a forward price of X. The value of this forward contract at time τ will 58 5.6 OPTIONS ON FORWARDS AND FUTURES be (F T τ − X )e −r(τ −T ) . Payoff = max  (F τ T − X )e −r(T −τ ) , 0  r Options on the Futures Price: as in the last case, this is an option to enter a futures contract at time τ and price X; however, futures are marked to market daily so that a profit of  τ T − X would immediately be realized within one day of time τ . Payoff = max[( τ T − X ), 0] (iii) The forward price is given by F tT = S t e (r−q)(T −t) , and if interest rates are constant, we also have F tT =  tT . We may therefore write Volatility of F tT = volatility of  tT =  var  ln S t  = σ In general, the volatility of the forward price equals the volatility of the spot price; the volatility of the futures price equals the volatility of the underlying stock if the interest rate is constant. (iv) Black Scholes Equation for Forwards/Futures: We shall now repeat the analysis of Sec- tion 4.2(i)–(iii), but with a forward or futures price replacing the stock price of the underlying equity stock. We use the notation V tT to denote the forward/futures price and v tT as the value of the contract. Using the same construction as before, we suppose that we have a small portfolio containing a forward/futures option plus  units of forward/futures contracts, such that the portfolio is perfectly hedged against market movements. The value of the portfolio is f t −  t v tT = f t The key difference between this and the previous analysis lies in this expression. For a forward contract, v tT is the value at time t of a contract to buy a unit of commodity at time T for a price equal to the time t forward rate; but such a contract has zero value at time t. Similarly, a futures contract at time t has zero value. Now consider an infinitesimal time interval δt during which the forward/futures contract changes in value by δv tT . It follows from Section 5.5(iii) that δv tT =  e −r(T −t) δF tT forward  t+ δ tT −  tT = δ tT futures Either way, we can make the undemanding assumption that δv tT = A(V tT , t)δV tT . The increase in value of the hedged portfolio over time t can now be written δ f t −  t δv tT = δ f t −  t A t δV tT The arbitrage condition corresponding to equations (4.5) is δ f t −  t A t δV tT f t = r δt (5.5) It is assumed that forward and futures prices follow a similar Wiener process to a stock price: δV tT V tT = µδt + σ δW t 59 5 The Black Scholes Model Substituting this into equation (5.5) and using Ito’s lemma for δ f t gives  ∂ f t ∂t + µV tT ∂ f t ∂V tT + 1 2 σ 2 V 2 tT ∂ 2 f t ∂V 2 tT  δt + σ V tT ∂ f t ∂V tT δW t − V tT A t ( µδt + σ δW t )  t = rf t δt The coefficient of δW t must equal zero, since the portfolio is perfectly hedged, so that A t  t = ∂ f t ∂V tT Substituting this into the remaining terms gives ∂ f t ∂t + 1 2 σ 2 V 2 tT ∂ 2 f t ∂V 2 tT = rf t (5.6) (v) Significance of the Simplified Black Scholes Equation: The equation which has just been derived holds for forward prices and for futures prices. In the case of futures contracts, it does not depend on the idealized assumptions which were used to equate the forward and futures prices, i.e. constant interest rates. The equation is simpler than the Black Scholes equation for options on an equity stock. The reason can be traced to equation (5.5): the cost of entering a forward or futures contract is zero, and these instruments have no dividend throw-off. Consequently, the financing costs for the hedge are zero and the financing term reduces merely to the cost of carrying the option itself. This becomes immediately plain by examining the Black Scholes equation written in the form of equation (4.20). The partial differential equation for forwards/futures has the same form as the general Black Scholes equation for an equity stock, in which one has set q = r. This is in line with the properties of forwards and futures with which we are already familiar. Consider first the forward price: from equation (3.4) we have E t [S T ] risk neutral = S t e (r−q)(T −t) = F tT where the symbol E t [·] risk neutral indicates that the expectation is taken at time t and risk neutral means that we have set µ → r. Then E t [F τ T ] risk neutral = E t  S τ e (r−q)(T −τ )  risk neutral = e (r−q)(T −τ ) E t [S τ ] risk neutral = e (r−q)(T −τ ) S t e (r−q)(τ −t) = F tT The risk-neutral expected growth rate of F tT is therefore zero, which is the same as for an equity where q = r. Clearly, this same result would hold for a futures price when  tT = F tT , i.e. when interest rates are constant. However the result is more general, and holds for variable interest rates also. The reason is that a futures contract costs nothing to enter so that arbitrage assures that the expected profit from the contract must be zero: E t [ τ T ] risk neutral =  tT (vi) Black ’76 Model: We have established that the Black Scholes equation for an option on a forward/futures price can be obtained from the general equation for an option on the equity 60 [...]...5.6 OPTIONS ON FORWARDS AND FUTURES price by setting q → r ; therefore, the Black Scholes formula for an option on a forward or futures price can be obtained from the general Black Scholes formula by just the same procedure: f t = e−r (τ −t) φ{Vt T N[φd1 ] − X N[φd2 ]} √ 1 Vt T 1 + σ 2 (τ − t) ; ln d2 = d1 − σ τ − t; d1 = √ X 2 σ τ . Solve the Black Scholes equation subject to the appropriate boundary conditions. Either of these methods can be used to derive the Black Scholes model for the. derived the Black Scholes model) , and to the Black Scholes equation. The two approaches should therefore lead to the same final conclusions: now comes the acid