Advanced Derivatives Course Chapter 8

Week 8: Black-Scholes Equation For Financial Derivatives (see also Wilmott, Chapter 5)

Lecture VII.1 Risk-Free Portfolio

Perhaps some of you have become desperate and frustrated because of somewhat heavy mathematical stuff we have encountered up to this moment. Now all your efforts are going to be rewarded. The goal of the next two lectures is to see how bits and pieces of math we have been hard learning in the previous lectures are beautifully combined in to the famous Black-Scholes model. This week is probably the most important in this course. Although financial derivatives are centuries old, until recently pricing some of them had been a pure guesswork. The breakthrough came in the early seventies when Fisher Black, Myron Scholes, with help from a third, Robert Merton, developed an algorithm to do it instead. Their work swiftly became important, because demand for options and derivatives in general soared in the 1970s. That was partly because the breakdown of the Bretton Woods exchange-rate regime, and two oil-price shocks, led to huge swings in the prices of financial assets. It may also have reflected growing market sophistication. This course is evidence to the latter. As we all already know financial derivatives are deals written on other securities (bonds, shares, etc.) In most of the cases 1 these contracts have a specific termination date – the expiry time, which we denoted as T. At the expiration time, the option should depend only on the value of the underlying asset S(T) and the time T . The value of the option at time T is called payoff and this can be represented as a function V(S(T),T), which is known exactly at time T. For example, for a European call V(S(T),T) = max( S(T)-E,0). For times t other than T (that is, t
1

In the future lectures we will be discussing exotic derivatives, which do not have a specific expiration time. These derivatives are in perpetuity.

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Remarkably, those simple properties of calls and puts can be exploited to construct a very special portfolio2. We shall denote the value of this portfolio as Π.

: One long option position and a short position in some quantity ∆ of the underlying: Π = V(S,t)- ∆ · S. The first term on the right-hand side is the option position and the second term is the asset position. Notice the minus sign in front of the second term indicating that the asset is a short position. For the time being, ∆ is an unspecified coefficient, which we will determine in a moment. Now we make a crucial assumption that the underlying asset, S, follows a geometric Brownian motion: dS = ìSdt + óSdW. A natural question to ask is how the value of the portfolio changes from time t to t+dt? Since both the option value and the asset price will change, we have to use the chain rule to find the change in the portfolio value: dΠ = dV(S,t) - ∆ · dS. The important point to be made is that the coefficient ∆ in the above formula has not changed during the time step dt. Intuitively, this is because the values of V(S(t),t) and S change unexpectedly and when we set up the portfolio we could not anticipate these changes. This is why we still have the same quantity of the underlying at the moment t+dt. The above definition of the infinitesimal increment of the value of the portfolio, based on practical experience, is consistent with the mathematical definition of the Ito integral. Indeed, we can write the following logical statement: d Π = Π (t + dt) − Π (t ) =

t + dt

t +dt

t

t

∫ dΠ = ∫ ( dV − ∆dS).

Now, if you check your lecture notes with the definition of the Ito integral, you can clearly see that the coeffic ient ∆ must be a non-anticipating function. As a matter of fact, all stochastic differential equations are understood with respect to the Ito integral. The sole reason for studying the Ito integral is to make sense out of stochastic differential equations. In finance, the given portfolio is called a self-financed portfolio which means that once set it changes its value without any external injections of cash. This requirement also leads to the above formula for the change in the portfolio value. Indeed, the above mentioned portfolio can also be presented as follows 2

Do you remember, one of your computer exercises was to analyse a portfolio insensitive to the volatility?

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π = V − ∆ S − ∆ B B, where B is a bond and ∆B is Π is presented as π + ∆BB. change as well. However, if position must be equal to the of the two vanishes:

the size of our short position in the bond. Our original portfolio When time t changes to t + dt our positions ∆ and ∆B may our portfolio is self-financed than the change in the share opposite change in the bond position, so that the combination d∆ ⋅ S + d∆ B ⋅ B = 0.

In other words, when we sell bonds, we simultaneously buy shares and vice versa. This equation gives rise to the above presented formula for the change in the portfolio value: d Π = dV − ∆dS . Ito’s Lemma

Since the option price, V(S(t),t), is a function of both ht e share and time, the change dV(S,t) is also determined by using the chain rule and the Taylor expansion: dV(S,t) = Vtdt + VSdS + (1/2)VSSd 2S. Note that we expanded V to the second order in dS. Now remember that the share follows geometric Brownian motion. Therefore, in the third term on the right-hand side we have d 2S = (ìSdt + óSdW)2 = ó2S2d2W. Here we have dropped all terms with d2t and dtdW, because they can be considered as negligible for infinitesimally small dt and dW. However, d2W is not negligible as we have shown in the previous lectures. In fact, we have established that in the mean square limit sense, we can use the magic formula d2W = dt. Thus, we arrive at the conclusion that d 2S = ó 2S2dt. Correspondingly for the increment of the option value we obtain the following formula dV(S,t) = [Vt+ ìSVS + (1/2)VSS ó 2S2]dt + óSVSdW. This is the Ito formula and the stochastic version of the chain rule used in derivation of the Ito formula is known as Ito’s lemma. It plays a crucial role in option pricing. In situations where one has to apply the Ito formula, one will in general be given a stochastic differential equation that drives the process, like eq. (11). In a sense, the Ito formula can be seen as a vehicle that takes the SDE for S(t) and determines the SDE that corresponds to V(S(t),t).

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Using the Ito formula we can determine the SDE for financial derivatives once we are given the SDE for the underlying asset. For a market participant who wants to price an option, but who is willing to take the behaviour of the underlying as in-put, Ito’s formula is a necessary tool. Now we can put all bits together to derive the formula for the value change of the portfolio: dΠ(S,t) = [Vt+ ìSVS + (1/2)VSS ó 2S 2]dt + óSVSdW – Ä(ìSdt + óSdW) .

Delta Hedging When you look at the above equation for the change in the portfolio value, you can see that the right-hand side contains two types of terms, deterministic and random. The deterministic terms are those with the dt, and the random terms are those with the dW. Due to the random terms, the right-hand side of the above equation is unpredictable. These random terms are the risk in our portfolio. Is there any way to reduce or even eliminate this risk? Our experience with the binomial tree model tells us that this can be done in theory (and almost in practice) by carefully choosing the coefficient Ä. The random terms (with dW) are óS(VS – Ä)dW. Thus, if we choose Ä = VS, then the randomness is completely eliminated in the equation. The perfect elimination of risk, by exploiting correlation between two instruments (in this case an option and its underlying) is generally called delta hedging. Delta hedging is an example of a dynamic hedging strategy. From one time-step to the next the quantity VS changes, since it is, like V, a function of the ever-changing variables S and t. This means that the perfect hedge must be continually rebalanced. This is difficult to achieve in practice because of the cost of such a procedure. Normally, a portfolio is reshuffled on a daily basis. In the last lecture of this course, I hope to return to this problem again. The irony of delta hedging is that it has been known for a very long time, well before Black and Scholes work. However, only Black, Scholes and Merton have realised to make the next (Nobel prize winning) step.

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Arbitrage After choosing the coefficient Ä as above, we hold a portfolio whose value changes by the amount dΠ(S,t) = [Vt+ (1/2)VSS ó 2S2]dt. This change is completely risk-less. But if we have a completely risk-free change dΠ in the portfolio, then it must be the same as the growth we would get, if we put the equivalent amount of cash in a risk-free interest-bearing account. In other words, the following equality must hold: dΠ = rΠdt. This is nothing but the no-arbitrage principle. To see why this should be so, consider in turn what might happen, if the return on the portfolio were, first, greater and, second, less than the risk-free rate. If we were guaranteed to get a return of greater than r from the delta-hedged portfolio, then we could borrow from the bank, paying interest at the rate r, invest in the risk-free option/share portfolio and make a profit. If, on the other hand the return were less than the risk-free rate, we should go short the option, delta hedge it, and invest the cash in the bank. Either way, we make a risk-less profit in excess of the risk-free rate of interest. At this point we say that, all things being equal, the action of investors buying and selling to exploit the arbitrage opportunity will cause the market price of the option to move in the direction that eliminates the arbitrage. The Formula

By putting together the delta hedging and the no-arbitrage principle, we obtain [Vt+ (1/2)VSS ó 2S2]dt = r(V – SVS)dt. By moving all terms to the left and on dividing by dt, we get ∂V 1 2 2 ∂ 2V ∂V + 2σ S + rS − rV = 0. 2 ∂t ∂S ∂S This is the famous Black-Scholes equation, which has changed the world (at least the world of finance). Throughout the rest of this course we will familiarise ourselves with the advantages of the Black-Scholes theory of option pricing.

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Lecture VII.1 Black-Scholes World In the previous lecture, we have finally derived the main equation of this course – the Black-Scholes equation: ∂V 1 2 2 ∂ 2V ∂V + 2σ S + rS − rV = 0. 2 ∂t ∂S ∂S This equation is obtained by combining various partial derivatives of a function and then setting the combination equal to zero. In mathematical language, this is a linear parabolic partial differential equation. In fact, almost all partial differential equations in finance are of a similar form (those of you who wrote essays, could come accross a few equations of the BS type for various derivative models.) These equations are almost always linear, meaning that if you have two solutions of the equation, then the sum of these solutions is itself a solution. One good thing about parabolic equations is that they are relatively easy to solve numerically. The fact that the BS equation has only partial derivatives is quite natural, because as you may still remember, only partial derivatives of standard calculus can be used in financial calculations with random variables. It is clear that the BS equation by itself does not represent any financial interest. What is important are the solutions to this equation, which give the values of various financial derivatives. Therefore, without knowing how to solve this equation, we will not achieve anything. The rest of this course will be dedicated to the analysis of financially valuable solutions of the BS equation. The BS equation contains all the obvious variables and parameters such as the underlying, S, time, t, volatility, ó, and the risk-free rate, r, but curiously there is no trace of the drift ì. Why is this? Any dependence on the drift dropped out at the same time as we eliminated the random dW component of the portfolio discussed in the previous lecture. The economic argument for this is that since we can exclude all risk via perfectly hedging the option with the underlying, we should not be rewarded for taking unnecessary risk, that is, only the risk-free rate of return needs to be used. The fact that the BS equation is independent of risk preferences is a powerful argument. If risk does not enter the equation, it cannot affect its solution! Any set of risk preferences can, therefore, be used when evaluating V. In particular, the very simple assumption that all investors are risk neutral can be made. This must ring a bell! Indeed, the same assumption was made when we discussed the risk-neutral valuation in the “tree-model lectures”. The BS equation gives a continuous time generalisation of this technique. Thus, in the stochastic differential equation for the underlying (i.e. geometric Brownian motion), one can replace ì with r. Let us now talk about some characteristic features of the BS world.

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The BS Assumptions 1) The underlying follows a geometric Brownian motion. This is a reasonable conjecture for the behaviour of equities. However, for many other derivative products, like interest rate derivatives, this is an unnecessary assumption and we will discuss other types of random walks for the underlying. 2) The risk-free interest rate is a known function of time. This restriction is made only to simplify the task of solving explicitly the BS equation. If r were constant, this job would be even easier. In practice, the interest rate is often taken to be time dependent but known in advance (some of you might have already encountered such a thing as the term structure). In reality, the rate r is not known in advance and is itself stochastic (this is the reason for the existence of interest rate derivatives). This is why there is a huge research activity ‘beyond Black3) There are no dividends on the underlying. This assumption can be easily dropped, because it is very straightforward to take dividends into account. 4) Delta hedging is done continuously. This is definitely impossible. Hedging must be done in discrete time. Often the time between re-hedges will depend on the level of transaction costs in the market for the underlying; the lower the costs, the more frequent the re-hedging. We shall return to the issue of a discrete hedging in the last lectures. 5) There are no transaction costs on the underlying. The dynamic business of delta hedging is in reality expensive since there is a bid-offer spread on most underlying assets. This is related to the previous assumption. 6) There are no arbitrage opportunities. Without any doubt there are arbitrage opportunities in real world. This is how some people make their living by exploiting them. However, by making this assumption, we are ruling out any model-dependent arbitrage. In spite of some imperfectness of the BS assumptions, the model gives rise to pretty good values for various derivative instruments. In many cases, it is known how to correct the above assumptions to fit better into reality. Final Conditions We have derived the BS equation for a generic derivative product without specifying what kind of option we are valuing, whether it is a call or a put, or what the strike price and the expiration date are. The individual features of the option are specified by the final condition. We must specify the option value V as a function of the underlying asset at the expiry time T. In other words, we must prescribe V(S T ,T), the payoff. For example, if we are valuing a European call option, then we know that V(ST,T) = max(ST – E, 0). For a European put we have

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V(ST,T) = max(E - ST, 0). We shall come back to the discussion of the final conditions later on. Thanks to the final condition, the option value depends on the strike price. Options On Dividend-Paying Equities Now I would like to discuss how the BS equation could be generalised to value options on shares paying dividends. This might be just about the simplest generalisation of the BS model. In some cases dividends are paid only several times per year, and therefore need to be treated discretely, but the large number of dividend payments on an index such as the S&P 500 are so frequent that it may be best to regard them as a continuous payment. To keep things simple, let us assume that the asset receives a continuous and constant dividend yield, D. Thus, in infinitesimal time interval dt each asset receives an amount DSdt. This must be tailored into the derivation of the BS equation. Let us look at the change in the value of the portfolio: dΠ =

∂V ∂V ∂ 2V dt + dS + 12 σ 2 S 2 2 dt − ∆dS − D∆Sdt . ∂t ∂S ∂S

The last term on the right-hand side is simply the amount of the dividend per share, DSdt, multiplied by the number of the asset held, -Ä. Note that when we short the asset, the dividend does not belong to us and has to be paid to the asset owner. This is why we subtracted the dividend from the portfolio value. It is easy to check that if the Ä is still given by the rate of change of the option value with respect to the underlying, then the risk-dependent terms with dW cancel out, like in the non-dividend-paying case, and the portfolio becomes risk-less. Applying the noarbitrage principle, we now get a BS equation generalised for dividends Vt + (1/2)ó 2S 2VSS + (r – D)SVS – rV = 0. Currency Options Options on currencies are dealt with in exactly the same fashion. In holding the foreign currency, we receive interest at the foreign rate of interest rf. This is just like receiving a continuous dividend. Using the previous analysis, we readily find that the corresponding BS equation is Vt + (1/2)ó 2S 2VSS + (r – rf)SVS – rV = 0. Commodity Options When we deal with commodities, we have to take into account a cost of carry. That is, the storage of commodities costs money. It is convenient to introduce q as the fraction of the value of a commodity that goes towards paying the cost of carry. This means that just holding the commodity will result in a gradual loss of wealth, even if the commodity price remains fixed. Mathematically, for each unit of the commodity held,

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an amount qSdt will be required during short time dt to finance the holding. This is just like having a negative dividend. Therefore, we simply use the previous formulas to get Vt + (1/2)ó 2S 2VSS + (r + q)SVS – rV = 0. Options On Futures Our final modification of the BS equation is an equation for options on futures. Recall that the future price of a non-dividend paying equity F is given as follows F = er(T - t)S t, where T is the maturity date of the futures contract. The BS equation is written in terms of S and t. Now we want to rewrite this equation in terms of F and t. In other words, instead of V(S,t) we look for a function U(F(S,t),t) = V(S,t). You can notice that the function U is an example of composite functions which were discussed in week V lecture notes. Therefore, in order to differentiate this function, we have to use the chain rule. For example, VS = UFFS = er(T - t)UF, Vt = Ut + UFFt = Ut - rFUF, and so on. All in all, we find the following equation for a derivative on futures Ut + (1/2)ó 2F2UFF – rU = 0. Note that the futures can be understood as an asset paying a continuous dividend equal to the risk-free rate r. The equation for an option on futures is actually simpler than the BS equation. I think we have seen enough of differential equations. Now it is time for solutions.

Try To Answer The Following Questions 1) Can you explain what the risk-less portfolio is? 2) Can you apply the Ito lemma to the option on an asset following geometric Brownian motion? 3) Can you derive the Black-Scholes equation using the no-arbitrage principle? 4) What are the main assumptions of the Black-Scholes world? Which of them can be easily amended? 5) Can you explain how currency options are rela ted to options on dividend-paying equities? 6) Can you give examples of any other options?

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