Advanced Derivatives Course Chapter 11

Continues Time Methods In Finance

Week 11: American Style Options (see also Wilmott, Chapter 9)

Lecture X.1 The Perpetual American Put So far we have been mostly discussing European style options, that is, derivatives, which can be exercised only at a certain date. Due to this condition, the pricing problem was simplified to a large extent, allowing us to find the value of the options in exact form. American style options are much more flexible compared to European ones, since they can be exercised at any time prior to the expiration date. This flexibility might be a big bonus for the option holder, but it is certainly a big pain for those who have to value them. So far the only suitable method for pricing American options in our disposal was the binomial tree approach (‘back through the tree technique’). The goal of the next two lectures is to discuss two particular examples of American style options, which are not traded much in the market, but which are simple enough to illustrate the non-trivial aspects of the pricing problem for general American options. As we already know, American options are contracts that may be exercised early, prior to expiry. For example, if the option is a call, we may hand over the exercise price and receive the asset whenever we wish. Most traded shares and futures options are American style, but most index and currency options are European. The right to exercise at any time at will is clearly valuable. Because of that, the value of an American option cannot be less than an equivalent European option. We have already made this point before. But as well as giving the holder more rights, they also give him more headaches: when should he exercise? Part of the valuation problem is deciding when is the best to exercise. This is what makes American options much more interesting than their European cousins1. Perpetual Options There is a very simple example of an American style option that we can examine for the insight that it gives us in the general case. This simple example is the perpetual American put which was first considered by Merton. This contract can be exercised for a put payoff at any time. There is no expiry: that is why it is called a ‘perpetual’ option. So we can, at any time of our choosing, sell the underlying and receive an amount E. That is, the payoff is max(E – S, 0). We want to find the value of this option before exercise as well as the strategy for deciding when to exercise. •

The first point to note is that the value of the option should be independent on time, V(S). It can depend only on the price of the underlying, S. Indeed, as long as the

1

There is another type of options called Bermuda options. These options can be exercised prior to expiry but only on pre-agreed specific days. Because these options are between European and American styles, they are called Bermuda options.

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•

underlying price is the same, it does not matter when we enter this contract: the option must have the same value. Thus, only the share price may affect the value of the option. This is a property of perpetual options when the contract details are time-homogeneous, provided that there is a finite solution. If a contract has no finite solution, this contract has no financial significance. When we come to the general, non-perpetual, American option, we unfortunately loose this property. (‘Unfortunately’, since it makes it much easier to find the solution in this special case.) The second point to make, which is important for all American options, is that the option value can never go below the early-exercise payoff. We have proved this in week 2 lectures (page 14 of the lecture notes). In the case under consideration V ≥ max(E – S, 0).

While the option value is strictly greater than the payoff, it must satisfy the BS equation, like all the options on equities do. Since the value of the perpetual option does not depend on time, it must satisfy 1 2

σ 2S2

d 2V dV + rS − rV ≤ 0 . 2 dS dS

Note that the term with the time derivative dropped out and the partial derivatives with respect to S are replaced with total derivatives. This is the ordinary differential equation you get when the option value is a function of S only. The general solution of this secondorder ODE is V (S ) = A ⋅S + B ⋅ S

−

2r σ2

,

where A and B are arbitrary constants. (Exercise: verify that V(S) satisfies the BS equation.) The first part of this solution (that with coefficient A) is simply the asset: the asset itself satisfies the BS equation. If we can find A and B we have found the solution for the perpetual American put. We have to analyse the boundary conditions. We know that as the share value goes up, the value of the put goes down. That is, in the limit S →∞, the option must tend to zero. Clearly, in order to satisfy this condition, the coefficient A must be zero. Otherwise, V(S)→∞, as S→∞. So we have fixed one constant. What about B? Let us postulate that while the asset value is ‘high’, we wont exercise the option. But if it falls too low, we immediately exercise the option, receiving E – S. (Common sense tells us: we don’t exercise when S>E.) Suppose that we decide that S = S* is the value at which we exercise, i.e. as soon as S reaches this value from above, we exercise. How do we choose S*? When S = S*, the option value must be the same as the exercise payoff:

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V(S*) = E – S*. It cannot be less, as that would result in an arbitrage opportunity (as we have already established before), and it cannot be more or we would not exercise (if we did, we would loose money on the difference V(S*) – (E – S*).) Continuity of the option value with the payoff gives us one equation: −2 r

B( S *)

σ2

= E − S *.

But since both B and S* are unknown, we need one more equation. Let us look at the value of the option as a function of S*, by eliminating B using the above equation. We find that for S>S*  S  V ( S ) = ( E − S *)   S *

−

2r σ2

.

For S≤S* the solution is V(S) = E – S, since the option will be exercised for this range of the asset value. We are going to choose S* to maximise the option value at any time before exercise. In other words, what choice of S* makes V given by the formula above as large as possible? The reason for this is obvious: if we can exercise whenever we like, then we do so in such a way as to maximise our worth. We find this value by differentiating V(S) with respect to S* and setting the resulting expression equal to zero: 2r 2r  −  − ∂  2r  S  σ  1  S  σ (E − S *) =    ( − S * + 2 (E − S *)) = 0. ∂S *  σ  S *   S * S *   2

2

We find that S* =

E . σ2 1+ 2r

You can also check that the second derivative of V(S) with respect to S* is negative when S* takes the above value. Therefore, this choice of S* maximises V(S) for all S≥S*. Thus, we have found the value of the American perpetual put for all S≥S* with the given choice for S*: σ2 E    P( S ) = 2 r  1 + σ2 r  2

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1+

2r σ2

−

S

2r σ2

.

Continues Time Methods In Finance

We exercise the option as soon as the asset price reaches the level at which the option price and the payoff meet. This position, S*, is called the optimal exercise point. A Numerical Example Suppose that the share price, the strike, the volatility and the risk-free rate are €100, €110, 30%, 4%, respectively. Then the perpetual American put is worth     0.3 2  110  P= 2 × 0 .04  0 .3 2  1+   2 × 0 .04 

1+

2×0 .04 0.3 2

× 100

−

2×0. 04 0.3 2

= 32 .43 .

As you can see, the option is quite expensive. The Perpetual American Call With Dividends It is curious to find the value of a perpetual American call with dividends. The methodology is very similar to the one we used to price a perpetual American put. So I just write down the answer: 1 E  V ( S ) =  1  α 1− α 

1−α

Sα ,

 2 4 8r α = 12 − 2 (r − D − 12 σ 2 ) + (r − D − 12 σ 2 ) 2 + 2 4 σ σ  σ

 . 

It is optimal to exercise the option as soon as S reaches S* =

E 1 − α1

from below. An interesting special case is when D = 0, that is, when there is no dividend. Then the solution is V = S and S* becomes infinite. Thus, when there are no dividends on the underlying, it is never optimal to exercise the American perpetual call, irrespectively of the strike price. The value of the option simply coincides with the value of the share: the option and the share become indistinguishable. In other words, the share can be understood as an American perpetual call! ‘Ordinary’ American Options Let us try to apply some of the previous insight to general (non perpetual) American options. In this case, the option value is a function of both the share price S and time t.

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The pricing equation is exactly the same as for European options, that is, the BS equation. This is true, if we assume that the holder of the option always exercises optimally. However, if the holder makes a mistake, the writer of the option will make more than the risk-free return. The writer also makes more profit, if the holder has a poor estimate of the volatility of the underlying and exercises in accordance with that estimate. There are three conditions to be fulfilled. •

If the payoff for early exercise is P(S,t), possibly time-dependent, then the noarbitrage constraint V(S,t) ≥ P(S,t),

•

must apply everywhere. At expiry we have the final condition V(S,T) = P(S,T).

•

The option value is maximised, if the owner of the option exercises so that ∆=

∂V ∂S

is continuous. The American option valuation problem consists of the BS equation and the three (boundary) conditions mentioned above. Now if we substitute the BS solution for a European call in the absence of dividends, then the BS equation is clearly satisfied. Moreover it will satisfy all the boundary conditions for an American call without dividends. The conclusion is that the value of an American call option is the same as the value of a European call option, when the underlying pays no dividends. This is just another way of arriving at the well known result. In the given circumstances, to exercise an American call before expiry would be ‘subNone of this is true, if there are dividends on the underlying. Again, to see this, simply substitute the expression for the European call on a dividend-paying asset into the ‘noarbitrage constraint’ V(S,t) ≥ P(S,t). Since the European call option has a value which approaches Se-D(T – t) as S→∞, there is clearly a point at which the European value fails to satisfy the given constraint (because Se-D(T – t) < S). If the constrain is not satisfied somewhere, then the problem has not been solved anywhere. This is very important: our solution must satisfy the inequalities everywhere or the solution is invalid. Mathematically, the problem for the American option is what is known as a free boundary problem. In the European option problem, we know that we must solve for all

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values of S from zero to infinity. When the option is American, we do not know a priory where the BS equation is to be satisfied: this must be found as part of the solution. This means that we do not know the position of the early exercise boundary. Moreover, except in special and trivial cases, this position is time-dependent. For example, we should exercise the American put, if the asset value falls below S*(t), but how do we find S*(t)? Not only is this problem much harder than the fixed boundary problem, but this also makes the problem non-linear. That is, if we have two solutions of the problem, we do not get another solution, if we add them together. The reason is that two contracts put together have only one exercise opportunity. However, what might be optimal for one solution, might not be optimal for the other. Therefore, the optimisation problem for two options in one solution is different from two options in two different solutions. If the contracts were both European, then the sum of the two separate solutions would give the correct answer: the European valuation problem is linear.

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Lecture X.2 A Bit Of Exotics: Russian Option

In a Horizon program on BBC2 (2.12.99), you could watch what a dramatic role Russia played in the fate of the LTCM hedging fund and the worldwide derivative trade as a whole. Perhaps because of this, currently, any combinations of the words ‘Russia’ and ‘derivatives’ created (until recently) a slight panic among derivative traders. Imagine now a financial instrument called ‘Russian option’… This will be the subject of the present lecture. A Russian option2 is a perpetual American3 lookback option, which, at any time chosen by the holder, pays out the maximum realised asset price up to that date. Such options give the holder an extremely advantageous payoff and they are therefore relatively expensive. However, the valuation of this option will allow us to touch upon many features of more ‘practical’ lookback derivatives. To make the problem interesting, we assume that there is a continuously paid constant dividend yield: without dividends the problem is trivial. As the time horizon is infinite, the option is independent on time and is a function of the share price, S, and the maximum realised value of the asset price, which we will denote J. The value of the option satisfies the time-independent Black-Scholes equation: 1 2

σ 2S 2

d 2V dV + (r − D )S − rV ≤ 0 2 dS dS

with the following boundary condition dV =0 dJ on J = S. This condition means that the value of the option does not depend on J when the share price is equal to its maximum realised so far. Let us explain this condition. The maximal share value realised so far, J, by definition is greater than S. Otherwise, J would not be the maximum. The derivative of V with respect to J is the speed with which the option value changes when J changes. As J approaches S, this speed must vanish. Otherwise, it will be possible for the function V(J) to change through the point J = S. This will contradict the assumption that J is the maximal value realised so far. Thus, we come to the conclusion that the value of the option must not depend on J when S = J. A quick prove of the above formula is to note that, because the payoff of V is equal to J, at any given share value S the function V(J) must reach its minimum for J = S. This proves the above formula. 2

The name of the option is after the mathematical method used for valuing this derivative developed by a Russian mathematician. 3 Since the option is both Russian and American, it should probably be called Alaskan option.

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The option value must also satisfy V ≥ J, since the option is American and the right-hand side of this inequality is the ‘early’ exercise payoff (since the option is perpetual, any exercise is an early exercise.) Otherwise, there would be an arbitrage opportunity. There must exist a free boundary, i.e. a certain value of the share, S*, at which the option is exercised, since this option is useless, if it is never exercised. On the boundary both V and dV/dS must be continuous. Let us seek a solution in the form V = J × W(ξ), where ξ = S/J. Then the BS equation can be represented in the following form (1/2)σ2 ξ2W′′ + (r – D) ξW′ – rW = 0, where ′ denotes d/dξ. Suppose that the free boundary is at ξ = ξ0. Then the boundary conditions become W – W′ = 0

at ξ = 1

and W = 1,

W′ = 0 at ξ = ξ0.

The latter comes from continuity of dV/dJ at S = S*. Otherwise, there will be arbitrage opportunities. The general solution is given by W = A × ξα for constant A and α. Taking into account the boundary conditions, the solution for the Russian option is found to be α  J   S α α +   − α −  S   V ( S , J ) =  α + − α −   S *   S *   J, −

where

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+

 , S * ≤ S ≤ J ,  ,  0 ≤ S ≤ S *.

Continues Time Methods In Finance

α ± = σ1 [− r + D + 12 σ 2 ± (r − D − 12 σ 2 ) 2 + 2σ 2 r ], 2

α (1 − α − )  α S* = J  +  α − (1 − α + ) 

1 − −α +

.

When the dividend yield is zero, i.e. D = 0, the problem does not have a solution. It is, clearly, never optimal to hold such an option, when the underlying does not pay dividends. A Numerical Example Suppose that the share price, J, the dividend, the volatility and the risk-free rate are €108, €110, €5, 30%, 4%, respectively. Then the boundary is at S* = € 105.31 and the Russian option is worth €110.10. It is an awful amount of money for one option! The Russian option is extremely expensive. This is why it is not traded much in the market.

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Try To Answer The Following Question Have I learnt anything useful in this course?

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