A Review and Implementation of Option Replication in Discrete Time with Transaction Costs T.A. McWalter 9 December 2002

Abstract The problem of option pricing and replication in the presence of transaction costs is considered in this report. A brief literature survey identifies the major approaches to the problem. Initially an error analysis is performed revealing the effect of discrete-time hedging updates. Two of the dominant paradigms to hedging in the presence of transaction costs are then presented. A Monte Carlo implementation with a Value at Risk comparison of these algorithms is performed thus providing insight into the problem of hedging under a more realistic set of market assumptions.

Contents 1 Introduction 1.1 Literature to be Reviewed . . . . . . . . . . . . . . . . . . . . . . 1.2 Document Structure . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5

2 Discrete Time 2.1 Hedging Error Over a Discrete Time Step . . . . . . . . . . . . . 2.2 Discrete Hedging Over Multiple Time Steps . . . . . . . . . . . .

6 6 9

3 A Local-in-Time Method 12 3.1 The Method of Leland . . . . . . . . . . . . . . . . . . . . . . . . 12 4 A Global-in-Time Method 4.1 The Hamilton-Jacobi-Bellman Equation . . . . . . 4.1.1 An Intuitive Proof . . . . . . . . . . . . . . 4.2 The Merton Problem . . . . . . . . . . . . . . . . . 4.3 An Optimal Control Approach to Pricing Options 4.4 The Whalley-Wilmott Hedging Algorithm . . . . . 4.5 An Extended Algorithm . . . . . . . . . . . . . . .

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17 17 18 19 20 24 25

5 Numerical Comparisons 5.1 Parameters used in Monte Carlo Simulations 5.2 Value at Risk . . . . . . . . . . . . . . . . . . 5.3 Results . . . . . . . . . . . . . . . . . . . . . . 5.4 The Choice of Risk Aversion Constant . . . .

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27 27 27 28 32

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6 Conclusion 33 6.1 A Control Theory Derivation of the Black-Scholes Delta . . . . . 34 6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1

List of Figures 2.1 2.2 2.3

The distribution of hedging error for a single time step . . . . . . The distribution of Profit and Loss for multiple updates . . . . . Standard deviation of P&L as a function of strike Price . . . . .

8 9 11

3.1

Comparison of P&L distributions for Black-Scholes and Leland .

15

4.1 4.2

Decision space indicating buy, sell and no-transaction regions . . 23 Comparison of P&L distributions for Leland and Whalley-Wilmott 25

5.1 5.2 5.3 5.4 5.5 5.6

Comparison of P&L distributions showing VAR . . . . Mean P&L as a function of number of updates . . . . VAR as a function of updates . . . . . . . . . . . . . . Cumulative transaction costs as a function of updates Mean P&L excluding costs as a function of updates . . VAR as a function of risk aversion . . . . . . . . . . .

2

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29 30 30 31 31 32

Chapter 1

Introduction In the derivation of the Black-Scholes-Merton differential equation a number of simplifying assumptions are necessary [1][11, page 245]. In particular two of these assumptions are that ‘There are no transaction costs or taxes’ and ‘Security trading is continuous’. These are particularly problematic in the real-world pricing and hedging of options since they are mutually antagonistic concepts. Assuming transaction costs, the closer continuous rebalancing of the hedge portfolio is approximated, the more expensive it becomes to maintain the hedge. In fact, due to the infinite variation of the Brownian motion process, rebalancing the portfolio continuously using the Black-Scholes hedging scheme results in an infinite cost of hedging. This leads to an infinite option price which seems paradoxical. In the case of a call option for instance, it would be cheaper to buy a single share for every contract held as this dominates the pay-off function. On the other hand allowing the intervals between rebalancing to become larger incurs smaller total transaction costs. This however produces larger errors in the hedging portfolio due to hedge slippage and a situation where the writer is left exposed to the risk of the market. In this report these concepts and the practical approaches to creating realistic hedging strategies under the relaxed assumptions will be explored.

1.1

Literature to be Reviewed

Initially it is necessary to understand and analyze hedging in a discrete time framework without transaction costs [4] [19]. Boyle and Emanuel [2] in particular examine the distribution of errors in the hedging portfolio for short fixed intervals and show that, ignoring certain higher order terms in the Taylor expansion, the expected value of the hedging errors is zero. They also indicate that the error distribution follows that of a chi-squared distribution with one degree of freedom. When considering multiple (n) updates, the distribution is specified by a chi-squared distribution with n degrees of freedom. The variance

3

of the distribution is investigated in the paper by Kamal and Derman [12] 1 and shown to be proportional to the Vega of the option. Armed with the statistical insight from these papers it is now possible to tackle the issue of transaction costs. These discrete-time results were expanded by Leland [14] to incorporate the effect of proportional transaction costs resulting in a modified volatility option pricing formula. Since his seminal paper on the pricing and replicating of options in the presence of transaction costs there have been a large number of publications on the issue. In general three distinct approaches to the problem have been proposed [25]: super-replication, imperfect replication, and utility maximization. The super-replication approach does not attempt to replicate the option exactly but proposes strategies that provide a replication portfolio that dominates the option value at maturity almost surely. In an article by Soner, Shreve & Cvitani´c [20] it is proved, assuming transaction costs, that the cheapest portfolio to dominate the value of a call option at maturity is a single stock of the underlying held long for the duration of the option. Although useful as a limiting case, this is of course not practically applicable as an option pricing or replicating strategy. In essence this paper indicates that no perfect replication strategy exists when transaction costs are incurred and that there is always an element of risk that the writer of an option contract is exposed to. Imperfect replication methods introduce criteria for the tracking of hedging errors. If at discrete time steps these errors can be shown to be independent, then it is possible to formulate strategies which allow replication with zero expected error. Although the Black Scholes pricing is the basis for this approach, substantial deviations are proposed, in particular a number of these methods propose modified volatilities to be used in pricing. Papers that typify this approach are those by Leland [14], the binomial model of Boyle & Vorst [3] and Hoggard, Whalley & Wilmott [10] who have modified the Leland approach to a more general cost structure. By specifying a utility function that relates risk and return in a quantifiable fashion it is possible to repose the hedging problem as a stochastic control problem. Hedging then occurs when the risk of loss outstrips the transaction costs required to rebalance the portfolio. In contrast to imperfect replication techniques, this allows the error of hedging to be specified in an ‘exogenous’ fashion using a risk aversion parameter. This approach was originally proposed for the problem of portfolio selection under transaction costs by Davis & Norman [5]. It was then first applied to option replication under transaction costs by Hodges & Neuberger [9] and later expanded by and Davis, Panas & Zariphopoulou [6]. The latter model was then further simplified from a computational perspective by Whalley & Wilmott [21] [22] and more recently Monoyios [18] has provided another computational framework for the model. An alternative way to classify these hedging algorithms is on the basis of the update scheme used [21]. In general imperfect replication schemes like that of Leland are local-in-time models. This means that a fixed time update scheme 1 Thankyou

to Keith Mitchell for pointing out this reference.

4

is assumed, in particular a ∆t is selected and the portfolio is updated at these fixed intervals. The pricing and hedging regimen is thus directly affected by the choice of time increment. Alternatively, the control theory approach leads to a global-in-time model in which the decision to update is independent of the time period. This implies that the hedge portfolio and underlying stock price are monitored in as close to a continuous fashion as is practically possible. A decision criterion indicates when updates to the portfolio should be made.

1.2

Document Structure

The remainder of this document is structured in the following manner. Chapter 2 performs an error analysis on the hedging inefficiency introduced as a result of moving from a continuous case to a discrete-time update. Both the single period and multiple update cases are investigated. In Chapter 3 the idea of local-in-time methods is discussed and a presentation of the modified variance method of Leland is given. Chapter 4 introduces the control theory approach to option pricing and replication. The fundamental Hamilton-Jacobi-Bellman equation is introduced with an intuitive proof and an initial application to the Merton problem is provided as an example of its use. Then a presentation of the Davis Panas Zariphopoulou paper is given. Finally an asymptotic approximation to this approach is presented. In order to compare the approaches on a rational basis, the Value at Risk methodology is employed in Chapter 5. A discussion of the results analyzes the performance of the different hedging schemes implemented. Chapter 6 concludes the report summarising major results and proposing further work in the area.

5

Chapter 2

Discrete Time The original Black-Scholes option price derivation employs a continuous time update of the hedging portfolio to ensure the removal of all risk. The moment discrete update points are specified, the portfolio becomes risky again. If these updates are numerous and performed at fixed intervals, the expected return of the hedge portfolio will be zero, in other words the mean price is still the same as the Black-Scholes value [4]. However, unless the number of updates tends to infinity, there remains a distribution of profits and losses. In this chapter an analysis of the error introduced as a result of updating the hedge discretely in time is presented. This is performed for a single time step and is then generalised to multiple time steps.

2.1

Hedging Error Over a Discrete Time Step

In the following analysis the approach of Boyle and Emanuel [2] has been followed, although the notation has been changed to allow for the continuity of development. It is assumed that a call option and an associated proportion of stock and cash are held in a portfolio. The standard (discrete) log-normal stock price process is also assumed √ ∆S = µ∆t + σ ∆t S

(2.1)

where  is drawn from a normal distribution with zero mean and unit variance (i.e.  ∼ N (0, 1)). Given a call option C(S, T ) with strike price K on the underlying stock price process, the Taylor series expansion of the option, in terms of changes in stock and time, is given by ∆C =

∂C 1 ∂2C 1 ∂2C 1 ∂2C 2 ∂C 2 ∆S + ∆t + ∆S + ∆S∆t + ∆t + · · · (2.2) ∂S ∂t 2 ∂S 2 2 ∂S∂t 2 ∂t2

6

substituting equation (2.1) into (2.2) and eliminating higher order terms, the following representation for a change in the call option is obtained ∆C =

∂C 1 ∂2C 2 2 2 ∂C ∆S + ∆t + S σ  ∆t + O(∆t3/2 ) ∂S ∂t 2 ∂S 2

(2.3)

Now consider a portfolio π consisting of the call option held long and a short position of ∂C ∂S (= N (d1 )) shares in the underlying. The change in value of this portfolio over a small increment of time ∆t is

⇒

π

=

∆π

=

∂C S ∂S ∂C ∆C − ∆S ∂S C−

(2.4)

The initial investment required for π is borrowed at the riskless rate r, thus the hedging error (HE) over the time period is given by HE

= π + ∆π − πe−r∆t = ∆π − πr∆t + O(∆t2 )

(2.5)

where the small value expansion for the exponent was used. Substituting equation (2.4) into (2.5) gives HE = ∆C −

∂C ∂C ∆S − (C − S)r∆t + O(∆t2 ) ∂S ∂S

(2.6)

It is now possible to use the Black-Scholes formula (C = SN (d1 )−Ke−rT N (d2 )) and the fact that ∂C ∂S = N (d1 ), to show that equation (2.6) becomes HE = ∆C −

∂C ∆S + rKe−rT N (d2 )∆t + O(∆t2 ) ∂S

Substituting expression (2.3) for ∆C into this equation gives HE = (

1 ∂2C 2 2 2 ∂C + rKe−rT N (d2 ) + S σ  )∆t + O(∆t3/2 ) ∂t 2 ∂S 2

Now

∂C 1 ∂2C 2 2 =− S σ − rKe−rT N (d2 ) ∂t 2 ∂S 2 Thus neglecting higher powers of ∆t HE =

1 ∂2C 2 2 2 S σ ( − 1)∆t 2 ∂S 2

(2.7)

This illustrates that the hedging error over a single time period is proportional 2 to the Black-Scholes Gamma (Γ = ∂∂SC2 ) and the time increment (∆t). It is also proportional to the square of the stock price and the volatility of the stock price process. 7

Distribution of Hedging Error (Single Step)

0.45

Equation 2.7 Monte Carlo Simulation

0.4

0.35

Probability

0.3

0.25

0.2

0.15

0.1

0.05

0 −0.2

0

0.2

0.4

0.6 Hedging Error

0.8

1

1.2

1.4

Figure 2.1: The distribution of hedging error for a single time step The stochastic component of this quantity is dependent on the square of a normal variable and is distributed as chi-squared with one degree of freedom. In order to verify this a simulation was written which performed a Monte Carlo experiment in which fully hedged portfolios were allowed to drift under the influence of the stock price process for a small time increment, in this case a single day1 . The distribution of profits and losses were then compared with the distribution as derived in equation 2.7. Figure 2.1 shows the output from the simulation. As can be seen in the figure, the returns are highly skew since the hedging error is negative when |  |< 1. This ensures that −1 ≤ HE < 0 about 68% of the time. In the event that |  |> 1, corresponding to a relatively large stock price movement over the time period, the hedging error is positive and may correspond to a larger amount. This of course may be significant when considering the fact that real stock price processes are leptokurtic (i.e. have fatter tails). It is important while interpreting this analysis to remember that in general the writer of an option would be holding the option short. The profit and loss would therefore be reversed (i.e. the losses are in fact gains and vice versa). In this context there is a large probability of making a small profit and a small probability of making larger losses. 1 See

Chapter 5.1 for a discussion on the values of the parameters used in this simulation.

8

Distribution of Hedging Error (Multiple Steps)

0.12

0.1

Probability

0.08

0.06

0.04

0.02

0 −4

−3

−2

−1

0 1 Hedging Error

2

3

4

5

Figure 2.2: The distribution of Profit and Loss for multiple updates

2.2

Discrete Hedging Over Multiple Time Steps

An extension of the analysis of the hedging error to multiple time steps is now possible. This analysis is based on the paper by Kamal and Derman [12] and as before the notation has been partially changed to facilitate continuity. Assume that at each of n time steps the hedge portfolio is rebalanced, the final profit and loss (P &L) can be expressed as an extension of equation (2.7), being the sum of the hedging errors at each update. F inal P &L =

n X 1 i=1

2

2 Γi−1 Si−1 σ 2 (2i − 1)∆t

where Γi and Si are evaluated at time ti = i∆t and the i ∼ N (0, 1) are assumed independent and identically distributed (i.i.d.). Figure 2.2 shows the output of a Monte Carlo experiment that simulates this for the call option rehedged on a daily basis over the life of the option. The distribution looks similar to a normal distribution but is skewed in the positive direction. It is in fact a chi-squared distribution, since the sum of n chi-squared distributions with one degree of freedom is a chi-squared distribution with n degrees of freedom [8, page 122]. It is now possible to consider the variance of the P &L. This is expressed as the following expectation   n X n X σP2 &L = E  Ai Aj (2i − 1)(2j − 1) (2.8) i=1 j=1

9

where Ai =

1 2 Γi−1 Si−1 σ 2 ∆t 2

From the properties of the Brownian motion, i and j are independent for i 6= j and Ai is independent of i since it depends on the previous time step. Thus only the diagonal terms of (2.8) remain. Noting also that E[2i ] = 1 and E[4i ] = 3 we have the following expression σP2 &L

=

n X

2E[A2i ]

i=1

=

n X 1 i=1

2

  4 (σ 2 ∆t)2 E Γ2i−1 Si−1

(2.9)

Performing Gaussian integration2 , it can be shown that s  2 4 T2 4 2 E Γi S i = S 0 Γ0 φ(ti ) 2 T − t2i where φ(ti )

=

  (µ − r)2 t2i (µ − r)ti √ − exp 2µt − 2d1 σ2 T σ T    (µ − r)ti ti (µ − r)2 t2i √ exp d22 + 2d2 + σ2 T T + ti σ T

Thus equation (2.9) becomes σP2 &L

n X 1 2 2 2 = (S0 Γ0 σ ∆t) 2 i=1

s

T2 φ(ti−1 ) T 2 − t2i−1

(2.10)

For small values of µ and r and strikes close to S0 , so that d1 and d2 are small, φ(ti ) ' 1. It is now possible to use the following approximation3 s Z Tr n X 1 πT π T2 T2 ' dt = = n 2 2 2 2 T − t ∆t T − t 2∆t 2 0 i−1 i=1 Thus equation (2.10) becomes σP2 &L = Since S02 Γ0 =

π n(S02 Γ0 σ 2 ∆t)2 4

1 ∂C σT ∂σ ,

σP2 &L

=

π n 4



σ∆t ∂C T ∂σ

2

2 This is an involved algebraic calculation and has not been verified here, see Kamal and Derman [12] for details. R further
3 Note that √ dx +C = sin−1 x a 2 2 a −x

10

⇒

σP &L

=

π 4n

=

r



∂C σ ∂σ

2

σ π (κ0 ) √ 4 n

(2.11)

Thus it has been shown that the profit and loss is directly proportional to the option’s Vega and volatility and inversely proportional to the square root of the number of updates. As Kamal and Derman point out, the intuitive way of understanding this result is that when the hedging updates are preformed discretely the portfolio is in some sense sampling the underlying volatility directly and there is likely to be approximation error in the sampling of the stock process volatility. The hedging error is thus sensitive to the change in option price √ with respect to a change in volatility. This directly relates to the factor of 1/ n. As the number of updates increases, the error in estimation of sample volatility decreases proportionally to the inverse root of the number of updates. This is analogous to the same factor that appears in the central limit theorem [8, page 194] In order to verify the results derived, a Monte Carlo experiment was performed and the standard deviations of the cumulative P&Ls were compared to the expression derived. The results are shown in figure 2.3. Note that the approximation 2.10 consistently underestimates the standard deviation of P&L while equation 2.11 overestimates at strikes close to initial stock price and underestimates at strikes with a larger deviation from initial stock price. Standard Deviation of Hedging Error

1

0.95

Standard Deviation

0.9

0.85

0.8

0.75

0.7

0.65 95

Equation 2.10 Equation 2.11 Monte Carlo Simulation 100

Strike Price (Note So=100)

105

110

Figure 2.3: Standard deviation of P&L as a function of strike Price

11

Chapter 3

A Local-in-Time Method Transaction costs introduce an interesting tradeoff. It has been previously mentioned that in order to eliminate risk in a hedging portfolio, continuous trading is necessary. Continuous adjustment of the portfolio in the presence of non-zero transaction costs results in unbounded prices for options since the Brownian motion processes have infinite variation resulting in an infinite cost of rebalancing. This leads to the obviously untenable situation that the transaction costs may become larger than the price of the underlying stock. The obvious solution to this problem is to assume that trading happens at discrete intervals and to assume that the error analysis put forward in the previous chapter holds. This will allow for a modification of the Black-Scholes strategy that imposes a risk versus cost tradeoff resulting in a limit on the total transaction costs incurred. Local-in-time methods employ a fixed discretization of time as an approach to limit costs. At the outset a fixed time hedging regimen is determined and on the basis of the time period between updates (∆t) the price of the option is determined. Any change in this time period leads to a different option price. The seminal work in this field was published by Leland [14] He assumes fixed update periods and proposes a modified volatility related to the proportional transaction costs and the size of update period in order to bound the maximum total transaction costs. This chapter will review the method of Leland.

3.1

The Method of Leland

One approach to hedging in the presence of transaction costs could be to charge a premium for the fact that transaction costs are incurred (perhaps the expected transaction cost over the number of updates) and use the standard Black-Scholes update at each time step. Leland criticizes this approach on three grounds • The expected transaction cost has no known closed form computation and is difficult to determine through other means

12

• Transaction costs are correlated with changes in stock price (i.e. transaction costs are path dependent) • Transaction costs and their uncertainty become very large as ∆t → 0 Leland proposes a modified variance approach to the problem. Let k represent the proportional symmetric transaction cost measured as a fraction of turnover. The same notation and stock price process (equation 2.1) used in Chapter 2 are assumed here. Now define a modified variance   ∆S k 2 2 σ ˆ = σ 1+ 2 E (3.1) σ ∆t S p √ ∆S ' 2/πσ ∆t where E S

The expression for the expected value is derived as follows √ ∆S = E µ∆t + σ ∆t E S √ ≤ |µ|∆t + σ ∆tE|| Z ∞ √ x2 2 √ = |µ|∆t + σ ∆t xe− 2 dx 2π 0 p √ = |µ|∆t + 2/πσ ∆t

It is possible to ignore the first term (|µ|∆t) since the second term will dominate in value when considering small ∆t. This is also convenient since it is in general very difficult to measure the drift of real stock price processes. Now price the option using the modified variance ! ∂ Cˆ rT S Cˆ = SN (dˆ1 ) − Ke N (dˆ2 ) +k ∂S

The term in brackets indicates that an additional amount for the cost of purchasing the number of shares in the initial hedge portfolio needs to be factored into the option price. It can now be shown that the expected error of the hedging portfolio (π) is zero, where ∂ Cˆ S − Cˆ π= ∂S In a similar manner as before (cf. equation 2.6) the hedging error is specified. Note that the signs have been changed since the portfolio now contains a short position in the option and a long position in the stock. HE =

∂ Cˆ ∂ Cˆ ˆ ∆S − ∆Cˆ − ( S − C)r∆t − T C + O(∆t2 ) ∂S ∂S

13

(3.2)

The extra term T C is the transaction cost incurred when rebalancing the portfolio at the next time step and is given by ! 1 ∂ Cˆ TC = k ∆ (S + ∆S) 2 ∂S 1 ∂ 2 Cˆ k 2 ∆S(S + ∆S) + O(∆t3/2 ) = 2 ∂S 1 ∂ 2 Cˆ 2 ∆S = + O(∆t3/2 ) (3.3) k S 2 ∂S 2 S In this instance it is convenient to use the following Taylor expansion as an approximation for the change in value of the option.  2 ∂ Cˆ ∂ Cˆ 1 ∂ 2 Cˆ 2 ∆S ˆ ∆C = + O(∆t3/2 ) ∆S + ∆t + S ∂S ∂t 2 ∂S 2 S

(3.4)

Substituting (3.3) and (3.4) into (3.2) gives

but

 2 ∂ Cˆ 1 ∂ 2 Cˆ 2 ∆S ∂ Cˆ S)r∆t − ∆t − S HE = (Cˆ − ∂S ∂t 2 ∂S 2 S 2ˆ 1 ∂ C ∆S − k 2 S 2 + O(∆t3/2 ) 2 ∂S S (Cˆ −

(3.5)

1 ∂ 2 Cˆ 2 2 ∂ Cˆ ∂ Cˆ S)r = S σ ˆ + ∂S 2 ∂S 2 ∂t

so (3.5) becomes " #  2 ∆S 1 ∂ 2 Cˆ 2 2 ∆S + O(∆t3/2 ) S σ ˆ ∆t − HE = − k 2 ∂S 2 S S

(3.6)

Substituting expression (3.1) for σ ˆ 2 and reordering terms, (3.6) becomes "  #  2  ∆S ∆S ∆S 1 ∂ 2 Cˆ 2 2 − + O(∆t3/2 ) (3.7) S σ ∆t − + k E HE = 2 ∂S 2 S S S

Ignoring O(∆t3/2 ) terms and taking an expectation gives the result E(HE) = 0. As with the standard Black-Scholes strategy, this replicating strategy yields max[S − K, 0]. Analyzing the terms in square brackets of equation 3.7 the expression 2

σ ∆t −



14

∆S S

2

is O(∆t) while the expression   ∆S ∆S − k E S S

is O(∆t1/2 ). This means that as ∆t → 0 the transaction costs will tend to dominate the hedging error. Consider the limiting cases. Under arbitrarily small time updates this approach is better behaved than a standard Black-Scholes hedging regime. When a continuous update is approximated (∆t → 0), the modified variance becomes unbounded (ˆ σ → ∞). This means that the price predicted is equal to the current price of a single share and that at each time step the modified Delta is equal to one share (Since dˆ1 → ∞ and dˆ2 → 0). This indicates that in the limit of continuous updates the method enforces a situation where a single share is held for every option contract. This strategy naturally dominates the payoff of the option and anticipates the super-replication result of Soner Shreve and Cvitani´c [20]. In the case where transaction costs are made arbitrarily small (k → 0) this approach becomes the standard Black-Scholes approach and the option price is not affected by the choice of ∆t. Figure 3.1 shows the distribution of HE on a portfolio which includes proportional transaction costs of 1%. The initial premium used in both cases is the amount predicted by the modified volatility method of Leland. The price of

0.12

Distribution of Hedging Error Black−Scholes Leland

0.1

Probability

0.08

0.06

0.04

0.02

0 −5

−4

−3

−2

−1 Hedging Error

0

1

2

3

Figure 3.1: Comparison of P&L distributions for Black-Scholes and Leland

15

the option1 was 5.90 corresponding to a difference of 23% on the Black-Scholes price for the same option. The modified volatility was 0.496 compared with the stock price process of 0.4. The number of updates was 60, corresponding to a twice daily update period over the one month life of the option.

1 See

Chapter 5.1 for a discussion on the values of the parameters used in this simulation.

16

Chapter 4

A Global-in-Time Method In the previous chapter local-in-time methods were described and it was noted that the updates were directly dependent on the time increment factor selected. Global-in-time methods use discriminating factors that are not dependent on a fixed time update. In this chapter one such method will be reviewed. This algorithm overcomes the problem of large cumulative transaction costs by providing a decision rule that is used to monitor the share price at every time instant and decide whether or not a hedge update should be implemented. It should be noted that the Leland algorithm, although more efficient than Black Scholes is not optimal in any well defined mathematical sense. The method described here uses a control optimization procedure to provide an optimal trade-off between the level of risk selected and cost. In order to derive the equations necessary for the global-in-time method, a review of the Hamilton-Jacobi-Bellman [7][13][16] equation of optimal control will be necessary. The Merton problem is solved as an example to illustrate the use of the HJB equation before the relevant theory relating to options with transaction costs is developed.

4.1

The Hamilton-Jacobi-Bellman Equation

The HJB equation is one of the fundamental equations of dynamic control theory. The derivation given here, which is similar to that presented by Kao[13, page 409], is simple and may lack some of the finer points of a more rigorous analysis but it shows the basic principles in a succinct fashion. Given a Brownian motion W on a probability space (Ω, F, P ), consider the following optimization problem over [0, T ] (Z ) T

J(0, x0 ) = max E π

subject to

f (X, π, s)ds + F (X(T ), T )

0

dX = µ(X, π, t)dt + σ(X, π, t)dW

17

and

X(0) = x0

(4.1)

where f (.) is a running reward function, F (.) is a terminal reward function and π is an admissible trading strategy. The HJB equation, subject to the relevant boundary conditions, is given by   1 2 −Jt (t, x) = max f (x, π, t) + Jx (t, x)µ(x, π, t) + Jxx (t, x)σ (x, π, t) (4.2) π 2 Here the subscripts represent the partial derivative with respect to the subscripted variable. This equation provides the optimal solution to the system 4.1 in the form of a partial differential equation. It is still necessary to solve the relevant free boundary problem in order to arrive at a final solution.

4.1.1

An Intuitive Proof

For any 0 < t < T , we can define J(t, x) similarly, being the optimal reward over [t, T ] (Z ) T

J(t, x) = max E

f (X, π, s)ds + F (X(T ), T )

π

subject to

t

dX = µ(X, π, t)dt + σ(X, π, t)dW

and

X(t) = x

Using the principle of optimality in dynamic programming this equation can be rewritten recursively as ) (Z t+∆t

f (X, π, s)ds + J(t + ∆t, x + ∆x)

J(t, x) = max E π

(4.3)

t

The integral is approximated by Z t+∆t f (X, π, s)ds = f (x, π, t)∆t + O(∆t2 )

(4.4)

t

and J(.) is approximated by the Taylor series expansion J(t + ∆t, x + ∆x)

=

J(t, x) + Jt (t, x)∆t + Jx (t, x)∆X 1 1 + Jtt (t, x)∆t2 + Jxx (t, x)∆X 2 2 2 +Jxt (t, x)∆t∆X + O(∆t2 )

Substituting (4.4) and (4.5) into (4.3) gives   f (x, π, t)∆t + J(t, x) + Jt (t, x)∆t + Jx (t, x)∆X J(t, x) = max E + 21 Jxx (t, x)∆X 2 + Jxt (t, x)∆t∆X + O(∆t2 ) π Which can be rewritten as   f (x, π, t)∆t + Jt (t, x)∆t + Jx (t, x)E[∆X] 0 = max + 12 Jxx (t, x)E[∆X 2 ] + Jxt (t, x)∆tE[∆X] + O(∆t2 ) π 18

(4.5)

Now E[∆X] = µ(x, π, t)∆t and E[∆X 2 ] = σ 2 (x, π, t)∆t thus   f (x, π, t)∆t + Jt (t, x)∆t + Jx (t, x)µ(x, π, t)∆t 0 = max + 21 Jxx (t, x)σ 2 (x, π, t)∆t + O(∆t2 ) π Dividing through by ∆t and ignoring higher order terms (since ∆t → 0) gives the result.

4.2

The Merton Problem

In order to illustrate the use of the HJB equation it is now applied to the Merton investment and consumption problem. One of the seminal works in the area of mathematical finance, the Merton portfolio selection problem [15] and its solution were a necessary contribution for the Black-Scholes theory to arise. The problem as stated here significantly deviates from the original presentation by Merton but leads to the same result. An investor trades to maximize expected utility of wealth at some terminal time T V (x) = sup E[U(XTx,π )] π

XTx,π

where is the wealth at T when using trading strategy π and starting with initial wealth X0 = x. The ‘two-asset’ market consists of a risky stock (St )0≤t≤T following the process dSt = µSt dt + σSt dWt and a riskless asset with constant interest rate r. Let the admissible trading strategy πt = ∆Xt St t be the fraction of wealth in the stock, where ∆t is the number of shares held at time t, then dXtx,π = dXt

= =

(Xt − ∆t St )rdt + ∆t dSt [r + (µ − r)πt ]Xt dt + πt σXt dWt

This simple system can be written in terms of equation (4.1) as f (X, π, t) F (X(T ), T ) µ(X, π, t) σ(X, π, t)

=

0

= U(XTx,π ) = [r + (µ − r)πt ]Xt =

πt σXt

Thus the HJB equation (4.2) is given by   1 2 −Jt (t, x) = max Jx (t, x)[r + (µ − r)πt ]Xt + Jxx (t, x)(πt σXt ) πt 2 with the boundary condition J(T, XT ) = U(XT ). Rearranging terms gives   1 max Jxx (t, x)πt2 σ 2 Xt2 + Jx (t, x)(µ − r)πt Xt + Jx (t, x)rXt + Jt (t, x) = 0 πt 2 (4.6) 19

Equation (4.6) is maximized by the following choice of πt π∗ =

−Jx (t, x)(µ − r)Xt Jxx (t, x)σ 2 Xt2

Taking a trial solution of J(t, x) = b(t)U(x) with x = Xt and b(T ) satisfying the boundary condition b(T ) = 1 and assuming a log utility function U(x) = log x gives Jx (t, x) =

b(t) , Xt

Jxx (t, x) = −

Thus π∗ =

µ−r σ2

More generally, for a utility function of U(x) = is the risk aversion parameter, then π∗ =

b(t) Xt2

(xγ −1) , γ

where γ < 1 and γ 6= 0

µ−r σ 2 (1 − γ)

This is the now famous Merton proportion.

4.3

An Optimal Control Approach to Pricing Options

This section explores a significantly different approach to pricing options first presented by Hodges and Neuberger [9] and later expanded by Davis, Panas and Zariphopoulou [6]. The derivation here follows the same approach as that of Davis, Panas and Zariphopoulou (DPZ). Although it has been mentioned that the hedging of options in the presence of market frictions and in discrete time subjects the writer to elements of risk, nowhere has there been a characterization of the writer’s attitude to risk. Thus far, only a preference-independent approach to pricing has been presented 1 . In the presentation of a control theory derivation of option pricing and hedging, a writers attitude to risk will be characterized. This is achieved by basing the outcome on the maximization of utility of wealth and allowing risk to be characterized as part of the specification of the utility function. A summary of the DPZ paper approach is now presented. In order to provide continuity with the mathematical development thus far, the notation has been slightly changed from the original. On the time interval t ∈ [0, T ] consider a market in which a stock St is a stochastic process on (Ω, F, P ). A set T (B0 ) denotes the admissible trading 1 perhaps this is not strictly true since the writer of an option has at his disposal the ∆T parameter as a measure of risk

20

strategies available to the investor when starting with an initial wealth B 0 . An element π ∈ T (B0 ) is a vector (B π (t), y π (t)) ≡ (Btπ , ytπ ) where B π (t) is a function giving the amount of cash held at time t and y π (t) is a function giving the number of shares held at time t. Given λb and λs , the proportional costs incurred when buying and selling, define the liquidated cash value of a portfolio after costs as a function of the number of shares and the current share price  (1 + λb )yt St if yt < 0 c(yt , St ) = (1 − λs )yt St if yt ≥ 0 Given the option writer’s utility function U(x), we define two valuation functions, one function includes the option while the other is exclusively a market portfolio Vw (B) = sup E{U(BTπ + I(ST ≤K) c(yTπ , ST ) π

+I(ST >K) [c(yTπ − 1, ST ) + K])} and V1 (B) = sup E{U(BTπ + c(yTπ , ST ))} π

where I(A) is the indicator function of the event A. The minimal initial endowments that generate positive utility may be defined as Bw = inf{B : Vw (B) ≥ 0},

B1 = inf{B : V1 (B) ≥ 0}

Intuitively (−B1 ) should be thought of as an ‘entry fee’ that the writer is prepared to pay to participate in the market. The price of the option is then given by the difference pw = B w − B 1

At this price the writer is indifferent between going into the market to hedge the option and going into the market on his own account. Note that this is not a market equilibrium price, since each writer has a different risk profile and hence a different utility function. It is now possible to consider the limiting cases. DPZ show that in the event that continuous updates without costs are possible, the model above reduces to Black-Scholes. It is presumed that when costs are incurred and the writer is not prepared to assume any risk that the super-replication result [20] can be derived in a similar manner2 . Thus if the writer is unwilling to assume any risk, then B1 = 0 and λb = λ s = 0 λb , λ s > 0

⇒ ⇒

pw = Bw = Black-Scholes value pw = Bw = S0 (Super-replication)

Now that the system has been specified it is possible to formulate the control problem that needs to be solved in order to maximize the utility of wealth. 2 It

may be an interesting exercise to prove the super-replication result using this approach.

21

Consider two functions that will govern the portfolio selection. L(t) and M (t) are the cumulative number of shares bought or sold at time t over the interval [0, T ] and may be written as Z t Z t m(τ )dτ l(τ )dτ and M (t) = L(t) = 0

0

where l and m are bounded by k < ∞. The differential equations governing the amount of cash, the number of shares and the stock price process are given by dBt

=

dyt dSt

= =

rBt dt − (1 + λb )St dL(t) + (1 − λs )St dMt

dL(t) − dM (t) µSt dt + σSt dWt

Dropping the subscripts indicating time dependence and rearranging terms, the system above can be rewritten as dB

=

[rB − l(1 + λb )S + m(1 − λs )S]dt

dy dS

= [l − m]dt = µSdt + σSdW

The control equations for the valuation functions Vw and V1 may now be derived. Notice that the system has been described by two partial differential equations and one stochastic differential equation. The multi-variate version of the HJB equation is thus needed, but since there is only one stochastic component no cross-terms develop and the situation is almost as simple as equation 4.2 presented earlier. The valuation functions Vj , where j ∈ {w, 1}, are given by ( ∂Vjk ∂Vjk = max [rB − l(1 + λb )S + m(1 − λs )S] + 0≤l,m≤k ∂t ∂B ) ∂Vjk ∂Vjk 1 ∂ 2 Vjk 2 2 (l − m) + µS + σ S (4.7) ∂y ∂S 2 ∂S 2 Notice the superscript k on the valuation functions indicating that they are constrained by the limit imposed on the cumulative integrals specified earlier. Rearranging terms ( ! ! ) ∂Vjk ∂Vjk ∂Vjk ∂Vjk − (1 + λb )S − (1 − λs )S max l− m 0≤l,m≤k ∂y ∂B ∂y ∂B +

∂ 2 Vjk ∂Vjk ∂Vjk ∂Vjk 1 + rB + µS + σ2 S 2 = 0 (4.8) ∂t ∂B ∂S 2 ∂S 2

Letting f1 =

∂Vjk ∂Vjk − (1 + λb )S , ∂y ∂B

f2 = 22

∂Vjk ∂Vjk − (1 − λs )S ∂y ∂B

Optimal control is achieved by examining the following three cases f1 ≥ 0,

f1 < 0, f1 ≤ 0,

f2 > 0 f2 ≤ 0 f2 ≥ 0

⇒

⇒ ⇒

l = k,

m=0

l = 0, l = 0,

m=k m=0

All other combinations are inadmissible since the valuation functions are increasing functions of B and y. Under some technical conditions specified in the DPZ paper it is possible to allow k → ∞ in which case the following partial differential equations ensue (   ∂Vj ∂Vj ∂Vj ∂Vj − (1 + λb )S , − − (1 − λs )S max , ∂y ∂B ∂y ∂B ) ∂Vj ∂Vj ∂Vj 1 2 2 ∂ 2 Vj + rB + µS + σ S =0 (4.9) ∂t ∂B ∂S 2 ∂S 2 Intuitively this indicates that our decision space is split into three regions, a buy region, a sell region and a no-transaction region. If the state of the portfolio is in the no-transaction region then it drifts under the influence of the stock price diffusion process. If the state is in the buy or sell region then the investor immediately performs the minimum transaction to bring it back into the no-transaction region. It turns out that this effectively implements a hedging bandwidth region around the Black-Scholes delta. This is depicted diagrammatically in figure 4.1 In order to solve the system it is necessary to specify a utility function. DPZ use an exponential utility function given by U(x) = 1 − e−γx . By doing a transformation of variables incorporating the utility function   B Vj (t, B, y, S) = 1 − exp −γ −r(T −t) Qj (t, y, S) e

Figure 4.1: Decision space indicating buy, sell and no-transaction regions 23

the transformed partial differential equations are given by (   γ(1 + λb )S ∂Qj γ(1 − λs )S ∂Qj min + −r(T −t) Qj , − + −r(T −t) Qj , ∂y ∂y e e ) ∂Qj ∂Qj 1 2 2 ∂ 2 Qj + µS + σ S =0 ∂t ∂S 2 ∂S 2

(4.10)

subject to the boundary conditions Q1 (T, y, S) = exp(−γc(y, S)) Qw (T, y, S) = exp(−γ{I(S≤K) c(y, S) + I(S>K) [c(y − 1, S) + K]}) The price of the option is given in terms of the solution to this system as   Qw (t, 0, S) e−r(T −t) pw (t, S) = ln γ Q1 (t, 0, S) Thus the optimization problem has been specified as a free boundary problem in three dimensional space (t, y, S) ∈ [0, T ] × R × R+ The DPZ paper goes on to analyze this system of equations in terms of viscosity solutions and provide numerical solutions by recasting the differential equations into Markov chains and utilizing the discrete time dynamic programming algorithm.

4.4

The Whalley-Wilmott Hedging Algorithm

It is obvious from the previous section that the analysis accomplished thus far is not easy to implement, especially when considering a real world environment in which traders are given limits on the amount of risk they are allowed to assume. It is fortuitous that the preceding analysis can be made easy to implement by an approximation derived through an asymptotic analysis of the system of differential equations 4.10. Whalley and Wilmott [22][23] perform this asymptotic analysis on the system and propose an algorithm to approximate hedging which is easy to implement. The results of this analysis are presented here without proof. In order to reduce the three dimensional free boundary problem into a one dimensional free boundary problem, an assumption of small transaction costs needs to be made. Furthermore if the assumption of proportional and symmetric costs is made λb = λ s = λ then the hedging bandwidth is easily calculated by the following expression B(t) =



3λSt e−r(T −t) Γ2t 2γ

1/3

where and Γt is the Black-Scholes Gamma and γ is the risk aversion parameter. When pricing the option, the Whalley-Wilmott algorithm specifies that the 24

0.16

Distribution of Hedging Error Leland Whalley−Wilmott

0.14

0.12

Probability

0.1

0.08

0.06

0.04

0.02

0 −6

−4

−2

0 Hedging Error

2

4

6

Figure 4.2: Comparison of P&L distributions for Leland and Whalley-Wilmott Black-Scholes price should be corrected by the square root of this amount (i.e. p add B(t) to the Black-Scholes price). It is important to remember that the local-in-time methodology requires that the hedge updates should be performed as often as possible. At each opportunity to rebalance the portfolio, the proportion of shares held yt is adjusted using the following equation  if yt−1 < ∆t − B(t)  ∆t − B(t) ∆t + B(t) if yt−1 > ∆t + B(t) yt =  yt−1 if ∆t − B(t) ≤ yt−1 ≤ ∆t + B(t)

where ∆t is the Black-Scholes Delta. Figure 4.2 shows a comparison between the P&L distributions of the Leland and Whalley-Wilmott algorithms. It should be noted that in order to make the comparison fair, the same initial premium was allocated in both cases being that of the Leland modified variance price3 .

4.5

An Extended Algorithm

Symmetric proportional costs are an idealistic assumption in the real world. In most situations transaction costs consist of fixed and proportional components 3 See

Chapter 5.1 for a discussion on the values of the parameters used in this simulation.

25

and are asymmetric4 . Whalley and Wilmott [22] provide an extended algorithm which can be used to determine a hedging bandwidth and rebalance points for more complicated cost structures. Although this algorithm has not been used in the numerical results presented in this report, it is mentioned here since it is likely to be of practical use in real-world situations. Unlike the simple algorithm presented earlier the hedging bandwidth and rebalance points need not coincide. As an example, if the transaction costs consist exclusively of a fixed amount regardless of volume or value of shares, it is more efficient to rebalance exactly to the Black-Scholes delta when the hedging bandwidth is exceeded. Explicit rebalance points should in general be calculated. When the transaction costs are given by the function K(S, n) where n is the number of shares (bought or sold) with share price S, then the hedging bandwidth is given by ∆t − A(S, t) ≤ y ≤ ∆t + A(S, t) The optimal rebalance points are given by y = ∆t ± B(S, t) To find A(S, t) and B(S, t) solve the following system of equations ∂K γ (S, A − B) AB(A + B) = 3δΓ2t ∂n γ (A + B)3 (A − B) = K(S, A − B) 12δΓ2t where δ = e−r(T −t) At each opportunity to rebalance the portfolio, the proportion of shares held y t is adjusted using the following equation  if yt−1 < ∆t − A(t)  ∆t − B(t) ∆t + B(t) if yt−1 > ∆t + A(t) yt =  yt−1 if ∆t − A(t) ≤ yt−1 ≤ ∆t + A(t)

4 For example in the South African case, MST (Marketable Securities Tax) is only levied on the purchase of shares.

26

Chapter 5

Numerical Comparisons In this chapter the performance of the hedging strategies is reviewed. The approach followed is similar to that of Mohamed [17]. Monte Carlo experiments are used as a means of comparing the algorithms. In order to provide a useful comparison criterion, a Value at Risk methodology is employed. This measures the ability of the algorithms to deal with the risk associated with hedging.

5.1

Parameters used in Monte Carlo Simulations

In order to preserve consistency throughout this document, all Monte Carlo simulations have been performed using the same basic parameter values. These parameters are for an at the money call option S0 K

= 100 = 100

µ = 0.05 σ = 0.4 T = 1/12 r

=

0.05

Furthermore, all Monte Carlo experiments were performed with 50000 trials to ensure stability in the results. Even with this number of trials a small amount of randomness is present in some of the output.

5.2

Value at Risk

Previously it was shown that when moving from a continuous hedging scheme to a discrete time scheme with transaction costs that the party writing and hedging options is implicitly exposed to risk. It is therefore not enough to evaluate the methods reviewed on the basis of expected returns since, excluding the effect of 27

transaction costs, these should naturally be zero. When the effect of transaction costs is observed, expected returns decrease with an increase in the number of updates. It is therefore necessary to measure the potential to lose money as a result of the risk involved in the process of hedging. In this case a measure dependent on the variation of hedging error is needed. When frictions are present it is of course not possible to determine the variance directly as in Chapter 2. Since it has become standard banking practice to evaluate the risk of investment portfolios on the basis of Value at Risk it would seem suitable to apply the same measure to evaluate the risk in a hedging portfolio. Common practice is to select the VAR measurement to be the maximum loss not to be exceeded with a 95% confidence level. In the case presented here, this was performed over the life of the option. The profit and loss of a particular strategy was calculated at the date of expiry as the final hedge portfolio less the exercise payment, if one was due. The hedge portfolio starts initially with no holdings except an initial endowment being the Black-Scholes value of the option at inception. This corresponded to an amount of 4.81 for the option parameters given above. It should be noted that each of the methods priced the option differently and that it was expected that in all cases the portfolios would make a loss due to the effect of the transaction costs. The most efficient portfolio strategy is therefore the one that makes the least loss. A distribution of profits and losses was built up by taking a sample of 50000 Monte Carlo trials. From this sample it was possible to calculate the mean P&L, the VAR being the ninety-fifth percentile of the P&L sample and the mean total transaction costs present-valued to the date of expiration. Figure 5.1 shows an example of a comparison between the P&L distributions for the three methods and also indicates the corresponding VAR of the P&L represented as a vertical line.

5.3

Results

In order to evaluate the effectiveness of the methods a Matlab program was written that compared the hedging performance as a function of the number of updates performed over the life of the option. The following values were measured for comparison • The mean P&L as a function of number of rehedging updates (Figure 5.2) • The VAR as a function of updates (Figure 5.3) • The cumulative transaction costs as a function of updates (Figure 5.4) • The mean P&L excluding costs as a function of updates (Figure 5.5) As predicted, the mean P&L is not a very useful measure for evaluating the performance of the hedging strategies. It can be seen from the graph that 28

Distribution of P&Ls and VAR

0.18

Black−Scholes Leland Whalley−Wilmott

0.16

0.14

Probability

0.12

0.1

0.08

0.06

0.04

0.02

0 −6

−4

−2

0 P&Ls

2

4

6

Figure 5.1: Comparison of P&L distributions showing VAR the mean is a strictly decreasing function of the number of updates. In the case of the Whalley-Wilmott (WW) algorithm a bound on the mean can be clearly seen. On the other hand the Black-Scholes (BS) mean can be seen to be divergent. Although there is a theoretical limit on the loss attributable to the Leland algorithm, it is clearly still decreasing over the interval of hedge updates measured. It should be noted however that as a profitability measure, the mean of the WW algorithm is far superior to that of the BS and Leland algorithms. The VAR of the P&Ls is a far more effective way of comparing the different hedging methods. As can be seen from Figure 5.3, the VAR for each method decreases as the number of updates increases. In the case of the BS and Leland algorithms a critical point is reached where the transaction costs begin to cause the VAR to increase again. The VAR of WW algorithm continues to decrease on the whole interval, this is of coarse due to the fact that it is a global-in-time method. The WW algorithm is thus better able to create a tradeoff between the hedge-risk and the increasing costs of hedging more often. In doing so it is able to perform approximately 10% more effectively than the Leland algorithm and 25% more effectively than a standard Black-Scholes approach when VAR is used as the metric. Figure 5.4 shows that as the number of updates increases the total transaction costs incurred rise. In the case of the Black-Scholes update this is unbounded. Although the costs in the case of the Leland algorithm are bounded they are much larger than the WW algorithm and do not appear to be converging in the intervals measured.

29

Mean P&L vs no. of Rehedges

−0.8

−1

−1.2

Mean P&L

−1.4

−1.6

−1.8

−2

−2.2

−2.4

−2.6 10

Black−Scholes Leland Whalley−Wilmott 20

30

40

50 60 70 Number of Rehedges

80

90

100

110

Figure 5.2: Mean P&L as a function of number of updates

VAR vs no. of Rehedges

−2.4

−2.6

−2.8

VAR

−3

−3.2

−3.4

−3.6

−3.8 10

Black−Scholes Leland Whalley−Wilmott 20

30

40

50 60 70 Number of Rehedges

80

90

Figure 5.3: VAR as a function of updates

30

100

110

Mean Total Costs vs no. of Rehedges

0

Mean Total Transaction Costs

−0.5

−1

−1.5

−2

−2.5 10

Black−Scholes Leland Whalley−Wilmott 20

30

40

50 60 70 Number of Rehedges

80

90

100

110

Figure 5.4: Cumulative transaction costs as a function of updates

Mean P&L excluding costs vs no. of rehedges

0.05

0

Mean P&L excluding costs

−0.05

−0.1

−0.15

−0.2

Black−Scholes Leland Whalley−Wilmott

−0.25

−0.3 10

20

30

40

50 60 70 Number of Rehedges

80

90

100

110

Figure 5.5: Mean P&L excluding costs as a function of updates

31

Figure 5.5 shows in the case of the BS and Leland algorithms that when the transaction costs are added back to the mean P&L, the mean hedging error is zero regardless of how many updates are performed. Perhaps a more interesting observation is the fact that the mean P&L for the WW algorithm when costs are added back is negative (at a value of approximately -0.255). This shows that this algorithm is prepared to make a small loss on average as a result of hedge slippage in order to save on the cost of rehedging.

5.4

The Choice of Risk Aversion Constant

One of the most important factors to consider when using the WW algorithm is the choice of risk aversion constant (γ). In order to gauge the effect of this constant on the VAR of the hedge portfolio, a Monte Carlo simulation was written to produce a graph of VAR as a function of risk aversion. The results are displayed in figure 5.6. VAR is minimized for values 1 ≤ γ ≤ 10. A value of γ = 5 was used for all the simulations on which the comparisons above were made.

VAR as a function of Risk Aversion

0

VAR

−5

−10

−15 −4

−3

−2

−1

0

x

1

2

3

γ (*10 )

Figure 5.6: VAR as a function of risk aversion 32

4

Chapter 6

Conclusion This report has presented an introduction to the theory of pricing and hedging of options in the presence of transaction costs. This has been accomplished by relaxing the Black-Scholes assumptions of continuous time and no market frictions. Local-in-time and global-in-time methodologies have been explored. Numerical experiments were performed and the most efficient method reviewed as measured by the Value at Risk methodology was the Whalley-Wilmott asymptotic algorithm of the Davis, Panas and Zariphopoulou control optimization approach. This algorithm proved easy to implement and provided a 10% and 25% improvement on the Leland and Black-Scholes algorithms respectively. It would be interesting to test these algorithms on market data and perform a similar VAR analysis. Real-world data would introduce a number of factors which have not been modelled here including market incompleteness, stochastic volatilities, bid-ask spreads and jumps. It would be interesting to see how well the hedging algorithms perform with these deviations from the theory. This report is concluded with a conjecture that could lead to further work on the area. It has been shown that in the event of assuming discrete-time and transaction costs that the writer of an option is exposed to market risk. This implies that the writer is necessarily taking a speculative position in the market. Generally when a speculative position is being taken, it is necessary to move from the risk-neutral world and consider the expectation of the stock price process under real-world probabilities. This involves estimating the drift of the stock price process one is speculating on. If the writer of an option is exposed to market risk, it may be better (under the assumption of Brownian motion) to incorporate the drift of the price process into the hedging portfolio. We have seen in the Merton problem, that the optimal investment choice under the consideration of Brownian motion is to select the Merton proportion. Consider now a similar control theory derivation of the Black-Scholes Delta.

33

6.1

A Control Theory Derivation of the BlackScholes Delta

Suppose a portfolio (X) consists of some cash, a certain proportion (πt ) of shares (S) and a single option on the same share (V (S, t)) held short. What is the optimal portfolio choice? As before the expected utility of wealth at final time T needs to maximized U (x) = sup E[U(XTx,π )] π

XTx,π

where is the wealth at T when using trading strategy π and starting with initial wealth X0 = x. Again, a risky stock (St )0≤t≤T following the process dS = µSdt+σSdW and a riskless asset with interest rate r are the instruments available for investment. In this case (contrasting with the Merton problem) πt is defined to be the number of shares held at time t. Then dXtx,π = dX = (X − πt S)rdt + πt dS − dV Now by Itˆo the change in the value of the option is given by 1 dV = Vt dt + µSVS dt + σ 2 S 2 VSS dt + σSVS dW 2 Substituting this and the SDE for the stock price process into the expression for the portfolio gives dX = (X − πt S)rdt + πt (µSdt + σSdW ) 1 −(Vt dt + µSVS dt + σ 2 S 2 VSS dt + σSVS dW ) 2 Rearranging terms leads to the following expression for the change in value of the portfolio 1 dX = (rX − πt rS + πt µS − VS µS − Vt − σ 2 S 2 VSS )dt 2 +(πt σS − VS σS)dW Thus the HJB system can be described as follows f (X, π, t) F (X(T ), T )

= 0 = U(XTx,π )

µ(X, π, t)

=

σ(X, π, t)

=

1 rX − πt rS + πt µS − VS µS − Vt − σ 2 S 2 VSS 2 σS(πt − VS )

 1 2 2 −Jt (t, x) = max Jx (t, x) rX − πt rS + πt µS − VS µS − Vt − σ S VSS πt 2  1 2 + Jxx (t, x)[σS(πt − VS )] (6.1) 2 



34

Rearranging terms again gives   1 max Jx (t, x)(µ − r)πt S + Jxx (t, x)[πt2 σ 2 S 2 − 2πt σ 2 S 2 VS ] πt 2 1 + Jxx (t, x)σ 2 S 2 VS2 + Jt (t, x) 2   1 2 2 +Jx (t, x) rX − Vt − µSVS − σ S VSS = 0 2 ⇒ max πt



 1 Jxx (t, x)σ 2 S 2 πt2 + [Jx (t, x)(µ − r)S − Jxx (t, x)σ 2 S 2 VS ]πt 2 +... = 0

This is maximized for the portfolio choice π∗ =

Jxx σ 2 S 2 VS − Jx (µ − r)S Jxx σ 2 S 2

Take a trial solution of J(t, x) = b(t)U(x) with x = X and b(T ) satisfying the boundary condition b(T ) = 1. Then assuming the utility function U(x) =

xγ − 1 , γ

γ < 1 and γ 6= 0 (In which case U(x) = log x)

gives Jx (t, x) = b(t)X γ−1 , Thus π ∗ = VS +

6.2

Jxx (t, x) = b(t)(γ − 1)X γ−2 (µ − r) X σ 2 (1 − γ) S

Discussion

This is an interesting expression and deserves some attention. It appears to be a portfolio choice that includes the Black-Scholes delta and a Merton proportion 1 . In the case where no risk is assumed, γ → −∞ and only the Black-Scholes delta is left. It is interesting that the Black-Scholes PDE arises as a direct result of this formulation when the initial portfolio value is assumed to be the value of the option (i.e. X0 = V ). Substituting π ∗ = VS into equation 6.1 gives 1 −Jt (t, x) = rV − rSVS − Vt − σ 2 S 2 VSS Jx (t, x) 2 1 Note that the Merton proportion is not specified in the same manner as in Chapter 4 since our choice of π is different here.

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As a result of the choice of risk aversion constant, the right hand side of the equation is zero, therefore the PDE on the left hand must also be equivalent to zero. This of course is just the Black-Scholes PDE. On the other hand, what happens if the writer wishes to assume some risk? (This is true of the situation where hedging occurs in discrete time with transaction costs.) In this case the choice of risk aversion constant must be something other than γ = −∞. This constrains the portfolio choice to include a Merton proportion commensurate with the risk selected. It is interesting that in at least one approach to the problem of hedging in discrete time with transaction costs, Wilmott [24] mentions ‘. . . the assertion that µ must sometimes be measured. The discrete hedging of options is one of the times;. . . ’ It is thus conceivable that the inclusion of a drift dependent factor into the hedge portfolio may lead to better performance. Perhaps this could be considered as an extension to the work presented here. Of course this may only be of theoretical interest as it is in general very difficult to accurately measure the drift of real world processes.

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