Monte Carlo pricing & hedging of a discrete arithmetic Asian option.

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Monte Carlo Pricing & Hedging

of a Discrete Arithmetic Asian Option

Georgios Tsiounis

0 50 100 150 200 250 300 2 4 6 8 10 12 14 16

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University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Georgios Tsiounis Student nr: 10273077

Email: georgios.tsiounis@gmail.com Date: April 25, 2014

Supervisor: Prof. Dr. M.H.Vellekoop Second reader: J.P. de Kort

Supervisor: J.P. de Kort

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Abstract

In this thesis we explore the implications in pricing and hedging of a discrete arith-metic Asian option. We address the various computational challenges and implement techniques to reduce the variance and increase the accuracy of the Monte Carlo es-timates. We perform various hedge tests of our Black-Scholes assumed methods on synthetical data from the Heston model and numerically report the resulting hedging error from the discretization of the delta hedging strategy and the induced model mis-specification. We relate to the relevant work on hedging under misspecified models and approach the question of optimal volatility. Throughout, we maintain the point of view of an insurer by considering the Asian option as a proxy for more complex insurance products and by referring to the associated regulatory constraints.

Keywords Model Misspecification, Hedging, Asian Options, Monte Carlo, Pathwise Deriva-tives

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Preface

I would like to thank my parents Heracles and Ioanna for their unlimited love and support and my supervisors M.H.Vellekoop and J.P. de Kort for their insightful guidance and feedback.

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Introduction

In the last decades, derivatives’ trading has had the lion’s share in global financial activity. Traded in standardized forms in exchanges around the world, over-the-counter with tailor-made specifications, or embedded in other contracts, used as an investment vehicle, hedge instrument, or for speculative purposes, they can be found everywhere. The significant boost was given by Black, Scholes and Merton whose work allowed a better understanding and handling of derivatives. Since then, the field has concentrated much scientific attention. It has benefited from tools from Stochastic Calculus and the advancements in the processing power of computers.

From the point of view of an insurer, short positions in contingent claims are au-tomatically entered into by selling insurance contracts with embedded options. The consequent exposure to the risks inherent in the short positions can prove significantly harmful to the insurer. Therefore, the issue of hedging against them is raised. Hedging is the construction of a strategy which neutralizes or mitigates the exposure to one or more targeted risks. Hedging entails working in a holistic framework, starting from the pricing of the hedged item, recognizing its risk profile and stretching all the way to setting and maintaining the hedge strategy.

The embedded options found in insurance contracts usually have complicated fea-tures, such as path dependence. They are also not traded in the market. Therefore, analytical pricing formulas may not be available and, instead, simulation techniques need to be used as an alternative. Simulation introduces an extra computational bur-den and, very importantly, it is subject to error. The lack of closed form pricing formulas complicates also the calculation of the sensitivities, which, when based on simulation, are subject to error as well. The sensitivities are of utmost importance in building the hedge. Incorrectly estimated sensitivities jeopardize the efficiency of the hedge.

The prime driver of risk is the underlying asset. A static hedge against this risk is possible when another product with the same sensitivity is available in the market. The complexity of the considered options renders the possibility of a static hedge using similar and tradable products unlikely. In theory, there exists a dynamic strategy that replicates the payoff perfectly. In practice, rebalancing can only be done in discrete points in time and the portfolio is expected to deviate from the targeted payoff, resulting in a profit or a loss.

After the Black-Scholes model, many others have been developped with the purpose of capturing better the dynamics and evolution of an asset. Black-Scholes is limited in its assumptions, especially due to the constant volatility. Different models including stochastic volatility address the observed varying volatility and the volatility skew/smile. However, combined with the aforementioned, the use of such a model in hedging com-plicates even further the calculations and introduces model risk. This points to the question of how a hedge based on a Black-Scholes constant volatility would perform, in a setting where stochastic volatility is assumed to prevail in reality. In other words, what is the impact of model misspecification on the hedge efficiency.

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2 Georgios Tsiounis — Pricing & Hedging of an Asian Option

In the following, we approach the question by performing hedge tests where the real world is modeled by Heston’s stochastic volatility model and the hedge model is Black-Scholes, using different proposed volatilities. We adopt an insurer’s point of view and consider the regulatory constraints regarding the exposure to market risk. We choose to work with a discrete Asian option as a proxy for insurance products. We start with pricing using Monte Carlo Simulation and measure its accuracy. We present and extend methods for calculation of the targeted sensitivities and measure their accuracy. We also perform hedge tests without model misspecification in order to measure the impact of the previous methods, the rebalancing frequency and the choise of strategy.

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Pricing a Discrete Arithmetic

Asian Option

Asian options belong to the broader category of path dependent options. The payoffs of these products are determined not only by the underlying asset price at maturity but also by a set of asset prices, henceforth fixings, observed prior to maturity and during the lifetime of the option. The payoffs of Asian options in particular, involve an average of these fixings. Further subcategorization considers the type of the average (arithmetic, geometric), the set of fixings (discrete, continuous) and the timeframe relevant to the fixings.

When adopting an insurer’s point of view, the choice to employ an Asian option as a proxy for insurance products is rather reasonable. Complex contract specifications, such as path dependence, are typical of insurance products. For example, consider a profit sharing scheme with a guaranteed minimum annual return equal to the average U-rendement during that year. The annual profit sharing can be interpreted as an Asian option and the entire scheme as a basket of Asian options.

Path dependence, as we will see below, complicates the pricing, sometimes disal-lowing the derivation of closed form formulas. Instead, approximative methods are put in use. Consequently, the error resulting from such methods, as well as techniques to improve the estimations, become meaningful considerations. Especially when, for our purposes of the hedge tests, the calculated prices will be combined with other variables which themselves are a source of error.

In this chapter we present the pricing method for a discrete arithmetic Asian option based on the classic Monte Carlo simulation, an improved method using Antithetic Variates sampling as a Variance Reduction technique and the calculation method for the Standard Error in both cases. The chapter is based on Glasserman (2004) and Boyle, Broadie & Glasserman (1997).

2.1 Product Payoff and Price

The payoff at maturity of a discrete arithmetic Asian option depends on the arithmetic average of a finite set of fixings observed at predetermined times. It can be expressed as:

Pasian(T, St1, ..., Stn) = maxST − K, 0 , (2.1)

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where,

T is the maturity of the option, K is the strike of the option, ST =

n

X

i=1

Sti

n is the arithmetic average of the fixings, Sti is the fixing at time ti and

ti, i = 1, ..., n are the predetermined observation times (satisfying: 0 ≤ t1 < ... < tn≤ T ).

We assume that the underlying asset S follows a Geometric Brownian Motion with constant drift and diffusion parameters, the well known Black-Scholes model:

dSt= µ Stdt + σ StdWt, (2.2)

where,

µ is the constant drift parameter, the annualized drift rate of S,

σ is the positive constant diffusion parameter, the annualized volatility of returns of S, Wt is a standard Brownian Motion.

Under the risk neutral measure Q, (2.2) becomes:

dSt= r Stdt + σ StdWt, (2.3)

where, r is the annualized continuously compounded risk free rate which is smaller than µ and, also, assumed to be constant.

In this Black-Scholes setting, the risk-neutral price of the Asian option at any time t ∈ [0, T ] is:

Vasian(t, St1, ..., Stj, St) = E

Q_{[ DF (t, T ) P}

asian(T, St1, ..., Stn)| Ft] (2.4)

where,

j is a positive integer such that tj ≤ t < tj+1,

DF (t, T ) is the discount factor corresponding to the time interval [t, T ] and {Ft} is the filtration generated by the Brownian Motion Wt.

The price at t depends on St as well as on the previous fixings {Sti, i = 1, ..., j}. This

information is included in the σ-algebra Ft, since the information on the Brownian

Motion is sufficient to know the past asset prices. The time t, here, is purposely allowed in the entire lifetime of the option to emphasize that constructing, monitoring and evaluating a dynamic hedge strategy entails calculating the price (and also the relevant sensitivities) across all t ∈ [0, T ].

Returning to (2.4) and because of our assumption of the risk free rate, we may write: DF (t, T ) = e−r(T −t) (2.5) and substitute (2.5), which is now deterministic in Ft, together with (2.1) in (2.4) to

get: Vasian(t, St1, ..., Stj, St) = e −r(T −t) EQ max ST − K, 0 F_t . (2.6) Deriving pricing formulas is not as straightforward as in the case of a European call or put option. Complications are introduced by the path dependent feature of the Asian option. Not only the terminal but also the intermediate fixings within the remaining time to maturity ({Stj+1, ..., Stn}) enter, through ST, in the calculation of the price.

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case with the distribution of a linear combination of lognormal variables. This makes the derivation of a Black-Scholes type formula for the arithmetic Asian impossible.

For another preliminary observation, we can say that, with t approaching maturity, there are more and more fixings already observed (the Sti, ti ≤ t are known). We can

think of this as the augmenting with time of the information in Ftleading to a gradual

‘dissolving’ of the uncertainty in ST. We will see later what is the impact of this on

pricing and also on the option delta.

2.2 Monte Carlo Pricing

The basic idea behind Monte Carlo is that an expectation can be estimated by a sample mean. The strong law of large numbers suggests that as the size of an i.i.d. sample from a variable X tends to infinity, its mean converges to the true value of the expectation of X. In option pricing, the quantity to be sampled is the discounted payoff. The pric-ing problem, then, reduces to generatpric-ing a large enough sample of possible discounted payoffs.

Not knowing the distribution of the discounted payoff is not limiting. It can be bypassed as soon as we can generate possible realizations of the relevant stochastic variables and model the product specifications to construct the payoffs. Usually, this is a relatively easy task and it is the reason why Monte Carlo has grown to a very practicable tool in option pricing.

In our case of the Asian option in the Black-Scholes setting, the unknown distribution of the arithmetic average is no longer a problem. What we do instead, is generate asset paths, containing the required fixings, determine the arithmetic average of these fixings pathwise, calculate the corresponding discounted option payoff and, finally, average over all simulations to get the estimated option price. We therefore have:

b Vasian(t, St1, ..., Stj, St) = k X i=1 DPi(t, St1, ..., Stj, St) k ≈ Vasian(t, St1, ..., Stj, St), DPi(t, St1, ..., Stj, St) = e −r(T −t)_Pi asian(t, St1, ..., Stj, S i tj+1, ..., S i tn), (2.7) where,

k is the number of simulations, b

Vasian(t, St1, ..., Stj, St) is the estimated option price at time t,

DPi(t, St1, ..., Stj, St) is the discounted payoff and

P_asiani (t, St1, ..., Stj, S

i

tj+1, ..., S

i

tn) is the payoff

both as calculated at time t and in the i-th simulation.

The set of fixings at time t and in the i-th simulation, given the information up to time t is {St1, ..., Stj, S

i

tj+1, ..., S

i

tn}. Notice that when pricing at an arbitrary t ∈ (0, T ]

this path can be part ‘historical’ and part generated, meaning, it contains the past observations which are known at t together with generated values for the remaining fixings until maturity. It appears in our notation as the St1, ..., Stj do not depend on

the i-th simulation and will be assigned the respective ‘historical’ values (Sti = sti, i =

1, ..., j). This is how the gradual ‘dissolving’ of the uncertainty in ST is modeled.

2.3 Black-Scholes Model Simulation

At every valuation time t ∈ [0, T ], in order to generate paths of the underlying asset for the remaining time to maturity T − t, we have to apply a method of discretization of the stochastic process (2.3). The most well-known method is the Euler Scheme. Generally,

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the method introduces a discretization bias to the approximation, which, for the case of the Geometric Brownian Motion, can be diminished by first taking the logarithm of the asset price. The discrete version of the asset price process on a set of times t = t0< t1< ... < tm= T is:

ln(Sti+1) = ln(Sti) + (r −

σ2

2 )(ti+1− ti) + σpti+1− tiZi+1, (2.8) where,

i = 0, ..., m − 1 and

Zi+1 are i.i.d. standard normal random variables.

The recursive expression (2.8) requires for the calculations to be performed in loops, which in MATLAB can be very time-consuming compared to working with matrices, therefore we replace it by:

ln(Sti+1) = ln(St0) + (r − σ2 2 )(ti+1− t0) + σ i X k=0 ptk+1− tkZk+1 . (2.9)

This expression allows for a total of k paths to be simulated without the use of loops. Notice that dependence of Sti+1 on Sti is eliminated and only the starting point of the

path is required. Instead of loops, we only need to generate an (m, k) matrix of standard normal random numbers, calculate the cumulative sums of these numbers column-wise and use matrix operations to implement (2.9). The outcome is an (m + 1, k) matrix of the (m + 1)-length paths starting at St, arranged in columns.

Suppose that an Asian option considers n fixings, including the asset price at t = 0. We start at t = 0 setting the time steps m so that at least n − 1 of them coincide with the fixing times. As t moves forward we can decrease the number of steps always making sure that, at every valuation point, concatenating ‘historical’ and simulated fixings gives n observations per path. Regardless of the valuation point though, we execute the same number of path simulations.

The implementation of (2.1) and (2.7) is, then, straightforward.

2.4 Monte Carlo Standard Error

Since only a finite number of simulations can be executed, the price derived by Monte Carlo simulation is an approximation of the true price of the option. It is, therefore, very important to provide a measure of the accuracy of the estimate. The standard devia-tion of the price is an appropriate measure and leads to the construcdevia-tion of confidence intervals, that is, the intervals wherein, with some certainty, a random variable lies.

Looking at 2.4 and 2.7, we see that the standard deviation of the estimated price is, essentially, the standard deviation of the mean of the discounted payoffs. With the distributional properties of the discounted payoffs being unknown, we turn to the sim-ulated payoffs sample and the Central Limit Theorem for an estimate of the standard deviation of the price.

The Central Limit Theorem suggests that, given a k-size sample from a distribution with mean µx and standard deviation σx and provided that the sample is i.i.d. and

k is large enough, the sample mean is approximately normally distributed with mean µx and standard deviation √σx_k. What we do is use the sample standard deviation of

the discounted payoffs as an estimate of the true standard deviation of the discounted payoffs and divide by the square root of the sample size to get the estimate of the

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Figure 2.1: A pair of stock paths generated by antithetic variates. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 90 95 100 105 110 115

standard deviation of the price. The latter is referred to as the standard error. We have:

c stdDP = v u u t 1 k − 1 k X i=1 DPi_{− b}_V2 _{≈ std} DP and d SEV := cstdV = stdc√DP k ≈ SEV (:= stdV) , (2.10) where,

DPi, V = Vasian and bV = bVasian as in 2.7,

c

stdDP is the estimated standard deviation of the discounted payoffs and

d

SEV is the estimated standard error of the option price

(all perceived as functions of {t, St1, ..., Stj, St}).

2.5 Antithetic Variates Sampling

Increasing the number of simulations, certainly, improves the accuracy of the estimates, however, the slow rate of convergence of Monte Carlo may require a very large number of simulations to be executed in order to approach satisfactory levels of standard error. Therefore, techniques have been developped to reduce the variance of the estimates. To name a few, Antithetic Variates, Control Variates and Stratisfied Sampling. We apply the Antithetic Variates method.

The idea behind the technique is that, since we generate the sample rather than just collect it, we can introduce an appropriate correlation among the simulated payoffs, so that the variance of the estimate is reduced. For the rest of the chapter, we drop again the arguments {t, St1, ..., Stj, St} without disregarding the dependence on time and past

fixings. Suppose we have already sampled k/2 discounted payoffs DPj, j = 1, ...,k₂ (k here is an even number), independent and identically distributed with variance σ_DP2 . Suppose further that for every DPj, we can choose another discounted payoff DPk2+j

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8 Georgios Tsiounis — Pricing & Hedging of an Asian Option ρj = Cov(DPj, DP k 2+j) < 0, ∀j. Then, by setting: Xj = DPj+ DPk2+j 2 , (2.11)

which are now i.i.d., we have a mean

X = k 2 X j=1 Xj k 2 = k 2 X j=1 DPj+ DPk2+j k = k 2 X j=1 DPj k + k X l=k₂+1 DPl k = k X j=1 DPj k = bVAV,

equal to the price estimated with Antithetic Variates ( bVAV) and variance

V ar(X) = 1 k2 4 k 2 X j=1 V ar(Xj) = 1 k2 k 2 X j=1

V ar(DPj) + V ar(DPk2+j) + 2Cov(DPj, DP k 2+j) = 1 k2 k 2 X j=1 σ_DP2 + σ_DP2 + 2ρj < 1 k2 k 2 2σ 2 DP = σ√DP k 2 ,

which is less than what would have been in the case of a size k i.i.d. sample. The reduction of the variance is achieved because of the negative correlation we induced.

The missing link is how to actually induce this negative correlation. The following suggests such a method and the sufficient condition which assures the negative correla-tion:

Let {z1, ..., zm} and {−z1, ..., −zm} be two sets of standard normal numbers (it is

straightforward that by changing signs the numbers of the second set follow again the standard normal distribution). Then:

Cov(φ(z1, ..., zm), φ(−z1, ..., −zm)) < 0,

as soon as φ(·) is monotone.

We can think of the set {z1, ..., zm} as the set of numbers needed to simulate a path

of fixings. What is more, looking at (2.9), we can see that the asset prices S (and, hence, the discounted payoffs DP ) are indeed monotone with respect to the standard normal random numbers. So, the proposition applies in our case, the discounted payoffs derived from antithetic pairs of standard normal numbers will indeed be negatively correlated, hence, the Antithetic Variates method indeed reduces the variance of the estimate.

2.6 Monte Carlo with Antithetic Variates Standard Error

By applying the Antithetic Variates method, the k-size discounted payoff sample is no longer i.i.d., however, the k₂-size sample of the Xj in (2.11) is still i.i.d., therefore we can

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adjust the estimators (2.10) of the standard deviation and standard error as follows: c stdX = v u u u t 1 k 2 − 1 k 2 X j=1 Xj− dVAV 2 and d SEVAV = c stdX q k 2 . (2.12)

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Chapter 3

Calculating Delta and Gamma

3.1 Definitions

The sensitivities, or Greeks as they are known, are defined as the first or higher order partial derivatives of the price function with respect to one or more parameters such as the underlying asset, volatility, interest rate, time etc. The Greeks reveal important information on the behaviour of the option under movements of these parameters and are used for risk management purposes. For the hedge strategies that we shall consider, the relevant Greeks are the delta and gamma which are defined as the first and second derivative of the price function with respect to the underlying asset:

∆asian(t, St1, ..., Stj, St) = ∂Vasian(t, St1, ..., Stj, St) ∂St (3.1) Γasian(t, St1, ..., Stj, St) = ∂2_V asian(t, St1, ..., Stj, St) ∂S2 t (3.2) The previously discussed dependence of the Asian price on past fixings is transfered onto the Greeks as well. Without disregarding this fact and only for brievity, in the discussion that follows, we shorten the notation by dropping the middle terms ({St1, ..., Stj}).

In the presence of a closed-form pricing formula, the Greeks can be calculated by differentiating the formula with respect to the relevant parameter (Carr [2000]). In the case of the Asian option, due to the lack of closed-form formulas, alternative methods based on simulation must be used to estimate the Greeks. First, we briefly present the Finite Differences method and next we concentrate on the Pathwise Derivatives method.

3.2 Finite Differences Method

The Finite Differences method stems from the intuitively more immediate view, the definition, of the derivative as the limit:

f0(x) = lim

h→0

f (x + h) − f (x)

h (3.3)

and of the second derivative as:

f00(x) = lim

h→0

f0(x) − f0(x − h)

h , (3.4)

which can be combined with (3.3), provided that the latter is defined, to give: f00(x) = lim

h→0

f (x + h) − 2f (x) + f (x − h)

h . (3.5)

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For appropriate choice of h, the limits can be approximated by: f0(x) ≈ f (x + h) − f (x) h f00(x) ≈ f (x + h) − 2f (x) + f (x − h) h . (3.6)

A first refinement, related to the Mean Value Theorem, which reduces the bias of the estimate of the first derivative is the central difference variant:

f0(x) ≈ f (x + h) − f (x − h)

2h . (3.7)

Finally, the finite difference estimates for delta and gamma are:

b ∆(t, St) = b Vasian(t, St+ h) − bVasian(t, St− h) 2h (3.8) b Γ(t, St) = b

Vasian(t, St+ h) − 2 bVasian(t, St) + bVasian(t, St− h)

h2 . (3.9)

The choice of h is of key importance because it has two opposite effects on the estimate. A small h decreases the bias of the estimate, but at the same time, increases its variance. Using the same set of random numbers in the calculation of the ‘bumped’ values is an easy way to decrease the variance. Of course, increasing the number of simulations k is also decreasing the variance. An optimal choice of h and k seeks to benefit from these opposite effects. We shall not enter this domain, but continue to a method which produces unbiased estimators.

3.3 Pathwise Derivatives Method

The Pathwise Derivatives method appears to have many advantages over the Finite Dif-ferences method. First, when pricing with Monte Carlo, all the data required to derive the pathwise estimates are available in the simulated paths, while, in the Finite Dif-ferences method, extra calculations have to be performed to give the ‘bumped’ values of the Greeks. Second, the pathwise estimators (the functions giving the values of the Greeks along a specific path) are unbiased, whereas, in the Finite Differences method, special care must be taken in assigning the bump, as it affects the bias. Third, the stan-dard error of the pathwise estimates can also be computed easily and give information on the accuracy of the calculations.

Expressions (3.1), (3.2) and (2.6) show that the delta and gamma are actually deriva-tives of an expectation, the calculation of which is halted by the lack of analytical formula for (2.6). The principle idea behind the Pathwise Derivatives method is to interchange the order of expectation and differentiation. This interchange results in the pathwise estimators, from which separate values of the Greeks are calculated along each simu-lated path of the Monte Carlo. The values are then averaged over all paths to give the method estimate.

We present the formulas derived in Broadie & Glasserman [1996] for the pathwise estimators of delta and gamma. In addition and as a requirement of constructing a dynamic hedge strategy, we extend them for arbitrary t ∈ [0, T ).

3.3.1 Delta

We start with the delta estimator for the discrete Asian option in Broadie & Glasserman (1996). The formula applies for t = 0 and assumes the Black-Scholes dynamics. We omit

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the proof since our extended proof for arbitrary t covers the case of t = 0 as well. The delta estimator is:

e−rT1 (SiT≥K) Si T S0 (3.10) where, 1_(Si

T≥K) is the indicator that the Asian finishes in-the-money in the i-th simulation,

Si_T is the arithmetic average of n fixings calculated in the i-th simulation and S0 is the asset price at t = 0

Averaging results in an estimate for delta at t = 0:

b ∆(0) = 1 S0erT Pk i=11_(Si T≥K) Si_T k . (3.11)

Following similar arguments as Broadie & Glasserman, we extend the formulas to apply for all t ∈ [0, T ).

Consider the discounted payoff as calculated at t ∈ [0, T ) and in the i-th simulation (see2.7). By interchanging the order of expectation and differentiation, the path estima-tor of the delta is the derivative of the discounted payoff with respect to the underlying asset. Making use of the chain rule of derivatives we get:

∂ e−r(T −t)P_asiani (t, St) ∂St = e−r(T −t) ∂P i asian(t, St) ∂Si_T ! ∂Si_T ∂St ! . (3.12)

The middle term is defined almost everywhere and equal to 1 when option is exer-cised, 0 elsewhere. In more detail, we can distinguish two cases. If the payoff is positive then it increases linearly with Si_T, while if it is zero then it can be assumed to stay zero for small increase of Si_T. It can be represented as the indicator function of the event that the option finishes in-the-money.

∂P_asiani (t, St)

∂Si_T

= 1_(Si

T≥K) (3.13)

The dependence of the indicator on time and asset path is also noted but appears in our notation only via Si_T.

Regarding the last term, we must first look at the connection between Si_T and the valuation time t. All future fixings,

n

S_ti_j, j = l + 1, ..., n, tl< t ≤ tl+1

o

, can be ex-pressed as a function of St based on our Black-Scholes assumption. All past fixings,

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Thus we have: ∂Si_T ∂St = 1 n   l X j=1 ∂stj ∂St + n X j=l+1 ∂Si_t_j ∂St   = 1 n n X j=l+1 ∂S_ti_j ∂St = 1 n n X j=l+1 ∂ Stexp n r −σ₂2 (tj− t) + σ √ tj− tZji o ∂St = 1 n n X j=l+1 exp r − σ 2 2 (tj− t) + σptj− tZji = 1 n n X j=l+1 S_ti_j St = 1 St n X j=l+1 Si tj n = 1 St n X j=1 Si_t_j1_(t_j≥t) n (3.14)

Notice in the fourth line of (3.14) that after the differentiation there is no dependence on St, since ST is linear to St. We will need this remark later, when we proceed with

the extension of the formula for gamma. We re-introduce the asset prices in the fifth line only for practical purposes and because the asset prices are data that are readily available in the simulations.

Combining (3.13) and (3.14) with (3.12) gives the extended pathwise estimator of delta: ∆i(t, St) = e−r(T −t) 1 (SiT≥K) 1 St n X j=1 S_ti_j1_(t_j≥t) n (3.15)

Notice that if we take t = 0, the sum equals the average ST and the expression is

identical to (3.10). So, the extended estimator can be used for all t ∈ [0, T ). The delta estimate in the extended setting is thus:

b ∆(t, St) = 1 Ster(T −t) k X i=1 1 (SiT≥K) Pn j=1 Si tj 1_{(tj ≥t)} n ! k . (3.16) 3.3.2 Gamma

For the gamma, one proposed alternative is to use a hybrid calculation, that is, a finite difference gamma of the actual pathwise delta and one ‘bumped’ pathwise delta:

b

Γ(t, St) =

b

∆(t, St+ h) − b∆(t, St)

h . (3.17)

Unfortunately for its simplicity, the method’s accuracy depends again on the choice of the bump size. Therefore, we turn to the pathwise derivatives method for the gamma as well. To do this we need to differentiate the true ∆, that is, the expectation of the unbiased extended estimator (3.15).

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14 Georgios Tsiounis — Pricing & Hedging of an Asian Option ∆(t, St) = EQ  e−r(T −t) 1_(S T≥K) 1 St n X j=1 Stj 1(tj≥t) n It  ,

where It = {St1 = st1, ..., Stj = stj, St = st} is the information available at time

t. We suppressed the path index as it is not needed in this context. Following again Broadie & Glasserman, we approach the differentiation by looking at the behavior of ∆ when St is changing. We already mentioned that only the indicator function depends

on the value of St. We may therefore write:

∆(t, St+ h) − ∆(t, St) h = EQ " e−r(T −t) 1_(S T(St+h)≥K)− 1(ST(St)≥K) 1 St Pn j=1 S_tj1_{(tj ≥t)} n It # h = EQ " e−r(T −t)1_(S T(St+h)≥K>ST(St)) 1 St Pn j=1 S_tj 1_{(tj ≥t)} n It # h = EQ " e−r(T −t)1_(S T(St+h)≥K>ST(St)) 1 St ST − Pn j=1 S_tj1_{(tj <t)} n ! I_t # h .

Here, the notation ST(·) is used to show the average’s dependence on the asset value

at t. Now, gamma equals the limit of the above as h tends to 0. The expectation, then, concentrates on the density of ST at K, which, standing at t, is conditional

on the past fixings being equal to their observed values. We denote this density as f (K) def= f ( ST = K

I_t). Moreover, re-writing the last term using S_T and the past fixings allows the immediate calculation of this term at the event ST = K, simply by

replacing ST by K and Stj = stj, tj < t. Thus, the resulting formula for gamma is:

Γ(t, St) = e−r(T −t) 1 St  K − n X j=1 Stj 1(tj<t) n   f (K). (3.18)

The issue with the calculation of the value of gamma is reduced to calculating the density of the option finishing at-the-money and is not a trivial issue, since the distribution of the sum of stock prices is unknown. We need to estimate this probability and that is where the Monte Carlo paths come into play. We first look at the probability PQ of the event ST ≤ K: PQ= PQ ST ≤ K I_t = PQ   Pn j=1Stj 1_(t_j≤t) +Pn j=1Stm 1_(t_j_>t) n ≤ K I_t   = PQ   Pn j=1stj 1_(t_j≤t) +Pn j=1Stj 1_(t_j_>t) n ≤ K   = PQ   n X j=1 Stj 1_(t_j_>t)≤ nK − n X j=1 stj 1_(t_j≤t)  

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Conditioning on the random variables {Stj, t < tj ≤ tn−1} and by the Law of Total Probability, we get: PQ= EQ  PQ   n X j=1 Stj 1(tj>t) ≤ nK − n X j=1 stj 1(tj≤t) Stj, t < tj ≤ tn−1     = EQ  PQ   n X j=1 Stj 1(t<tj≤tn−1) + Stn ≤ nK − n X j=1 stj 1(tj≤t) Stj, t < tj ≤ tn−1     = EQ PQ Stn Stn−1 ≤ x Stj, t < tj ≤ tn−1 , (3.19) where, x = 1 Stn−1  nK − n X j=1 stj 1(tj≤t) − n X j=1 Stm 1(t<tj≤tn−1)  .

We have found an unbiased estimator of the probability PQ. If all the information prior to tn is given, this probability is well understood since we know that the stock

returns follow a lognormal distribution. We define the conditional density of stock re-turns: ftn−1_(x)def₌ 1 xσ√∆tn2 √ π exp      − lnx −r − σ₂2∆tn 2 2σ2_∆t n      , (3.20)

where, ∆tn= tn− tn−1. Then (3.20) is an unbiased estimator of f (K). Suppose we can

sample xi pathwise. Then, combining with (3.18) we get the extended path estimator for gamma which can be used for all t ∈ [0, T ):

Γi(t, St) = e−r(T −t) 1 St  K − n X j=1 Stj 1(tj<t) n   ftn−1(xi) (3.21) Sampling the xi and evaluating the gamma estimator is quite straightforward since all the required information in (3.21), that is the current asset price and fixings until second to last, is available pathwise in Monte Carlo. Substituting the fixings, evaluating the density ftn−1_(xi_{) and then averaging results in the estimate for f (K) and gamma. The}

extended estimate for all t, is given by:

b Γ(t) = e−r(T −t) 1 St  K − n X j=1 Stj 1_(t_j_<t) n   k X i=1 ftn−1_(xi₎ k . (3.22) Last, note that by setting t = 0 in (3.21), the extended pathwise gamma estimator reduces to the estimator given in Broadie & Glasserman.

3.3.3 Greeks Standard Error

In the case of the ordinary Monte Carlo, the Greeks standard error is calculated similarly as the standard error of the price (see (2.10)). When the Antithetic Variates method is used, some attention is required. The condition for the method to reduce the estimate variance is the monotonicity of the estimated quantity with respect to the random numbers. The monotonicity condition holds for delta. We therefore calculate the delta standard error similarly as in (2.12). The gamma, however, is not monotone. therefore we cannot be sure of the effect of the Antithetic Variates method on its estimate. For

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this reason, we use only half of the sample, which is i.i.d. and for which the ordinary formula (2.10) for the standard error applies.

c stdY = v u u u t 1 k 2 − 1 k 2 X j=1 Yj− b∆AV 2 , d SE∆AV = c stdY q k 2 , c stdΓ= v u u u t 1 k 2 − 1 k 2 X j=1 Γj_{− b}_Γ2_, d SEΓ= c std(Γ) q k 2 , (3.23) where, Yj = ∆i_AV + ∆ k 2+i AV 2 .

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Introduction to Hedging

Entering a position in a contingent claim, entails the exposure in a number of risks inherit in the claim. Hedging is the procedure of immunizing this position against one or more of these risk factors. It is achieved by building and maintaining a strategy called hedge portfolio which, by construction, when held together with the position in the claim, forms a system that is neutral to changes in the value of the targeted risk factors.

In some cases, a static hedge, that is a portfolio of assets with fixed weights, can be a possibility, but this depends on the hedged item, the availability of appropriate assets to be included in such a portfolio etc. More frequently the hedge strategies are dynamic, meaning the hedge portfolios need to be monitored and adjusted.

4.1 Dynamic Hedging in Black-Scholes

The set of risk factors associated with a contingent claim includes all the variables entering the valuation of the claim and affecting by a change in their value the price of the claim. In the previous chapter we discussed the sensitivity to these changes expressed as the Greeks.

In the case of European options in a Black-Scholes economy, the only source of risk is the value of the underlying asset. Market completeness and no arbitrage pricing are connected to the existence of a self-financing and replicating portfolio comprising of the underlying risky asset and the riskless asset, the weights of which are dynamically adjusted so that the system is neutral to movements of the underlying asset. This continuously rebalanced procedure is called delta hedging.

In reality, the impracticability of a continuously rebalanced strategy implies a de-parture from a perfect hedge to an area of hedging errors. However, the price of an option, seen as a function of the underlying asset can still be locally approximated by a portfolio of market assets. Provided that there exist appropriate assets to be used in such a portfolio, the delta and also the gamma of the option are involved in assigning the weights in the hedge portfolio. For a delta neutral portfolio, this condition can be easily assumed to hold, since the underlying asset and a riskless asset can be thought to be available. But, to achieve a gamma neutral position, the existence of another deriva-tive written on the same underlying asset is required, so that it can contribute with its convex price function to the construction of the hedge, or in other words, to enable matching the gamma of the hedged item.

Assuming that there exists a European call option on the same underlying S, with maturity at least equal to the maturity of the Asian, we can employ a delta-gamma hedging strategy. To continuously maintain the neutral position, the weights in the

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hedge portfolio should, at any t ∈ [0, T ), be:

wc(t) = b Γasian(t) Γc(t) wS(t) = b∆asian(t) − wc(t) ∆c(t), (4.1) where,

C, ∆c and Γc are the Black-Scholes price, delta and gamma of the call option and

wc, wS are the weights of the call option and the underlying asset respectively.

At the maturity of the option the hedge portfolio is liquidated, hence we take that wc(T ) = wS(T ) = 0.

At inception, the amount of money required to match the value of the hedged item and the set up of the hedge is kept in a money market account:

M (0) = bVasian(0) − wS(0) S0− wc(0) C(0). (4.2)

Rebalancing of the hedge portfolio can, in reality, be performed only on a finite set of dates t0 < t1 < ... < ts = T during the lifetime of the hedged item. It is done by

reassessing the portfolio weights in order to neutralize again the delta and gamma. The weight adjustment at ti, i = 1, ..., s is done by bying or selling wc(ti) − wc(ti−1) units of

the call option and wS(ti) − wS(ti−1) units of stock. The balance is kept in the money

market account, which itself grows at the risk free rate:

M (ti) = M (ti−1) er(ti−ti−1)− (wc(ti) − wc(ti−1)) C(ti) − (wS(ti) − wS(ti−1)) Sti. (4.3)

The value of the hedge portfolio at any time t ∈ [0, T ], with tithe latest rebalancing

time is:

H(t) = M (ti) er(t−ti)+ wS(ti) St+ wc(ti) C(t). (4.4)

Setting wc(t) = 0, ∀t in (4.1), (4.2), (4.3) and (4.4), gives the respective expressions

for a delta-hedge.

No assumption has been made on replicating and self-financing properties of the hedge. In fact, equality (4.2) may not hold for t 6= 0, revealing a hedging error. Rebal-ancing frequency and also inaccurate estimates of the price and sensitivities, especially in hedging non-tradable items, can affect the hedge performance.

4.2 Measures of Hedge Performance

In our first set of hedge tests, we generate a number of asset paths to use as market data and perform 4 different strategies combining weekly or daily rebalancing with delta or delta-gamma hedging. Using Black-Scholes both as the market and as the hedge model our aim is to isolate and numerically illustrate the combined effect that the Monte Carlo estimation in pricing and sensitivities and the rebalancing frequency have on the performance of a delta and a delta-gamma hedge.

The results of these tests are reported in terms of the hedging error. At any time t ∈ [0, T ] it is defined as the difference between the value of the hedged item and the value of the hedge. The value of the hedge is market quoted while for a hedged item that is non-tradable market prices do not exist. This is a commonality for an insurer when hedging embedded options and guarantees. Therefore, only at maturity, where the price-payoff is certain (and certainly equal to the internally calculated price), we can

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observe the realized hedging error (essentially, the realized P &L). Prior to maturity, we shall use the internally calculated price. We define:

HE(t) = bVasian(t) − H(t), ∀t ∈ [0, T ]. (4.5)

Testing against multiple asset paths we are able to sample from the distribution of the realized hedging error. We shall report different statistics: the sample mean, the sample standard deviation, the 0.5% Value at Risk and the 0.5% Tail Value at Risk. For the latter two and for 1 000 paths we approximate as the 5th worst case scenario and the average of the 5 worst case scenarios respectively. These two are also highly relevant to the capital requirements prescribed by the regulator.

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Chapter 5

Hedging Under Misspecified

Model

So far, we restricted ourselves to the Black-Scholes setting, where the interest rate and the volatility of the asset are assumed to be constant throughout the lifetime of the option. These assumptions fail to prevail in reality. This fact has triggered the development of more sophisticated models that capture the variation in these quantities, expressing them as processes of time, asset price, or both.

Under such a model, the pricing and hedging methods have to be reviewed. Very often, the resulting pricing methods require complicated calculations that are, addition-ally, subject to issues of model’s goodness of fit and calibration. Such issues are especially important in the case of over-the-counter products whose prices and volatilities are not observed in the market.

The purpose of building an efficient hedge may, therefore, not be served as satisfac-torily as expected by choosing a more sophisticated model. Or, to invert the previous, using a hedge model different from the one we assume to be the description of the evolution of the asset asset may still produce efficient hedges.

In the following we present the work of Carr & Madan (2002), El Karoui, Jeanblanc-Piqu´e & Shreve (1998) and Schied & Stadje (2007) related to hedging under model misspecification. Particularly, the possibility to use the Black-Scholes model as the mis-specified hedge model when stochastic volatility is assumed to prevail in reality. In their work we find an impressive connection between the hedging error and the choice of volatility.

Inspired by these results we will perform hedge tests where different choices of Black-Scholes volatility are tested for their hedging performance and approach the question of whether there exists an optimal choice of volatility. We maintain the viewpoint of an insurer and consider a measure of performance and optimality that relates to the insurer’s functions and regulatory constraints.

5.1 Misspecified Volatility and Hedging Error

5.1.1 The European Option Case

Suppose a hedger wants to delta-hedge (continuously) a short position in a European option, assuming, incorrectly, a constant volatility. The misspecified volatility affects the price of the option as well as the weights in the hedge portfolio. Therefore, it is expected that the hedge portfolio will not satisfy the replicating condition and produce a hedging error.

Carr & Madan (2002), in a broader discussion on getting exposure to volatility, analyze three different approaches including the aforementioned. They quantify the way the resulting hedging error is driven by the difference of the realized volatility σt

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and the assumed constant volatility σ.

The asset price dynamics under the risk neutral measure in the two models are: d ˜St= r ˜Stdt + σtS˜tdWt and (5.1)

dSt= r Stdt + σ StdWt. (5.2)

The interest rate r is assumed to be constant in both models and the stochastic volatility σt follows an unknown stochastic process.

The price V (t, St) of the European option is known to solve the Black-Scholes partial

differential equation: ∂V (t, St) ∂t + r St ∂V (t, St) ∂St = r V (t, St) − 1 2σ 2_S2 t ∂2_{V (t, S} t) ∂S_t2 (5.3) subject to the terminal condition V (T, ST) = P (ST). Now suppose that we price the

option using V while observing the asset prices derived from the market model, thus obtaining the price V (t, ˜St).

Applying Itˆo’s lemma to V (t, ˜St), assuming (5.1) and given that the derivatives of

V satisfy (5.3) we get: dV (t, ˜St) = ∂V (t, ˜St) ∂t dt + ∂V (t, ˜St) ∂St d ˜St+ 1 2 ∂2V (t, ˜St) ∂S2 t d[ ˜St] (5.1) = ∂V (t, ˜St) ∂t dt + ∂V (t, ˜St) ∂St r ˜Stdt + σtS˜tdWt + 1 2 ∂2_{V (t, ˜}_S t) ∂S_t2 r ˜Stdt + σtS˜tdWt r ˜Stdt + σtS˜tdWt = ∂V (t, ˜St) ∂t + ∂V (t, ˜St) ∂St r ˜St ! dt +∂V (t, ˜St) ∂St σtS˜tdWt+ 1 2 ∂2_{V (t, ˜}_S t) ∂S2 t σ_t2S˜_t2dt (5.3) = r V (t, ˜St) − 1 2σ 2_S2 t ∂2V (t, ˜St) ∂S_t2 ! dt +∂V (t, ˜St) ∂St σtS˜tdWt+ 1 2 ∂2V (t, ˜St) ∂S2 t σ_t2S˜_t2dt = r V (t, ˜St) dt + 1 2 ˜ S_t2 ∂ 2_{V (t, ˜}_S t) ∂S2 t σ_t2− σ2 dt +∂V (t, ˜St) ∂St σtS˜tdWt. (5.4)

Considering the transformation er(T −t)V (t, ˜St) and because, obviously, d[er(T −t), V (t, ˜St)] =

0, we get: d er(T −t)V (t, ˜St) = = er(T −t)dV (t, ˜St) + V (t, ˜St) d er(T −t) = r er(T −t)V (t, ˜St) dt + 1 2e r(T −t)_S_˜2 t ∂2V (t, ˜St) ∂S2_t σ 2 t − σ2 dt+ ∂V (t, ˜St) ∂St σter(T −t)S˜tdWt− r er(T −t)V (t, ˜St) dt = 1 2e r(T −t)_S_˜2 t ∂2V (t, ˜St) ∂S2 t σ2_t − σ2 dt +∂V (t, ˜St) ∂St σter(T −t)S˜tdWt. (5.5)

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Now, let Ft = er(T −t)S˜t be the price in the market model of a futures contract on

the same asset S, maturing at T . We have: dFt = d er(T −t)S˜t = er(T −t)d ˜St− r er(T −t)S˜tdt (5.1) = er(T −t) r ˜Stdt + σtS˜tdWt − r er(T −t)S˜tdt = er(T −t)σtS˜tdWt. (5.6)

Substituting Ft and (5.6) in (5.5) gives:

d er(T −t)V (t, ˜St) = 1 2e −r(T −t) F_t2∂ 2_{V (t, ˜}_S t) ∂S2 t σ2_t − σ2 dt + ∂V (t, ˜St) ∂St dFt.

Integrating over the lifetime of the option yields:

V (T, ˜ST) − erT V (0, ˜S0) = Z T 0 1 2e −r(T −t) F_t2∂ 2_{V (t, ˜}_S t) ∂S2 t σ_t2− σ2 dt + Z T 0 ∂V (t, ˜St) ∂St dFt, (5.7)

and, finally, by rearranging we get the equation (9) in Carr & Madan:

P ( ˜ST) + Z T 0 e−r(T −t) F 2 t 2 ∂2_{V (t, ˜}_S t) ∂S2 t σ2− σ_t2 dt = erT_{V (0, ˜}_S 0) + Z T 0 ∂V (t, ˜St) ∂St dFt. (5.8) The right hand side describes the terminal value of a dynamically managed portfolio. The price at which the European option is sold is calculated in the misspecified model and the amount received is kept in the money market account for the entire lifetime of the option. The portfolio holds, also, at every time t ∈ [0, T ], ∂V (t, ˜St)

∂St futures contracts

on the same asset and with the same maturity. The left hand side suggests that the strategy has the option payoff P ( ˜ST) as its targeted value and that it misses the target

by an amount which can be regarded as the realized hedging error of the strategy based on the incorrect assumption of the volatility:

HE = Z T 0 e−r(T −t)F 2 t 2 ∂2V (t, ˜St) ∂S_t2 σ 2_{− σ}2 t dt (5.9)

Assuming Γ > 0, which is true for vanilla options, it is obvious that the hedging error can be zero only if σt = σ. Taking into account the convexity of the European

payoffs, we see, further, that the strategy results in a certain profit when the constant volatility dominates the true volatility.

A similar expression for the hedging error (5.9) is derived in El Karoui, Jeanblanc-Piqu´e & Shreve (1998). They prove that the formula for the hedging error holds in the case of a misspecified volatility that is stochastic and not path-dependent.

5.1.2 Application to Asian Options

Until now, the presented methodology revolved around European options. El Karoui, Jeanblanc-Piqu´e & Shreve take on the the case of the continuous Asian option and in (Lemma 11.1) they introduce a hypothetical asset whose value at maturity matches the asset price average in the Asian payoff. The Asian can, thus, be treated as a European option on this hypothetical asset and allowed the results obtain before.

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Under the same assumptions as in (5.1) for the asset dynamics, the risk-neutral price of the continuous Asian option is:

V (t) = e−r(T −t)EQ max 1 θ Z T T −θ S(u) du − K, 0 Ft , where, T − θ is the start of the averaging period.

Consider the hypothetical asset X:

Xt= e−r(T −t)EQ 1 θ Z T T −θ S(u) du Ft . Indeed the value of X at T is:

XT = 1 θ Z T T −θ Sudu,

so, a European call option on X has the same payoff as the continuous Asian option. By the addition rule and Fubini’s theorem we have:

Xt= e−r(T −t)EQ 1 θ Z T T −θ Sudu Ft = e−r(T −t)1 θ Z max(t,T −θ) T −θ EQ[ Su| Ft] du + e−r(T −t) 1 θ Z T max(t,T −θ) EQ[ Su| Ft] du

The first term corresponds to the part of the averaging period prior to t, if, of course, t > T −θ, hence, the max(·) and the second term to the remaining period until maturity. Since the interest rate is deterministic, forward prices equal futures prices (Black (1976)) and we can use:

EQ[ Su| Ft] = ( Ster(u−t) , t ≤ u Su , t > u to yield: Xt= e−r(T −t) 1 θ Z max(t,T −θ) T −θ Sudu + e−r(T −t) 1 θSt Z T max(t,T −θ) er(u−t)du = e−r(T −t)1 θ Z max(t,T −θ) T −θ Sudu + 1 rθ 1 − e−r(T −max(t,T −θ))St

Applying Itˆo’s formula, it can be shown that this process satisfies:

dXt= r Xtdt + σtρtStdWt, (5.10) where, ρt= 1 rθ 1 − e−r(T −max(t,T −θ)), (5.11) The volatility of X(t) is: γt = σt_XSt_tρt, which is positive and, in view of the process Xt,

it is: γt≤ σt , ∀t.

This suggests a super-replicating strategy for the continuous Asian option, presented in Theorem 11.2. The Black-Scholes price, CX of a European call option on X using a

volatility that dominates γ, paired with a hedging strategy that matches the initial call price and constantly holds ρt∂C_∂XX(t)_t units of the asset S, is a superstrategy for the Asian

option. The methodology leads to a super-replicating strategy using a scaled delta, we, however, need a result on the hedging error with non-scaled delta.

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5.1.3 A Direct Approach to Asian Options

Rather than working with a hypothetical asset such as X, Schied & Stadje (2007) approach the case of path dependent options directly. They extend the result of El Karoui, Jeanblanc-Piqu´e & Shreve and prove that robustness of a delta hedge based on a local volatility model, hence also the Black-Scholes model, applies also in the case of path dependent options, including the discrete arithmetic Asian. They define robustness as the existence of a superhedge for a short position when the local volatility overestimates the true volatility. Again, as in El Karoui, Jeanblanc-Piqu´e & Shreve, convexity is the sufficient condition permitting the result. Their contribution which is more relevant to our context, is the proof that the partial differential equation in Carr & Madan remains valid for path dependent options as well, thus, allowing the hedging error formula (5.9) to also apply in the case of path dependent options.

They start by assuming zero interest rates, without this limiting the extension to deterministic rates, and set the asset dynamics:

dSt= σ(t, St) StdWt (5.12)

By S and σ(t, S) they denote the underlying asset and local volatility in the misspecified model and distinguish from the true asset values ˜S for the evolution of which there is no need to consider a stochastic process. The local volatility is assumed to be contin-uously differentiable, bounded and non-zero and the product σ(t, S)S to be uniformly continuous in t and Lipschitz continuous in S.

In the model world, the delta hedging strategy can be thought to satisfy:

P = V (T, St1, ..., Stn) = V (0, S0) + Z T 0 ∂V (t, St1, ..., Stk, St) ∂St dSt, (5.13)

where k is such that tk ≤ t < tk+1. We have already used the same notation for the

Asian options. The option price at inception V (0, S0) depends only on the initial value

of the asset, the terminal price V (T, St1, ..., Stn) on all the fixings and the option delta

at t on St and the past fixings St1, ..., Stk, St, tk≤ t < tk+1.

To derive a similar expression for the delta hedging strategy using the true values ˜S without choosing a specific stochastic process, Schied & Stadje call forth F¨olmer and the pathwise Itˆo integral. The value of such strategy, then, results in the same expression (5.13) with true values ˜S replacing S, by taking the limit of the discrete rebalancing strategy as the interval length between rebalancing times tends to zero. We do not need to enter this domain, since we will specify a stochastic process for the evolution of the true asset prices.

Assuming bounded derivatives when using the true values ˜S so that the pathwise Itˆo integral is well-defined, we can apply Itˆo’s formula to get:

V (T, ˜St1, ..., ˜Stn) − 1 2 Z T 0 ∂2_{V (t, ˜}_S t1, ..., ˜Stk, ˜St) ∂ ˜S_t2 d[ ˜S]t− Z T 0 ∂V (t, ˜St1, ..., ˜Stk, ˜St) ∂t dt = V (0, ˜St1) + Z T 0 ∂V (t, ˜St1, ..., ˜Stk, ˜St) ∂ ˜St d ˜St, (5.14)

where [ ˜S]t denotes the quadratic variation of the true asset prices. Expression (5.14)

shows that the delta hedging strategy in the right hand side misses the targeted option payoff by a quantity which can be thought of as the realized hedging error:

HE = −1 2 Z T 0 ∂2V (t, ˜St1, ..., ˜Stk, Yt) ∂ ˜S2 t d[ ˜S]t− Z T 0 ∂V (t, ˜St1, ..., ˜Stk, ˜St) ∂t dt (5.15)

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Schied & Stadje continue to prove that, given a path dependent payoff function, the corresponding price function V (t, x1, ..., xk, x) is continuous on (t, x) ∈ [0, T ] × [0, ∞),

piecewise continuously differentiable in between observation times, twice continuously differentiable in x ∈ (0, ∞) and, for t ∈S

k(tk, tk+1), x ∈ (0, ∞), it satisfies the partial

differential equation: ∂V (t, x1, ..., xk, x) ∂t + 1 2σ(t, x) 2_x2∂2V (t, x1, ..., xk, x) ∂x2 = 0, (5.16)

∀k and ∀{x₁, ..., xk}. The proof makes use of the fact that the price functions of

Euro-pean payoffs which solve the partial differential equation (5.16) subject to the terminal condition V (T, x) = P (x) are continuous, twice differentiable and satisfy a polynomial growth condition. They work backwards in the intervals between observation times, by introducing functions depending on the observation at the right end of the intervals and proving them to be payoffs, hence, proving that the corresponding price functions satisfy (5.16) in between observation times (for the detailed derivation see the proof of Proposition 2.2. in Schied & Stadje (2007)).

Combining (5.15) with the partial differential equation (5.16) yields:

HE = −1 2 Z T 0 ∂2V (t, ˜St1, ..., ˜Stk, ˜St) ∂ ˜S_t2 d[ ˜S]t+ Z T 0 1 2σ(t, ˜S) 2_S_˜2 ∂2V (t, ˜St1, ..., ˜Stk, ˜St) ∂ ˜S_t2 dt (5.17) If we made the assumption that true asset prices follow a stochastic process then we could substitute the quadratic variation of ˜St, combine the integrals and result in an

expression similar to (5.9) which extends the application of the hedging error formula to path dependent options and to Asian options, in particular.

5.2 Possible Choices of the Misspecified Volatility

5.2.1 Asian Implied Volatility

The implied volatility is the value of the volatility such that the Black-Scholes price of an option matches the price observed in the market. With the other parameters of the Black-Scholes model being directly observable, it is the only variable that needs to be calibrated. Calculating the implied volatility by matching the Black-Scholes and market prices, serves also as the calibration of the Black-Scholes model. The calibration however may consider a range of products, which then means a solution in terms of minimizing the squared sum of pricing errors across these products. The range of products leads to a local or global calibration. The market price in our tests is actually derived from the Heston model. It is common, however, that one must rely on a model price for a non-tradable product to calculate the implied volatility.

Moreover, the implied volatility is a concept used extensively by practitioners. For example, Plat (2011) mentions a technique based on implied volatilities to be used when approximation is allowed or other complications impose it. The implied volatility for a tradable product is calculated from its market price at the time the pricing-hedging model for the insurance product is calibrated. Then, the implied volatility for the insurance product from its model price is calculated as well and the ratio of the two determines a scaling factor. At a later time and if no updated calibration of the model is available, this scaling factor is used to adjust the insurance product’s implied volatility given an updated implied volatility of the tradable product. The technique essentially assumes that the relationship between the implied volatilities of the tradable and the insurance product is unchanged for approximative purposes. From a practical perspective, implied volatilities for non-tradable products are necessarily deduced from model prices and this involves the frequent recalibration of the model to market prices

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of tradable assets, which, depending on the model, can be an arduous task for the frequency required.

The implied volatility makes a reasonable choice for our tests. We, however, restrict our choice of assumed volatility to remain the same for the entire lifetime of the Asian option and calculate it directly for this option. Apart from volatility dynamics, the characteristics of the respective product are involved in determining its implied volatility. And with a path dependent product like the Asian in our case, we could hope that the implied volatility can capture more information on the distributional properties of the underlying asset across the relevant period.

5.2.2 Integrated Stochastic Volatility

The motivation behind this approach arises from observing the hedging error formu-las presented in the previous section. There, we find that a scaled difference of the squares of the misspecified and the true volatilities appears as a reoccuring term. Sup-pose, although without proving, that, with changing volatility, the forward asset price and option gamma do not vary much. Taking a constant volatility as the misspecified volatility and, then, taking the expectation of the hedging error to be zero, results in:

E Z T 0 σ2− σ_u2 du ≈ 0 ⇒ E Z T 0 σ2du − Z T 0 σ2_udu ≈ 0 ⇒ E Z T 0 σ2du ≈ E Z T 0 σ_u2du ⇒ σ2T ≈ E Z T 0 σ_u2du ⇒ σ ≈ v u u tE h RT 0 σ2udu i T (5.18)

We can say informally that the choice of integrated volatility aims for a replication in expectation. The constant volatility in this case can be interpreted as the expected average true volatility over the lifetime of the option. The expression already holds for true volatility that is time but not path dependent. Overlooking the fact that our selected real world model will have path dependent volatility, we can still use the average over the generated volatility paths.

5.3 Description of the Conducted Hedge Tests

The ultimate goal of this thesis is to explore the existence of an ‘optimal’ constant volatility to be used for the hedging of a discrete arithmetic Asian option when stochastic volatility is assumed to prevail in reality. For this purpose we perform a series of hedge tests using synthetical market data from a stochastic volatility model, the Heston model in particular. The hedge strategy that will be employed is a daily rebalancing delta hedge based on the misspecified volatility. In this context, but in a far more limited scale, our tests bear similarities to the tests conducted by Schr¨oter and Monoyios [2012]. Results on the employed strategy in a non-misspecified setting will be drawn by the tests described in the previous chapter and will serve as a reference for the impact of the introduced misspecification.

Both the proposed volatilities of the previous section are considered in addition to randomly selected values within a meaningful range. In order to explore the influence of the Heston calibration on the existence of an ‘optimal’ volatility, we repeat our

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experiments for different parameter sets. For every parameter set and choice of volatility we sample from the distribution of the hedging error using 1 000 realizations of asset paths and the corresponding hedges for a maturity of 1 year and considering 250 trading days and equal fixings for the Asian option.

5.4 Measures of Hedge Performance

We have already mentioned in the previous chapter the statistics that we will report on the hedging error. In our final tests, apart from these measures, we need to identify a single measure representing the hedge performance, which is appropriate for comparing the different tests.

We return to our insurer’s point of view. The price of the product determines to great extent the premium asked of the client. It is true that, for vanilla options in the Black-Scholes model, the price increases with volatility and, in our tests, the same relationship held for the Asian option as well. The price is also the set-up cost of the hedge portfolio which, performed with a different value of volatility, aims for a different hedging error, at least in expectation. The results on robustness of Black-Scholes establish that super-replication is achieved with volatility overestimation, but this is related to price overestimation. On the other hand, the remaining risk exposure of an insurer lays down the capital requirements for solvency purposes. Specifically, the 0.5% worst case scenario, expressed, for example, as the (Tail) Value at Risk. The Cost of Capital (6%) for the 0.5% worst case scenario is reserved and, in practice, charged in the premium. Therefore, we opt for the tradeoff between price (hedging costs) and Solvency Capital Requirements, reflected in the premium, as a performance measure.

In order to level the comparison among the different hedges we need to account for the fact that the different hedges result in different mean hedging errors calculated under the real world measure. We consider the initial value of the product derived in the misspecified model surcharged by the Cost of Capital corresponding to the (Tail) Value at Risk and adjusted for the mean hedging error. The adjustment takes also into account the time value of money. Note here that, since the interest rate is assumed constant, this adjustment indeed ‘centers’ the distribution at 0, thus levelling the comparison. Note also that the shift of the distribution affects the tail linearly, by the value of the mean hedging error. We, therefore, have the following expression for the premium:

P remium = bV (0) − e−rTEP[HE] − CoC (T )V aR(HE) − EP[HE] , (5.19) where, (T )V aR and HE are in their physical sign, meaning a loss is negative and (T )V aR denotes the use of either the Value at Risk or the Tail Value at Risk.

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Chapter 6

Real World Model

6.1 Heston Model

The Heston model is probably the most famous stochastic volatility model. Driven by previous research justifying the use of stochastic volatility and reported weaknesses of the Black-Scholes model attributed to its assumptions of the asset returns distribution, Heston (1993) proposed this model which allowed correlation between volatility and asset price, partially explained the volatility smile and admitted closed-form pricing formulas of the Black-Scholes type.

The asset price at time t is assumed to follow: dSt= µ Stdt +

√

vtStdWt(1), (6.1)

where,

µ is the constant drift parameter, the annualized drift rate of S, vt is the annualized variance of returns of S and

W_t(1) is a standard Brownian Motion.

The variance vt is assumed to follow the square-root process:

dvt= κ (θ − vt) dt + ξ

√

vtStdWt(2), (6.2)

where,

κ is the rate of mean reversion, θ is the long-term mean variance, ξ is the volatility of volatility and

W_t(2) is a standard Brownian Motion correlated with W_t1,d[W_t(1), W_t(2)] = ρ dt. The parameters of the model allow for a wide variety of asset returns distributions and implied volatility surfaces to be produced. For example, skewness is affected by the correlation ρ. A positive ρ is responsible for a fatter right tail associated with large variance and a negative ρ for a fatter left tail associated with small variance. Empirical evidence support the negative correlation between volatility and asset returns. The volatility of volatility ξ affects the kurtosis. In the special case of zero ξ and long-run mean variance equal to its initial value, we return to the Black-Scholes model and to log-normality which is often refuted by market data. If non-zero, ξ contributes to the thickening of the tails. We take the volatility of volatility to be non-zero and smaller than the initial volatility.

Pricing options requires shifting to risk-neutral probabilities. The stochastic process for the volatility takes the form:

dvt= κQ θQ− vt dt + ξ

√

vtStdWt(2), (6.3)

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where, kQ= k + λ and θQ= _k+λk θ.

Parameter λ is interpreted as the market price of volatility risk. Estimating the risk neutral parameters requires market data, such as option prices or returns of delta-hedged positions on options. Since our tests use synthetical data, we chose a small positive value for λ which resulted in a risk neutral price smaller than the price under the real world measure.

6.2 Heston Model Simulation

In this section, we address the issue of simulating volatility and asset paths based on the Heston model. For the purposes of our hedge tests we need to generate synthetical market data under the real world measure and also calculate our targeted volatilities (implied and integrated) under the risk meutral measure. We, therefore, need to employ a discretization scheme for the stochastic processes. We present the algorithm for the risk neutralized processes. The same method was used for the asset price process under the real world measure for the generation of the market pseudodata.

Van Haastrecht & Pelsser (2008) compare different discretization methods under the prism of both computational efficiency and performance. They show numerically that the Full Truncation scheme of Lord et al (2008), which is based on the Log-Euler scheme and is very easy in its implementation, performs better than other methods which appear to be more sophisticated. Although the biasedness of the method is unavoidable, they suggest that with a sufficient number of time steps, which, for their experiments was at least 32 for a yearly horizon, it can be restricted to a small value. In our tests, we shall use 250 time steps for a horizon of one year.

Consider, first, the variance process and its naive Euler discretization over a time interval [t, T ]. It is relevant to think of the interval as the time to maturity of an option. For a partition t = t0 < t1< ... < tm = T , we have:

vti+1 = vti− κ Q _θQ_{− v} ti (ti+1− ti) + ξ √ vtipti+1− tiZ (2) i+1, (6.4) where, i = 1, ..., m − 1 and

Z_i+1(2) are i.i.d. standard normal random variables.

The problem in the above expression is that, for non-zero ξ and even with small time step, there is positive probability of negative values of variance. Several treatments are proposed in order to eliminate the negative values. The one that performs the best is the so-called Full Truncation scheme:

vti+1 = vti− κ Q _θQ_{− v}+ ti (ti+1− ti) + ξ q v+_t ipti+1− tiZ (2) i+1. (6.5)

Here, x+ denotes the positive part function max(x, 0).

Regarding the asset process, the following Log-Euler scheme is adopted:

log(Sti+1) = log(Sti) +

r −1 2v + ti (ti+1− ti) + q v_t+ ipti+1− tiZ (1) i+1, (6.6) where, i = 1, ..., m − 1 and

Z_i+1(1) are i.i.d. standard normal random variables.

The Log-Euler scheme is known to introduce no discretization error for the asset price. As for the correlated random variables Z_i+1(1) and Z_i+1(2) we can use the Cholesky decomposition. Specifically, we, first, draw two independent standard normal random numbers N1 and N2 and then set Zi+1(2) = N2 and Zi+1(1) = ρ N2+

p

1 − ρ2_N 1.

Monte Carlo pricing & hedging of a discrete arithmetic Asian option.