Pricing of American Options by Least-Squares Monte Carlo with Control Variates

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Control Variates

David Bultsma (10200800) Supervisor: Andreas Rapp Second reader: prof. dr. Peter Boswijk

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Statement of Originality

This document is written by Student David Bultsma who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this docu-ment are original and that no sources other than those docu-mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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This thesis is about the pricing of American put options by Least-Squares Monte Carlo with control variates. The Heston model and the Sch¨obel-Zhu model are used to simulate the volatility and asset price. Interest rates are simulated using the model of Cox, Ingersoll and Ross. In order to improve the accuracy we use European put options as control variates. Both stochastic volatility models are calibrated to historical prices of put options on Facebook at 8 different dates over the years 2013-2016. It turns out that at almost every date all prices can be replicated quite well by both models. Next the model parameters obtained by calibration at one date are used to forecast option prices at another date 6 months later. These parameters still provide in some cases model prices close to the observed market prices.

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Introduction

The pricing and optimal exercise of options with American-style exercise features is an impor-tant and challenging problem in derivatives finance. Since these options can be exercised at any moment up to their expiration, the pricing of these is more difficult than pricing options that can only be exercised at one time. Many options that are traded are American options and can be found in all major financial markets including e.g. the equity, foreign exchange or commodity markets. Longstaff and Schwartz (2001) presented a simple and powerful approach for approxi-mating the value of American options by Monte Carlo simulation, called Least-Squares Monte Carlo. At any time at which the option can be exercised, the holder of an American option has to compare the expected payoff from continuation to the payoff from immediate exercise. Only if the latter is higher than the expected continuation value, the option would be exercised before maturity. The conditional expectation of the payoff from keeping the option alive is therefore crucial to the optimal exercise strategy. The idea behind the valuation algorithm of Longstaff and Schwartz (2001) is that this conditional expectation can be estimated using least-squares regression. The payoff from continuation is regressed on a set of functions of the state variables and an estimate of the conditional expectation is given by the fitted value from the regression. By estimating this conditional expectation at every time the option could be exercised, the op-timal exercise strategy for each simulated path is fully specified. Using this specification, the American option can be valued accurately.

This thesis addresses the valuation of American put options by Least-Squares Monte Carlo in the presence of stochastic interest rates and volatility. Two different stochastic volatility models are used to simulate volatility and asset price paths, namely the Heston model and the Sch¨obel-Zhu model. In order to increase the accuracy of the simulation and the Least-Squares Monte Carlo algorithm, European put options are used as control variates. In contrast to the American option, the true model value of the European option is known exactly and is deter-mined using Fourier transform methods and characteristic functions. However, an option pricing

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model is only useful if it is able to generate prices that are close to the prices that are observed in the market. An important task is therefore to calibrate the model. When calibrating a model the option prices as observed in the market are taken as given and one tries to find parameters such that the observed prices are replicated as closely as possible by the model. Both models are calibrated at 8 different dates in the beginning of January and June to investigate how well historical prices of put options on Facebook from the years 2013-2016 can be replicated by Least-Squares Monte Carlo with control variates, and whether there is a difference in how well the prices are replicated by both models. Next we forecast the option prices at 7 different dates in the beginning of January and June during the years 2013-2016, using some of the model parameters obtained by calibration half a year before the concerning date, to see whether these parameters still provide a decent fit 6 months later. At each date we analyze how well the prices are forecasted and provide an explanation why in some cases the estimates are further off and how the option prices are affected by the different model parameters.

This thesis is organized as follows. Chapter2discusses the theoretical background, consist-ing of some basic options theory, an explanation of the stochastic volatility and interest rate models that are used, the pricing of European options using Fourier transform methods and characteristic functions and finally the valuation of American options using the Least-Squares Monte Carlo algorithm with control variates. Chapter3discusses the research setup and results, starting with the calibration of the stochastic interest rate and volatility models and then the forecasting of the Facebook option prices at 7 different dates over the years 2013-2016. Chapter

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Theoretical Background

2.1 Options

An option is a derivative that gives the holder the right to buy or sell a certain underlying asset (e.g. stocks, currencies or indices) at a certain date for a certain price. Some options involve trading the actual stock when they are exercised, whereas an option on e.g. an index is settled in cash. The party selling the option is called the option writer and the party buying the option is called the option holder. In contrast to forwards and futures, where the holder is obligated to trade the underlying asset, the holder of an option does not have this obligation. A call option gives the holder the right to buy the asset and a put option the right to sell the asset. The price in the contract is called the strike price or exercise price and the date is called the exercise date or maturity (usually denoted K and T respectively). For any given asset at any given time, many different options with different strike prices and maturities are traded. Options can be classified into different styles, the two most common are European and American options. European options can be exercised only on the expiration date itself, whereas American options can be exercised at any time up to the expiration date. Due to this difference, the prices of European and American options for the same asset and with the same strike price and maturity are different. Many options that are traded are American style. Since American style options can be exercised at any time up to the expiration date, analyzing these options is more difficult than their European counterparts. This thesis focuses on the pricing of American put options. An option can be in the money (ITM), at the money (ATM) or out of the money (OTM). Denoting the asset price by S, a call (put) option is in the money when S > K (S < K), at the money when S = K and out of the money when S < K (S > K). Obviously, an option that is out of the money will not be exercised. The payoffs at time T of a call and a put option are given by

max {ST − K, 0} and max {K − ST, 0}

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respectively.

If we consider for instance a European call option on a stock with a strike price of 100, there are several factors that influence the today’s price of this option (see Hilpisch (2015)):

• Initial stock price (S0): if the initial stock price would be e.g. 80, it is more likely that

when the option expires the stock price is higher than 100 (so the option has a positive payoff) than when the initial stock price would be 50.

• Time-to-maturity (T ): if the initial stock price would be 80 again and the time-to-maturity is only one week, it is less likely that when the option expires the stock price has exceeded 100 than when the time-to-maturity would be e.g. 3 months or more.

• Interest rate (r): the payoff of exercising an option is received in the future and therefore needs to be discounted to today.

• Volatility (σ): the volatility is a measure for the randomness of asset returns (standard deviation of the returns). When the initial stock price would be 80 and there are no price movements, it is less likely that the strike price will be exceeded at maturity than when there are bigger price fluctuations (and therefore a higher volatility).

Black, Scholes and Merton (1973) derived a formula for the price of a European call option, where the today’s price is a function of the above factors

C₀∗ = CBSM(S0, K, T, r, σ)

This option pricing model is discussed in the next section.

2.2 Black-Scholes-Merton Model

One of the most well known option pricing models is that of Black, Scholes and Merton (1973), who derived a closed-form solution for the price of a European option. They use a model of stock price behavior where it is assumed that percentage changes in the stock price in a short period of time follow a normal distribution:

∆S

S ∼ N (µ∆t, σ

2_∆t)

where µ is the expected return on the stock per year, σ the volatility of the stock price per year and ∆S the change in the stock price S in time ∆t.

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The model implies that logST S0 ∼ N µ −σ 2 2 T, σ2T and log ST ∼ N log S0+ µ −σ 2 2 T, σ2T

so that ST has a lognormal distribution, since log ST is normally distributed (see Hull (2012)).

The differential equations for the model are

dBt= rBtdt (2.1)

dSt= µStdt + σStdZt (2.2)

where the volatility (σ), the risk-free rate of interest (r) and the expected return (µ) are as-sumed to be constant. Btis a riskless asset, Sta risky asset and Zta standard Brownian motion.

Although the Black-Scholes-Merton model is a famous and widely recognized model, some of its assumptions are not realistic. For instance the assumption of constant volatility, which is usually not constant over time. Volatility has some characteristics that are commonly observed (see Hilpisch (2015)):

• Stochastic volatility: in reality volatility is not constant or deterministic, but stochastic. • Volatility clustering: periods in which there is high volatility are likely to be followed by periods in which volatility is high as well. Similarly, periods of low volatility are likely to be followed by periods in which volatility is also low.

• Leverage effect: there is a negative correlation between asset returns and volatility. Typically, the volatility increases when the stock price decreases.

• Mean reversion: volatility exhibits mean reversion; it never reaches zero and does not go to infinity, but the mean can change over time.

The assumption of constant volatility clearly does not take into account the above char-acteristics. Moreover, it also observed that stock returns exhibit fatter tails compared to the normal distribution, which means that large positive and negative returns occur more frequently compared to the normal distribution.

The assumption of a constant interest rate is not realistic either. Just like volatility, interest rates also have some characteristics that need to be taken into account (see Hilpisch (2015)):

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• Stochasticity: interest rates and especially short rates usually move in random fashion. • Mean reversion: interest rates do not trend to zero or infinity in the long term such that

there must always be mean reversion.

• Term structure: the yields of benchmark bonds and rates in interbank lending vary with time to maturity. This implies different (instantaneous) forward rates, i.e. different future short rate levels.

A realistic option pricing model must therefore take into account the above characteristics of volatility and interest rates. In order to deal with the variability of volatility and interest rates, several approaches have been suggested, such as the use of stochastic volatility and interest rate models.

2.3 Stochastic Volatility and Short Rate Models

Because of the above mentioned features of volatility that are commonly observed, a model is needed that captures these features. Two processes for stochastic volatility are considered; the model of Heston (1993), a mean-reverting square root process, and the model of Sch¨obel and Zhu (1999), a mean-reverting Ornstein-Uhlenbeck process.

2.3.1 Heston Model

The model of Heston (1993) is a popular stochastic volatility model, where stochastic volatility is not modelled directly, but stochastic variance. It is assumed that both the asset price and the volatility are stochastic processes. Letting Vt= vt2 denote the variance, the stochastic volatility

model of Heston (1993) in risk-neutral form is given by the following stochastic differential equations (SDE) dSt= rtStdt + p VtStdZt1 (2.3) dVt= κ(θ − Vt)dt + σ p VtdZt2 (2.4)

for 0 ≤ t ≤ T where the variables and parameters have the following meaning • St the asset price at time t

• r_tthe risk-free short rate at time t • V_t the variance at time t

• θ the long-term average of the variance

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• σ the volatility coefficient of the variance, also referred to as the volatility of variance • Zn

t standard Brownian motions

with κ, θ and σ > 0 and where Z_t1 and Z_t2 are allowed to be correlated, i.e. correlations are given by dZ_t1dZ_t2≡ ρdt.

Only if κ and σ would be equal to zero, the volatility would be deterministic, like in the Black-Scholes model. In that case the volatility in the Black-Scholes model would be the square root of the initial variance V0 in the Heston model. The parameter κ indicates the degree of

volatility clustering. When ρ is negative there is a negative correlation between St and Vt and

the leverage effect is therefore captured by this model. The process specifying the variance is called a mean-reverting square root process. Mean-reversion is a desired property for stochastic volatility or variance. The first term on the right hand side of equation (2.4) indicates that when the current variance is larger (smaller) than the long-term average of the variance, i.e. Vt > θ

(Vt < θ), the next variance will be decreased (increased). The value of κ indicates how much

the variance will be decreased/increased.

It can be shown that if κ, θ and σ satisfy the condition 2κθ > σ2 and V0> 0

the variance Vt is always positive, such that the above variance process is well-defined. The

above condition is also known as Feller’s condition (Feller (1951)).

A disadvantage of this kind of models is that the model depends heavily on the input pa-rameters. It is therefore important to have parameters that generate prices that are close to the real observed market prices. This is achieved by calibrating the model to observed market data. The calibration of a model is the process of finding parameter values, five in the case of the Heston model and the Sch¨obel-Zhu model, for which the model prices are as close as possible to the observed market prices. This is further discussed in chapter 3.

2.3.2 Sch¨obel-Zhu Model

Besides the model of Heston (1993), the model of Sch¨obel and Zhu (1999) is another popular model in financial modeling. The process that is used to model stochastic volatility is called an Ornstein-Uhlenbeck process and is applied by many financial economists for modelling stochastic volatility, e.g. Wiggins (1987) and Stein and Stein (1991). However, Stein and Stein (1999) used the assumption that the underlying asset and the volatility are not correlated, which is very

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restrictive and usually not the case. Sch¨obel and Zhu (1999) extended the formulation of Stein and Stein (1999) to the case where correlation is allowed. The Sch¨obel-Zhu model in risk-neutral form is given by the following SDE’s (see Zhu (2009))

dSt= rtStdt + vtStdZt1 (2.5)

and

dvt= κ(θ − vt)dt + σdZt2 (2.6)

with vt the volatility at time t, σ the volatility of the volatility that controls the variation

of the volatility, θ the long-term average of the volatility and κ the parameter controlling the speed of the adjustment of the volatility to its long-term average. Z_t1 and Z_t2 are allowed to be correlated, i.e. dZ_t1dZ_t2 ≡ ρdt. In contrast to the Heston model, the volatility is modelled instead of the variance.

Conditional on vs, vt is normally distributed (see Zhu (2009))

vt∼ N (m1, m2) (2.7) with m1= E[vt|vs] = θ + (vs− θ)e−κ(t−s) m2= Var[vt|vs] = σ2 2κ(1 − e −2κ(t−s)₎

The conditional variance m2 does not depend on the initial value vs. There is the possibility

for vt to be negative. The probability that vt is negative is P = N (−m1/

√

m2). However, this

probability is very small for a wide range of reasonable parameter values and does not lead to serious problems. If we would set for instance κ = 4, θ = 0.2, σ = 0.1, T − t = 0.3 and v0 = 0.2,

this would give a probability P of 1.50 × 10−9 (see Zhu (2009)).

How does the Heston model compare to the Sch¨obel-Zhu model? Initially the volatility vt

in the Heston model follows an Ornstein-Uhlenbeck process with a mean-reversion level that is equal to zero, i.e.

dvt= −βvtdt + δdZt2 (2.8)

Then using Itˆo’s lemma the squared volatility, i.e. the variance Vt= vt2, follows the

square-root process (see Sch¨obel and Zhu (1999))

dVt= κh(θh− Vt)dt + σh

p

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with β = κh 2 , δ = σh 2 , θh = δ2 κh (2.10) The difference between the equations (2.6) and (2.8) is therefore the mean-reversion param-eter θ, which is always zero in equation (2.8), whereas in equation (2.6) it is generally not the case. Since the volatility level in the long run is given by θ, the process (2.8) does not seem reasonable. The solution of Heston (1993) is however based on the process of (2.9) and not the process of (2.8). The parameters of (2.9) are overdetermined by (2.10) and therefore the volatil-ity process of (2.9) cannot be derived from (2.8) for a wide range of values of the parameters κh,

σh and θh. For many parameter values the two processes (2.8) and (2.9) are thus not mutually

consistent (Sch¨obel and Zhu (1999)). In the special case that θ is zero, the Sch¨obel-Zhu model reduces to the Heston model for the parameters (see Zhu (2009))

κ = κh 2 , σ = σh 2 , θh = σ2 κh , θ = 0 (2.11)

According to Sch¨obel and Zhu (1999) a favorite property of their model is that both the volatility and the squared volatility exhibit mean-reversion, whereas in the Heston model this is only the case for the squared volatility.

2.3.3 Cox, Ingersoll and Ross Model

The stochastic short rate model of Cox, Ingersoll and Ross (1985) is of the same form as the Heston model for the variance and is given by

drt= κr(θr− rt)dt + σr

√

rtdZt3 (2.12)

where the variables and parameters have the following meaning • rtthe short rate at time t

• θr the long-term average of the short rate

• κr the parameter that controls the speed of the adjustment of rt to its long-term average

• σ_r the volatility coefficient of the short rate • Z3

t a standard Brownian motion

It is assumed that the short rate process is uncorrelated with the asset price and the vari-ance/volatility, i.e. correlations are given by dZ_t1dZ_t3 ≡ dZ2

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2.4 Pricing of European Options

In order to increase the accuracy of the simulation for pricing American options, European options are used as control variates, which is further discussed in section 2.5.4. The prices of European options according to the Heston model and Sch¨obel-Zhu model are determined using Fourier transform methods and characteristic functions. We first start with the problem formulation.

2.4.1 Problem Formulation

A market model that is used to price a derivative should meet at least the requirements of no arbitrage opportunities (NA) and no free lunches with vanishing risk (NFLVR) (Hilpisch (2015)). The Fundamental Theorem of Asset Pricing is an important theorem that links the requirements of NA/NFLVR to the existence of an equivalent martingale measure. A martingale is a sequence of random variables for which, at a particular time and given the prior observed values, the expected value for the next time is equal to the current observed value. If the funda-mental theorem of asset pricing applies, the absence of arbitrage implies and is implied by the existence of an equivalent martingale measure under which all discounted stochastic processes of the market model are martingales. Because of this result the discounted price process of an attainable1 option is a martingale as well. As a result we can determine the today’s price of a European option with a certain maturity by determining the expected payoff at maturity under the equivalent martingale measure and discounting it back to today using the risk-free short rate.

Consider an economy in which two assets are traded: a risky asset with price St, with

0 ≤ t ≤ T , and a riskless bond that pays one unit of currency at time T . With r denoting the short rate, the value Bt(T ) of the bond at time t is given by Bt(T ) ≡ e−r(T −t). As mentioned

above it is assumed that there is an equivalent martingale measure Q under which the discounted asset price process is a martingale. The value of an attainable call option at time t is then given by

Ct= e−r(T −t)EQt (CT) (2.13)

and the value of the option today is therefore equal to

C0 = e−rTEQ₀(CT) (2.14)

1

A contingent claim is attainable if it can be replicated via an admissible trading strategy in available securities and contingent claims. A trading strategy is admissible if it is predictable and self-financing and if its value is at all times bounded from below (Hilpisch (2011)).

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where for a call option the payoff at maturity is given by CT = max[ST − K, 0]. The today’s

value of a call option can thus be written as

C0 = e−rT

Z ∞

0

CT(s)q(s)ds (2.15)

with q(s) the risk-neutral probability density function (pdf) of the asset price ST. However,

a closed form for the pdf of ST is not always availabe and therefore another approach is used,

involving the use of the characteristic function (CF) of ST, called Fourier-based option pricing.

2.4.2 Fourier-based Option Pricing

Fourier-based option pricing makes use of the CF instead of the pdf and the Fourier transform of the payoff CT. In order to price European options using Fourier-based option pricing, the

following definitions are needed, starting with the Fourier transform (see also Hilpisch (2015)).

Definition 1 (Fourier Transform). The Fourier transform of the integrable function f (x) is given by ˆ f (u) ≡ Z ∞ −∞ eiuxf (x)dx u can be real or complex and eiux is called the phase factor.

f can be determined using ˆf via Fourier inversion. The function f (x) is then defined as f (x) = 1

2π Z ∞

−∞

e−iuxf (u)duˆ

when u is a real number. For the complex number u = ur+ iui, with ur the real part and

ui the imaginary part of u, f (x) is defined as

f (x) = 1 2π

Z ∞+iui

−∞+iui

e−iuxf (u)duˆ Another essential ingredient is the characteristic function.

Definition 2 (Characteristic Function). Let X be a random variable with density q(x). The characteristic function ˆq of X is the Fourier transform of the pdf of X

ˆ q(u) ≡

Z ∞

−∞

eiuxq(x)dx = EQ(eiuX)

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this definition and that of the moment generating function (mgf) is the imaginary unit i. The mgf of a continuous random variable X with density q(x) is defined as

MX(s) ≡ E[esX] =

Z ∞

−∞

esxq(x)dx

for every s ∈ R such that this integral converges. The mgf exists if the integral converges for s in an open interval that includes the origin, i.e. if there exists a h > 0 such that MX(s) < ∞ ∀s ∈

(−h, h). When the mgf of X exists, every moment of X is finite, i.e. ∀r ∈ R+, E[|X|r] < ∞. The moments of X are then given by E[Xn] = M(n)_X (0), where

M(n)_X (s) ≡ d n dsnE[e sX_{] = E[} dn dsne sX_{] = E[X}n_esX_]

The moments are thus given by evaluating the nth derivative of the mgf at zero.

However, the mgf can be used for more than only obtaining the moments of X. If it can be shown that the random variables X and Y have the same mgf, their distribution is the same as well. Unfortunately the mgf doesn’t always exist. As already mentioned, when the mgf exists, this also means that all moments of X are finite, so when this is not the case, the mgf does not exist. The CF on the other hand always exists. When the mgf exists, the CF and mgf are related via ˆq(s) = MX(is).

A popular approach to Fourier-based option pricing that is used here is the approach of Lewis (2001). Consider a European call option with payoff CT ≡ max[es− K, 0], with s denoting the

log stock price (s ≡ log S). For the complex number u = ur + iui with ui > 1, the Fourier

transform of the payoff CT of the call option at maturity is given by (see Hilpisch (2015) for

proof)

ˆ

CT(u) = −

Kiu+1 u2_{− iu}

Now use Fourier inversion, which yields CT(s) = 1 2π Z ∞+iui −∞+iui e−iusCˆT(u)du

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(2015)) C0 = e−rTEQ0(CT) = e −rT 2π E Q 0 Z ∞+iui −∞+iui e−iusCˆT(u)du ! = e −rT 2π Z ∞+iui −∞+iui EQ₀(ei(−u)s) ˆCT(u)du = e −rT 2π Z ∞+iui −∞+iui ˆ CT(u)ˆq(−u)du (2.16)

Definition 3 (L´evy process). A stochastic process X = {Xt : t ≥ 0} is a L´evy process if it

has the following properties: • X₀= 0 with probability one.

• Independent increments: for any 0 ≤ t₁ < t2 < ... < tn < ∞, Xt2 − Xt1, Xt3 −

Xt2, ..., Xtn− Xtn−1 are independent.

• Stationary increments: the probability distribution of any increment Xt− Xs is only

dependent on the length t − s of the time interval and increments on equally long time intervals have an identical distribution.

• Continuity in probability: for any > 0 and t ≥ 0 it holds that lim

h→0P (|Xt+h− Xt| > ) = 0

Examples of L´evy processes are the Wiener process, also known as the Brownian motion process, and the Poisson process.

Let St≡ S0ert+Xt with Xt a L´evy process and eXt a martingale with X0= 0, then ˆq(−u) =

e−iuyϕ(−u) with ϕ the characteristic function of XT and y ≡ log S0+ rT . Then

C0= e−rT 2π Z ∞+iui −∞+iui e−iuyC(u)ϕ(−u)duˆ

Let k = log(S0/K) + rT , using the derived call option payoff transform then leads to (see

Hilpisch (2015)) C0= − Ke−rT 2π Z ∞+iui −∞+iui e−iukϕ(−u) du u2_{− ui} (2.17)

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Proposition (Lewis (2001)). Assume ui ∈ (0, 1), the present value of a call option is then given by C0 = S0− Ke−rT 2π Z ∞+iui −∞+iui e−iukϕ(−u) du u2_{− ui} (2.18) Setting ui = 0.5 leads to C0 = S0− √ S0Ke−rT /2 π Z ∞ 0

Re[eizkϕ(z − i/2)] dz

z2_{+ 1/4} (2.19)

with Re[x] denoting the real part of x (see Hilpisch (2015) for proof).

For the model of Heston (1993) the price of a European call option with strike price K and maturity T is given by the similar expression (see Hilpisch (2011))

C₀H(K, T ) = S0− B0(T ) √ S0K π Z ∞ 0

Re[e−iukϕH(u − i/2, T )] du

u2_{+ 1/4} (2.20)

with k ≡ log(S0/K), B0(T ) the discount factor for the Cox, Ingersoll and Ross model and ϕH

the characteristic function of the Heston model, which is shown in appendixA. The expression above can be evaluated using numerical quadrature and is done using Python’s quad function. The price of the put option can then simply be derived using the put-call parity

P0(K, T ) = C0(K, T ) + B0(T )K − S0 (2.21)

The call option pricing formula for the Sch¨obel-Zhu model, as given by Sch¨obel and Zhu (1999), is of the form

C(S, v, t, T ) = StF1− e−r(T −t)KF2 (2.22)

with F1and F2the probability distribution functions that, given the CF’s, can be determined

using the Fourier inversion formula (for j = 1, 2)

Fj = 1 2 + 1 π Z ∞ 0 Re fj(φ) exp(−iφ log K) iφ ! dφ (2.23)

However, as also mentioned by Sch¨obel and Zhu (1999), numerical integration of the above integrals is not trivial. Care must be taken of the complex logarithm that occurs in the for-mula, because the complex logarithm is a multivalued function. If only the principal value of the complex logarithm would be used, this would lead to an incorrect integration of the CF

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and therewith to incorrect option values. Fortunately Lord and Kahl (2008) came up with a formulation of the characteristic function in which the principal branch is the correct one. Their version of the characteristic function ϕSZ will therefore be used and can be found in appendix

A. The call option value is then given by

C₀SZ(K, T ) = S0− √ S0Ke−rT /2 π Z ∞ 0

Re[eiukϕSZ(u − i/2)] du

u2_{+ 1/4} (2.24)

with in this case k = log(S0/K) + rT . When using the reformulated version of the CF of

Lord and Kahl (2008), the expression (2.24) above can also evaluated by numerical quadrature and is also done using Python’s quad function.

2.5 Pricing of American Options

Since American options have an early exercise feature, pricing these options is more difficult than their European counterparts and a closed-form solution is not available. Longstaff and Schwartz (2001) came up with a simple but powerful method called Least-Squares Monte Carlo (LSM) to approximate the value of American options using simulation.

The idea of their approach is to use least-squares regression to estimate the conditional expected payoff from continuation. At any moment at which the option can be exercised, the holder of an American option has to compare the expected payoff from continuation to the payoff of immediate exercise, and when the latter is higher than the expected continuation value, only then the option will be exercised. The conditional expectation of the payoff from continuation is therefore crucial to the decision whether or not to exercise the option before maturity. The key insight of Longstaff and Schwartz (2001) is that this conditional expectation can be estimated from the cross-sectional information in the simulation using least squares. Specifically, the pay-offs from continuation received at later times are regressed on a set of basis functions of the values of the relevant state variables. In this case ten different basis functions (functions of the simulated values for the variance/volatility, stock price and riskless rate) are used. The fitted value from this regression then gives us an estimate of the conditional expectation function. By estimating this function at each point in time at which the option could be exercised, the optimal exercise strategy for each path is fully specified. When the optimal exercise strategy for the option is known, the option can be valued accurately. An advantage of this approach is that it only requires least-squares regression. First we start by formulating the pricing problem of American options.

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2.5.1 Problem Formulation

The uncertainty in the model economy with final date T is represented by the filtered probability space {Ω, F , F, P }. Ω denotes the continuous state space, F a σ-algebra, F a filtration, i.e. a family of increasing σ-algebras F ≡ {Ft∈[0,T ]} with F0 ≡ {∅, Ω} and FT ≡ F , and P the real or

objective probability measure. The set of uncertainties for each time in our model economy is defined by Xt ≡ (St, vt, rt), consisting of the stock price, volatility/variance and interest rate.

By the Fundamental Theorem of Asset Pricing, the value at time t of an attainable and FT

-measurable contingent claim VT ≡ hT(XT) ≥ 0, under suitable integrability conditions, is given

by (see Hilpisch (2011))

Vt= EQt (Bt(T )VT)

with Q a risk-neutral probability measure equivalent to the real world measure P , Et(·) the

conditional expectation E(·|Ft) and Bt(T ) the discount factor for discounting from time T back

to time t. For the today’s value we consider the case V0 = EQ0(B0(T )VT).

The valuation of an American option with maturity T can be formulated as the following optimal stopping problem (see Hilpisch (2011))

V0 = sup τ ∈[0,T ]

EQ₀(B0(τ )hτ(Xτ)) (2.25)

where V0 is the today’s value of the American option, τ a F-adapted stopping time, B0(τ )

the discount factor for discounting from the stopping time τ back to today, h(τ ) a nonnegative, Fτ-measurable payoff function and Xτ the market model vector process stopped at t = τ .

In order to price American options by MCS, a discretized version of the above optimal stopping problem is needed (Kohler (2009))

V0= sup τ ∈{0,∆t,2∆t,...,T }

EQ₀(B0(τ )hτ(Xτ)) (2.26)

The continuation value Ct is the value of not exercising the option at time t, and is given

under risk-neutrality by

Ct(x) = EQt (e

−¯rt+∆t∆t_V

t+∆t(Xt+∆t)|Xt= x) (2.27)

using the Markow property of Xtand ¯rt+∆t≡ (rt+∆t+ rt)/2 (see Hilpisch (2011)).

The value of the option at time t is then equal to the maximum of the payoff of exercising immediately ht(x) and the expected discounted continuation value Ct(x) (Kohler (2009))

Vt(x) = max[ht(x), Ct(x)] (2.28)

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2.5.2 Least-Squares Monte Carlo

Whether or not an American option is exercised before maturity is dependent on the value of continuation. The idea of Longstaff and Schwartz (2001) is to estimate the continuation value Ct(x) by ordinary least-squares regression at each time t, starting at time T and iterating

backwards to t = 0. In order to value the American put options we simulate I paths for the three-dimensional market model process Xt = (St, vt, rt) with M + 1 time steps and results

Xt,i, with vt either the variance (Heston model) or volatility (Sch¨obel-Zhu model) and where

t ∈ {0, ∆t, 2∆t, ..., T }, i ∈ {1, ..., I} and ∆t = T /M . For each path i at time T the optimal exercise strategy is to exercise the option only if it is in the money. The payoff of the option at time T is therefore simply hT(XT ,i) = max[K − ST,i, 0]. We then iterate backwards to

t = 0 and define at each time t and for each path i the discounted value of continuation as Yt,i ≡ e−¯rt+∆t∆tVt+∆t,i, with Vt+∆t,i the cash flow received if we continue the life of the option

and ¯rt+∆t≡ (rt+∆t+rt)/2. We then regress the I continuation values Yt,iagainst the I simulated

values for Xt,i. Given D basis functions b for the regression, the estimation of the continuation

value Ct,i is given by

ˆ Ct,i = D X d=1 α∗_d,t· b_d(Xt,i) (2.29)

where the optimal parameters α∗_d,t in equation (2.29) above are determined by the solution to the minimization problem

min α1,t,...,αD,t 1 I I X i=1 Yt,i− D X d=1 αd,t· bd(Xt,i) !2 (2.30)

In this case we use ten different basis functions (functions of St, rt and vt, so D = 10), 35000

paths (I = 35000) and M = 20, so there are 21 time steps in total. Valuing American options by LSM is therefore done by performing the following steps, implemented under risk-neutrality (see Hilpisch (2011)):

1. Simulate I paths of Xtwith M +1 time steps and resulting values Xt,i for t ∈ {0, ..., T }, i ∈

{1, ..., I}, such that for each time t we simulate 35000 values for S_t, vt and rt

2. At t = T only exercise the option if it is in the money, so the value of the option is VT,i= hT(XT ,i), which is only positive if the option is in the money at maturity and zero

otherwise

3. Iterate backwards t = T − ∆t, ..., 0:

• Regress the discounted value of continuation Yt,i against the Xt,i for every path

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• Use the value of ˆCt,i as given by (2.29) to estimate the continuation value Ct,i, using

the optimal parameters a∗_d,t resulting from (2.30)

• Determine the maximum of the payoff of immediate exercise and the expected dis-counted payoff of continuing the option (see (2.28)) and set

Vt,i=

 



ht(Xt,i) if ht(Xt,i) > ˆCt,i (exercise the option)

Yt,i if ht(Xt,i) ≤ ˆCt,i (do not exercise the option)

continue until t = 0

4. At t = 0 calculate the LSM estimator as the average of the time zero values V0,i over all I

paths ˆ V₀LSM = 1 I I X i=1 V0,i (2.31)

It is to be noted that the estimated continuation values ˆCt,i are only used to make the

de-cision whether or not the option is exercised. In case the option is not exercised, the real value Yt,i should be used.

Since asset prices are simulated for every point in time, the simulated continuation value Yt,i is also known. However, these values cannot be used directly since this would translate into

perfect foresight, which in reality is not possible and thus not acceptable. If we would use these values, this would result in a better-than-optimal exercise policy and therewith a consistently high biased estimator (Hilpisch (2015)). Furthermore it is mentioned by Longstaff and Schwartz (2001) that including in the regression only the paths for which the option is in the money, significantly increases the efficiency of the algorithm and decreases the computational time. 2.5.3 Discretization

The continuous time versions of the models as given by the equations (2.3), (2.4), (2.5), (2.6) and (2.12) cannot be used to simulate the different values in the model economy and therefore a discretization is needed.

The time interval [0, T ] is divided in equally spaced sub-intervals ∆t such that t ∈ {0, ∆t, 2∆t, ..., T }, so there are M + 1 time points with M ≡ T /∆t. For the Heston model, the following discretiza-tion for the asset price (2.3) and the variance (2.4) is used (see Hilpisch (2011))

St= Ssexp (¯rt− Vt/2)∆t + p Vt √ ∆tz_t1 (2.32) ˜ Vt= ˜Vs+ κ(θ − ˜Vs+)∆t + σ q ˜ Vs+ √ ∆tz_t2 (2.33)

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Vt= ˜Vt+ (2.34)

and for the asset price (2.5) and the volatility (2.6) according to the Sch¨obel-Zhu model the discretization (see Zhu (2009))

St= Ssexp (¯rt− vt2/2)∆t + vt √ ∆tz_t1 (2.35) ˜ vt= ˜vs+ κ(θ − ˜vs+)∆t + σ √ ∆tz_t2 (2.36) vt= ˜vt+ (2.37)

since the Cox, Ingersoll and Ross model according to (2.12) is similar to the Heston model for the variance, a similar discretization is used for rt

˜ rt= ˜rs+ κr(θr− ˜r+s)∆t + σr p ˜ r+s √ ∆tz_t3 (2.38) rt= ˜r+t (2.39)

with s the time step before time t, i.e. s = t − ∆t, zn_t standard normally distributed random variables and ¯rt≡ (rt+rs)/2. zt1and zt2are correlated with ρ whereas the other random variables

are not. x+ _{is notation for max[x, 0]. The above discretization is an Euler discretization and is}

called a full truncation scheme.

2.5.4 Control Variates

There are several techniques to increase the accuracy of the simulation, such as the use of control variates (see also Glasserman (2004)). Suppose that we want to estimate the mean of the random variable X, i.e. θ ≡ E(X). Further suppose that there is also another random variable Y that is correlated with X and for which the mean is known, i.e. µ ≡ E(Y ). If we would draw n pairs (Xi, Yi), i.i.d across i, the control variate estimator is then given by

ˆ

θc≡ ¯Xn− c( ¯Yn− µ)

for a given constant c. If X and Y would be positively correlated and c > 0, then if ¯Yn > µ

( ¯Yn < µ) for a given set of replications, then it is likely that also ¯Xn > θ ( ¯Xn < θ) and the

estimator will be corrected downwards (upwards).

In the case of the Monte Carlo simulation and the LSM algorithm, we could use European put options in order to improve the accuracy. The European counterpart would be a good

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choice, because the values of the two options are highly correlated under certain circumstances. The degree of the correlation between the two options is mainly dependent on how much the option is ITM or OTM. The correlation increases generally the more the options are OTM. The more an American option is ITM, the more important becomes the possibility that the option can be exercised early and the lesser the correlation between the American and the European option. In contrast to the American put option, the true model value of the European put option is known exactly and can be determined by evaluating expression (2.20) or (2.24) and using the put-call parity (2.21), as discussed in section 2.4. The estimation error is therefore also known. Since the error for the control variate is known exactly, we could use this to correct the estimated value of the American put option by an appropriate amount.

When we have simulated I paths for St, vtand rtto price an American put option with strike

K and maturity T , we also have I simulated present values of the European put option. These values are given by P0,i = B0(T )hT(ST,i) for i ∈ {1, ..., I}, with hT(ST,i) ≡ max[K − ST ,i, 0].

Using the true model value P₀M OD, the estimator (2.31) is corrected as follows ˆ V₀CV = 1 I I X i=1 V0,i− λ · (P0,i− P0M OD) (2.40)

So with λ > 0, when the simulated value of the European put option would be too high (low), i.e. P0,i> P0M OD (P0,i< P0M OD), the obtained value V0,i would be corrected downwards

(upwards). We correct the estimator using λ = 1, which gave on average better calibration results than the statistical correlation between the American and European option.

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Research Setup and Results

A good option pricing model should be able to generate prices that are close to the real observed market prices. Since both the Heston model and the Sch¨obel-Zhu model are very dependent on the input parameters, it is important to have parameter values for which these models generate prices that are as close as possible to the real market prices. This chapter investigates how well both models perform when used for simulation to price options over the years 2013-2016 by LSM with control variates. Both the Heston model and the Sch¨obel-Zhu model are calibrated to historical option prices of options on Facebook, which are one of the most traded options. First the models are calibrated to a set of options on this company at the beginning of January and June of each of the years 2013-2016 to see how well the observed market prices can be replicated by both models at each date. Then the four parameter values for κ, θ, σ and ρ resulting from the calibration procedure at each date are used to price options with similar strikes and maturities at the next date half a year later. This is further explained in section 3.2. First we start with the calibration procedure itself. Because of the assumption of zero correlation between the short rate and asset price model, the calibration of the short rate model and the asset price model can be done separately.

3.1 Model Calibration

3.1.1 Calibration of the Heston and Sch¨obel-Zhu Model

The calibration of a model is the process of finding model parameters such that the observed market prices of the derivative are replicated as closely as possible by the model. In order to do this an error function to be minimized needs to be defined. Several error functions can be defined. Commonly used functions are for instance the mean squared error (MSE) of the price differences, the MSE of relative price differences or the MSE of the differences in implied volatility. According to Christoffersen and Jacobs (2004) the objective itself, for instance valuation or hedging, should

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be taken into account when choosing the error function to be used for calibration. Their findings indicate that, across different loss functions, the MSE of the price differences performs best. The MSE of the price differences would thus be a good error function to use for option valuation applications and will therefore be used for the calibration (see also Hilpisch (2015)). For options i = 1, ..., N , the minimization problem is defined as

min p 1 N N X i=1 (P_iM arket− P_iM odel)2 (3.1) with p the parameter vector.

The process of calibrating the Heston model and the Sch¨obel-Zhu model consists of two steps. The first step is to search for regions where the minimum could be by means of brute force. In this step we specify an interval for the parameters κ, θ, σ, ρ and the initial variance V0 (Heston model) or the initial volatility v0 (Sch¨obel-Zhu model) as well as a step size for this

interval. Then for every possible parameter combination in these intervals the function value is evaluated using Python’s brute function. The second step is to search for a local minimum, where the parameters found in the first step are taken as starting values. This is done using Python’s fmin function (see also Hilpisch (2014)).

3.1.2 Calibration of the Cox, Ingersoll and Ross Model

As mentioned in section 2.3.3 the short rate model of Cox, Ingersoll and Ross is given by the SDE

drt= κr(θr− rt)dt + σr

√ rtdZt3

The calibration procedure therefore consists of determining parameters κr, θr, σr and r0 for

which, at all considered times t, the differences between the current market implied forward rate and the current model implied forward rate for time t are minimized, i.e.

∆f_(0,t) ≡ fM arket

(0,t) − f(0,t)M odel

With Bt(T ) denoting the price at time t of a zero-coupon bond with maturity T that pays

one unit of currency at time T , with t < T , the (instantaneous) forward rate at time t for maturity T is given by

f(t,T ) ≡ −

∂Bt(T )

∂T

where for the short rate at time t it holds that rt = f(t,t). The price of a zero-coupon bond is

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Bt(T ) = exp − Z T t f_(t,s)ds

However, forward rates are usually not quoted directly and therefore other market data must be used. Data that could be used are for instance LIBOR, Euribor or Treasury yields. In this case the LIBOR (London Inter-bank Offered Rate) is used, which is a benchmark rate that some of the world’s leading banks charge for borrowing from each other. The model is calibrated to USD LIBOR rates for different maturities. The relationship between forward rates and zero-coupon bond yields is given by (see Hilpisch (2015))

f_{(0,T )}= Y_{(0,T )}+ ∂Y(0,T )

∂T · T (3.2)

for different maturities and with Y(0,T ) the yield today of a bond with maturity T . The

(continuous) yield for a zero-coupon bond follows from the equation BT(T ) = B0(T )eY(0,T )·T

which can be written as

Y(0,T ) =

log BT(T ) − log B0(T )

T

setting the final value of the bond equal to 1 leads to Y_{(0,T )}= −log B0(T )

T .

This equation can also be used for continuous LIBOR rates. Since the LIBOR serves seven different maturities (overnight, 1 week and 1, 2, 3, 6 and 12 months), it is necessary to interpolate between the single data points. The partial derivative in equation (3.2) can be derived using a continuously differentiable interpolating function (for instance from a cubic splines regression (Hilpisch (2015))) and subsequently be used to derive forward rates for arbitrary times T . Since the quoted rates are not continuous rates, they have to be transformed into these. For simplicity it is assumed that one month has 30 days2. Consider for instance the 6-month LIBOR rate from June 2015, which was 0.440%. Then 1 unit at time 0 is half a year later equal to the corresponding factor

f_s6m= 1 + 180/360 · 0.00440 the equivalent annualized continuous rate is then given by

f_c6m= 360/180 · log(f_s6m)

2_{Simplifying assumption, since the day-count convention for LIBOR for most currencies is actual/360, except}

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The overnight rate is taken as the time t = 0 short rate.

Finally the MSE of the market implied and model implied forward rates at selected discrete time points needs to be minimized. The interval [0, T ] is divided into sub-intervals of equal size ∆t (with M ≡ T /∆t). Given a fixed r0, the minimization problem is then defined as

min p 1 M M X m=0 (f_(0,m∆t)M arket− f_(0,m∆t)M odel )2 (3.3) with p the parameter vector. The minimization is done using Python’s fmin function. The formula for the forward rate in the model of Cox, Ingersoll and Ross (1985) can be found in appendix B.

3.2 Calibration Results for Facebook Options

For the options on Facebook we consider the years 2013-2016. The historical observed market price that is used is the lowest closing ask, since it is assumed that one would have to buy the option. At each date the models are calibrated to options with three different maturities up to about 7 months. For each maturity 5 strikes are used for which the option is OTM and 5 strikes for which the option is ITM, which means a total number of 30 options. The historical option prices can be found in appendixC. The three models are calibrated at the beginning of January and June of the years 2013-2016, thus at 8 different moments. The dates at which the models are calibrated are shown in the timeline below

2013 2 Jan 3 Jun 2014 2 Jan 2 Jun 2015 2 Jan 2 Jun 2016 4 Jan 3 Jun 2017

In order to simulate the asset prices using the Heston and the Sch¨obel-Zhu model, we first need to simulate values for rt, as can be seen from equations (2.32) and (2.35). Therefore first

the Cox, Ingersoll and Ross model is calibrated to historical USD LIBOR rates at the above 8 dates according to the procedure as explained in section 3.1.2. The historical LIBOR rates can be found in appendix Cas well. The calibration results are shown in table3.1below. Python’s fmin function requires starting values. The last column contains the starting values (κr, θr, σr)

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Calibration Results Cox, Ingersoll and Ross Model

Date κr θr σr r0 Starting values

2 Jan 13 0.7838 0.0463 0.2695 0.0016 (1.0, 0.02, 0.1) 3 Jun 13 0.5315 0.0521 0.2354 0.0013 (1.0, 0.02, 0.1) 2 Jan 14 0.1433 0.1437 0.2029 0.0009 (2.0, 0.02, 0.2) 2 Jun 14 0.2810 0.0727 0.0010 0.0009 (1.5, 0.02, 0.1) 2 Jan 15 0.0949 0.2273 0.0010 0.0012 (1.2, 0.02, 0.1) 2 Jun 15 0.3139 0.0943 0.0010 0.0012 (1.0, 0.02, 0.1) 4 Jan 16 1.4646 0.0403 0.0010 0.0037 (1.0, 0.02, 0.1) 3 Jun 16 2.0580 0.0383 0.3968 0.0039 (1.0, 0.02, 0.1)

Table 3.1: Calibration results Cox, Ingersoll and Ross model.

Next the Heston model and the Sch¨obel-Zhu model are calibrated at each of the 8 dates mentioned above. At each date we simulate values for rt using the parameters for the Cox,

Ingersoll and Ross model that were obtained by calibration at that date.

As already mentioned in section 3.1.1, the calibration procedure consists of two steps. In the first step a starting interval for the five parameters and a step size needs to be specified. Then for every possible parameter combination within the five intervals the corresponding MSE is determined in order to scan for regions where the minimum could be. The second step consists of local optimization, where the resulting parameters from the first step are taken as starting values. Since the process of determining option prices and the corresponding MSE by Least-squares Monte Carlo for every parameter combination in the first step can take quite some computational time, there is a trade-off between the size of the starting interval and the corresponding step size and therefore different starting intervals must be considered. Since the error function to be minimized usually has different local minima, the possibility exists that the found minimum is not the global minimum. Given this possibility, the resulting MSE’s are quite low at most dates and the market prices can be replicated quite well. The tables below show the calibration results. Tables 3.2 and 3.3contain the resulting parameter values for V0/v0, κ, σ, ρ

and θ for the Heston model and the Sch¨obel-Zhu model respectively, as well as the corresponding MSE’s of the calibration. For every date, the Feller’s condition as mentioned in section 2.3.1

for the Heston model is met. Table3.2also contains the square root of both the initial variance and the long-term average of the variance. The starting interval for the parameters V0/v0, κ, σ,

ρ and θ of the form (lower bound, upper bound, step size) that gave the best results was (0.05, 0.45, 0.1), (1, 7, 2), (0.05, 0.5, 0.15), (−0.75, −0.25, 0.25), (0.05, 0.5, 0.15)

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except for 2 January 2014, in which case

(0.05, 0.20, 0.05), (3, 6, 1), (0.90, 1.30, 0.10), (−0.90, −0.60, 0.10), (0.15, 0.30, 0.05) and

(0.15, 0.30, 0.05), (3, 6, 1), (0.50, 0.80, 0.10), (−0.90, −0.60, 0.10), (0.45, 0.60, 0.05) were used for respectively the Heston model and the Sch¨obel-Zhu model.

Calibration Results Heston Model

Date V0 √ V0 κ σ ρ θ √ θ MSE 2 Jan 13 0.3192 0.5650 2.3580 0.0514 -0.7839 0.0675 0.2598 0.0042 3 Jun 13 0.1877 0.4332 3.2323 0.3029 -0.7028 0.1458 0.3819 0.0018 2 Jan 14 0.0985 0.3139 5.1085 1.1894 -0.9023 0.2030 0.4505 0.4007 2 Jun 14 0.1862 0.4316 3.5662 0.2286 -0.3071 0.1585 0.3981 0.0210 2 Jan 15 0.1158 0.3402 0.5781 0.3062 -0.5635 0.0864 0.2939 0.0082 2 Jun 15 0.0530 0.2301 1.2216 0.3873 -0.4175 0.1284 0.3584 0.0016 4 Jan 16 0.1303 0.3610 2.1121 0.7290 -0.5051 0.1299 0.3605 0.0185 3 Jun 16 0.0701 0.2648 0.5247 0.2518 -0.9973 0.2209 0.4700 0.0169 Average 0.0591

Table 3.2: Calibration results Heston model.

Calibration Results Sch¨obel-Zhu Model

Date v0 κ σ ρ θ MSE 2 Jan 13 0.5614 1.1476 0.1239 -0.6216 0.1446 0.0037 3 Jun 13 0.4280 1.7649 0.2780 -0.2777 0.3401 0.0016 2 Jan 14 0.2665 5.1029 0.5155 -0.9877 0.4463 0.3944 2 Jun 14 0.4117 0.0374 0.1926 -0.2149 0.5137 0.0172 2 Jan 15 0.3291 0.6008 0.2738 -0.3693 0.2803 0.0040 2 Jun 15 0.2311 1.2130 0.3388 -0.2493 0.2571 0.0035 4 Jan 16 0.3640 1.2192 0.2510 -0.6192 0.2620 0.0262 3 Jun 16 0.2563 1.0667 0.3274 -0.4717 0.3528 0.0058 Average 0.0571

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Looking at tables 3.2 and 3.3 we see that the initial volatilities for both models are close to each other at most dates, with the biggest difference for 2 January 2014. Furthermore we see that at each date ρ is negative, meaning that the stock price and variance/volatility are negatively correlated and the leverage effect is therefore captured by both models.

When comparing the MSE at each date for both models, it is clear that overall the market prices are slightly better replicated by the Sch¨obel-Zhu model, except for 2 June 2015 and 4 January 2016. The prices are replicated best by both models at 3 June 2013, whereas the MSE is clearly the largest at 4 January 2014. On average there is not a big difference between the calibration results of both models. Figure 3.1 below shows the calibration results for 3 June 2013. Figures3.1aand3.1bshow the real price and the model price for both models and figures

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(a) Heston model (S0= 23.85) (b) Sch¨obel-Zhu model (S0= 23.85)

(c) Heston model (d) Sch¨obel-Zhu model

Figure 3.1: Model prices, real prices and differences for 3 June 2013.

It can be seen from figures 3.1a and 3.1b that all model prices are quite close to the real prices, with the biggest deviation of about 8 cents, according to figures 3.1c and 3.1d. The graphs of the calibration results of the remaining 7 dates can be found in appendix D.

Furthermore we could also look at how well the prices of options with different maturities are replicated by both models over the years. Figure 3.2 below shows the MSE for three different maturity categories over all 8 dates at which the prices were replicated. The three categories, from lowest maturity to highest, contain 90, 70 and 80 options respectively.

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Figure 3.2: MSE for different maturities over all 8 dates.

Looking at figure3.2we see that only for the shortest maturities the prices are slightly better replicated by the Heston model. The MSE’s for both models decrease when the maturities become longer.

3.3 Forecasting Prices of Facebook Options

Next we investigate how well option prices can be forecasted by both models, using parameters that were obtained by calibration half a year ago. At each date at which we forecast we analyze the results and compare the parameters that were used for forecasting to those that were ob-tained by calibration at that date. We then try to explain why in some cases the estimates are further off and what the effects of the different parameters are on the option prices. In section

3.2 both models were calibrated at 8 different dates over the years 2013-2016. Now we use the parameter values that were obtained for κ, σ, ρ and θ at one date, to forecast the prices at the next date half a year later. However, since it would not be reasonable to assume that the initial variance/volatility (V0/v0) obtained by calibration at the previous date would be the same at

the next date half a year later, we will use the value for V0/v0 that is obtained by calibration

at the same date at which we forecast. So when forecasting the prices at e.g. 3 June 2013 with the Heston model, we would use the parameters κ = 2.3580, σ = 0.0514, ρ = −0.7839 and θ = 0.0675, which were obtained at 2 January 2013, and V0 = 0.1877, which was obtained by

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2013 3 Jun 2014 2 Jan 2 Jun 2015 2 Jan 2 Jun 2016 4 Jan 3 Jun 2017

The first date at which we forecast is 3 June 2013, using four parameters obtained by cali-bration half a year before, i.e.

2013 Calibration (2 Jan) 2013 Forecasting (3 Jun) 2014 2 Jan

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Figure 3.3: Predicted prices, actual prices and differences for 3 June 2013.

The parameters that were used for forecasting and the parameters that were obtained by calibration at 3 June 2013 are shown in tables 3.4 and 3.5 below for respectively the Heston model and the Sch¨obel-Zhu model.

Figure3.3shows that for this date both models understate the prices of every option, where the deviation from the actual price is the biggest for the longest maturity, as can be seen in figures 3.3c and 3.3d. The differences between predicted price and actual price are somewhat larger for the Sch¨obel-Zhu model, which also translates into a higher forecasting MSE, as shown in tables 3.4and 3.5.

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Parameters used for forecasting and forecasting MSE Parameter V0 √ V0 κ σ ρ θ √ θ MSE 0.1877 0.4332 2.3580 0.0514 -0.7839 0.0675 0.2598 0.0228 Obtained at 3 Jun 13 3 Jun 13 2 Jan 13 2 Jan 13 2 Jan 13 2 Jan 13 2 Jan 13

Parameters obtained by calibration and calibration MSE

3 Jun 13 0.1877 0.4332 3.2323 0.3029 -0.7028 0.1458 0.3819 0.0018 Table 3.4: Forecasting at 3 June 2013 with the Heston model.

Parameters used for forecasting and forecasting MSE

Parameter v0 κ σ ρ θ MSE

0.4280 1.1476 0.1239 -0.6216 0.1446 0.0368 Obtained at 3 Jun 13 2 Jan 13 2 Jan 13 2 Jan 13 2 Jan 13

Parameters obtained by calibration and calibration MSE 3 Jun 13 0.4280 1.7649 0.2780 -0.2777 0.3401 0.0016

Table 3.5: Forecasting at 3 June 2013 with the Sch¨obel-Zhu model.

Although the prices were replicated quite well by both models at 2 January 2013 (calibration MSE’s of 0.0037 and 0.0042), these four parameters lead to option prices that are too low. From table 3.5 we see that the long-term average of the volatility (θ) used for forecasting is lower (almost 0.20) than the value that was obtained by calibration at 3 June 2013. The same holds for the Heston model. Moreover, it can also be seen from tables 3.4 and 3.5 that the volatility coefficient of respectively the variance and the volatility (σ) that was used for forecasting is lower than the value obtained by calibration at 3 June 2013 (almost six times lower in case of the Heston model), meaning that there will be a smaller spread and less variation between the different simulated variance/volatility paths. A reason for these understated prices could be that we simulated variances/volatilities that were too low, leading to stock prices that are too high, due to the negative correlation between these two, which in turn has a negative effect on the price of a put option and leads to put option prices that are too low. The prices are understated most for the longest maturity. By simulating only 15 paths for the variance/volatility for the longest maturity T = 0.55 and with 20 time steps, we can see these effects of the differences between the parameters used for forecasting and those obtained by calibration in figure3.4below.

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(a) Simulated variance using parameters from 2/1/13 (Heston model)

(b) Simulated variance using parameters from 3/6/13 (Heston model)

(c) Simulated volatility using parameters from 2/1/13 (Sch¨obel-Zhu model)

(d) Simulated volatility using parameters from 3/6/13 (Sch¨obel-Zhu model)

Figure 3.4: Simulated variance/volatility using parameters from 2 January and 3 June 2013.

Figure3.4aclearly shows that the simulated variances using the parameters from 2 January 2013 are lower and show less variation than the simulated variances in figure 3.4b. This is also the case for the simulated volatilities in figures 3.4c and 3.4d. In this case the parameters ob-tained at 2 January 2013 therefore don’t lead to an accurate estimation of the prices at 3 June 2013.

Next we move to 2 January 2014, where we forecast prices using four parameters obtained by calibration half a year before, i.e.

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2013 Calibration (3 Jun) 2014 Forecasting (2 Jan) 2014 2 Jun

Figure3.5 below shows the results.

Figure 3.5: Predicted prices, actual prices and differences for 2 January 2014.

Out of all dates at which we forecast the prices, the forecasting MSE is the largest for 2 January 2014. Figures 3.5cand3.5dshow that both models understate and overstate the prices of the same options.

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calibration at 2 January 2014 are shown in the tables 3.6 and 3.7 below for respectively the Heston model and the Sch¨obel-Zhu model.

Parameters used for forecasting and forecasting MSE Parameter V0 √ V0 κ σ ρ θ √ θ MSE 0.0985 0.3139 3.2323 0.3029 -0.7028 0.1458 0.3819 0.5833 Obtained at 2 Jan 14 2 Jan 14 3 Jun 13 3 Jun 13 3 Jun 13 3 Jun 13 3 Jun 13

2 Jan 14 0.0985 0.3139 5.1085 1.1894 -0.9023 0.2030 0.4505 0.4007 Table 3.6: Forecasting at 2 January 2014 with the Heston model.

0.2665 1.7649 0.2780 -0.2777 0.3401 0.8255 Obtained at 2 Jan 14 3 Jun 13 3 Jun 13 3 Jun 13 3 Jun 13

Parameters obtained by calibration and calibration MSE 2 Jan 14 0.2665 5.1029 0.5155 -0.9877 0.4463 0.3944

Table 3.7: Forecasting at 2 January 2014 with the Sch¨obel-Zhu model.

The Sch¨obel-Zhu model understates the prices somewhat more than the Heston model, which therefore leads to a higher forecasting MSE. From table 3.7we see that the values for the long-term average of the volatility (θ) and the speed of adjustment of the volatility to its long-long-term average (κ) are lower than those obtained by calibration half a year later. The same holds for the Heston model, for which we can also see from table 3.6 that the volatility coefficient of the variance (σ) is lower than the value that was obtained by calibration at 2 January 2014, which also holds for the Sch¨obel-Zhu model, but to a lesser extent. However, since the calibration results were also the least for 2 Janary 2014, it is difficult to say that these deviations from the parameters that were later obtained by calibration cause these overstated and understated prices. Nevertheless it is clear that the parameters from 3 June 2013 are not valid anymore at 2 January 2014.

We now move to 2 June 2014, where we forecast prices using four parameters obtained by calibration half a year before at 2 January 2014, i.e.

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2014 Calibration (2 Jan) 2014 Forecasting (2 Jun) 2015 2 Jan

Figure 3.6: Predicted prices, actual prices and differences for 2 June 2014.

Figures3.6cand3.6dshow that for this date both models overstate the prices of most OTM options and understate those of most ITM options, with the Sch¨obel-Zhu model overstating the prices of OTM options somewhat more, which translates into a slightly higher forecasting MSE. Nevertheless we see in figures3.6aand3.6bthat especially the prices of options that are slightly ITM and OTM are close to the real observed market prices.

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The parameters that were used for forecasting and the parameters that were obtained by calibration at 2 June 2014 are shown in the tables 3.8and3.9below for respectively the Heston model and the Sch¨obel-Zhu model.

Parameters used for forecasting and forecasting MSE Parameter V0 √ V0 κ σ ρ θ √ θ MSE 0.1862 0.4316 5.1085 1.1894 -0.9023 0.2030 0.4505 0.2579 Obtained at 2 Jun 14 2 Jun 14 2 Jan 14 2 Jan 14 2 Jan 14 2 Jan 14 2 Jan 14

2 Jun 14 0.1862 0.4316 3.5662 0.2286 -0.3071 0.1585 0.3981 0.0210 Table 3.8: Forecasting at 2 June 2014 with the Heston model.

0.4117 5.1029 0.5155 -0.9877 0.4463 0.2670 Obtained at 2 Jun 14 2 Jan 14 2 Jan 14 2 Jan 14 2 Jan 14

Parameters obtained by calibration and calibration MSE 2 Jun 14 0.4117 0.0374 0.1926 -0.2149 0.5137 0.0172

Table 3.9: Forecasting at 2 June 2014 with the Sch¨obel-Zhu model.

In both table3.8and3.9we see large differences between the values of the volatility coefficient of the variance/volatility (σ) that are used for forecasting and those obtained by calibration at 2 June 2014. This means that when we use the higher values of σ for forecasting, there is a much larger spread and more variation between the simulated variances/volatilities. Furthermore we see in table3.9a large difference between the two values of the mean-reversion speed (κ), which means that when we use the higher value of κ for forecasting, the volatility moves at a faster rate to its long-term average, which is somewhat lower than the value obtained by calibration in case of the Sch¨obel-Zhu model. Since the difference between the value used for forecasting and the one later obtained by calibration for both κ and σ is quite large, the simulated variances/volatilities are quite different when using the parameters from 2 January and 2 June 2014. Figure3.7shows these effects of the different parameters used for forecasting and those obtained by calibration for 15 simulated paths, 20 time steps and T = 0.55.

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(a) Simulated variance using parameters from 2/1/14 (Heston model)

(b) Simulated variance using parameters from 2/6/14 (Heston model)

(c) Simulated volatility using parameters from 2/1/14 (Sch¨obel-Zhu model)

(d) Simulated volatility using parameters from 2/6/14 (Sch¨obel-Zhu model)

Figure 3.7: Simulated variance/volatility using parameters from 2 January and 2 June 2014.

What are the effects on the simulated stock price? Figure 3.8 below shows the simulated stock price using the different parameters used for forecasting and those obtained by calibration for 15 simulated paths, 20 time steps and T = 0.55.

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(a) Simulated stock prices using parameters from 2/1/14 (Heston model)

(b) Simulated stock prices using parameters from 2/6/14 (Heston model)

(c) Simulated stock prices using parameters from 2/1/14 (Sch¨obel-Zhu model)

(d) Simulated stock prices using parameters from 2/6/14 (Sch¨obel-Zhu model)

Figure 3.8: Simulated stock prices (15 paths) using parameters from 2 January and 2 June 2014.

When comparing figure3.8ato figure3.8b, we see that in figure3.8athe spread between the simulated stock prices is somewhat smaller when we simulate using the parameters obtained at 2 January 2014, compared to when we simulate using the parameters obtained at 2 June 2014. We see the same in figures 3.8cand 3.8dfor the Sch¨obel-Zhu model. The simulated stock price paths are therefore more dense when using the parameters from 2 January 2014 and we will simulate stock prices that are less extremely high or extremely low relative to the initial stock price (S0). This has a positive effect on the prices of put options that are further OTM and a

negative effect on the prices of put options that are further ITM. This could therefore be a rea-son why both models respectively overstate and understate the prices of OTM and ITM options.

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The next date is 2 January 2015, where we forecast prices using four parameters obtained by calibration half a year before at 2 June 2014, i.e.

2014 Calibration (2 Jun) 2015 Forecasting (2 Jan) 2015 2 Jun

Figure 3.9: Predicted prices, actual prices and differences for 2 January 2015.

Pricing of American Options by Least-Squares Monte Carlo with Control Variates

Control Variates

Contents

Introduction

Theoretical Background

2.1

Options

2.2

Black-Scholes-Merton Model

2.3

Stochastic Volatility and Short Rate Models

2.4

Pricing of European Options

2.5

Pricing of American Options

Research Setup and Results

3.1

Model Calibration

3.2

Calibration Results for Facebook Options

3.3

Forecasting Prices of Facebook Options