UNIVERSITY OF GRONINGEN
Pricing Quanto Barrier Options Efficiently
Y.P. Mulder s2017504
Master Thesis Finance
Supervised by dr. Artem Tsvetkov (University of Groningen) Faculty of Economics and Business
Department of Economic, Econometrics and Finance
Abstract
Table of Contents
1
Introduction... 4
2
Background & Literature Review ... 7
2.1 Option Pricing ... 7
2.2 Quanto Barrier Options ... 9
2.2.1 Barrier Options ... 9
2.2.2 Quanto Options ... 10
2.2.3 Bringing Both Together ... 11
2.3 Volatility Models ... 12
2.3.1 Black-Scholes ... 12
2.3.2 Term Structure Volatility ... 12
2.3.3 Local Volatility ... 12
2.3.4 Stochastic Volatility ... 13
2.3.5 Stochastic Local Volatility... 13
2.4 Approximating the Price ... 14
3
Methodology ... 17
3.1 Numerical Methods ... 17
3.2 Variance Reduction ... 18
3.2.1 Antithetic Variates ... 18
3.2.2 Control Variates ... 18
3.2.3 Common Random Numbers ... 19
3.3 Brownian Bridge ... 20
3.4 Testing Approach ... 20
3.4.1 Underlying Asset Dynamics ... 20
3.4.2 Volatility Models ... 21
3.4.4 Robustness ... 23
3.4.5 Interpreting the Numbers ... 23
4
Results ... 25
4.1 Black-Scholes Constant Volatility ... 26
4.2 Term Structure Volatility ... Error! Bookmark not defined. 4.3 Local Volatility ... 28
4.4 Stochastic Local Volatility... 29
5
Conclusion ... 30
6
Bibliography ... 31
1 Introduction
Every day a bewildering amount of money is traded in the financial markets that exist all around the world. These financial markets perform an important function in the economy, both locally and globally. They connect the providers of funding to those in need of these funds, whether they are governments, companies or people. This is a very essential function for a healthy, growing economy. The capital markets are the most well known of these financial markets and trade in the basic products many are familiar with, such as stocks and bonds. But a large part of trading within these financial markets also occurs in the derivatives market. This market might be somewhat less recognizable for the layman, but has a very high trading volume nonetheless.
The derivatives traded on the derivatives market are assets whose payoff depends on the value of other, underlying assets traded on other markets, such as the stock market or the foreign exchange market. Many different types of derivatives exist each with a different set of conditions that govern the rights and obligations of the holder. This makes derivatives quite complex instruments and makes them less suitable for inexperienced traders looking to speculate. They are, however, very suitable for hedging risks when applied in a well constructed strategy. This is why especially for large institutional investors derivatives play an important role in managing the risks of their portfolio. Despite the heavy turmoil and crises on the stock markets the volume of derivatives that are traded each year keeps growing. (World Federation of Exchanges Ltd. , 2015)
Due to the growth that the global derivatives market exhibits the importance of derivative pricing theory is also growing. After all, the more trades in derivatives occur the more important it is for market participants to be able to price these derivatives accurately and quickly. It is in the interest of both participants that take part in the contract of the derivative for its value to be determined accurately. This prevents a derivative from having intrinsic value for one of these parties due to a pricing error, an error that is paid out of the other parties’ pocket. Also, some parties, such as banks, deal in a large volume of derivatives and pricing all of these derivatives requires a lot of computational power. Therefore, there is great value in the further development of derivative pricing theory to enable the efficient and accurate calculation of derivative prices.
almost always serves as a vehicle currency as the large market for it keeps the transaction costs as low as possible. Often it is cheaper to use a large currency as a vehicle to trade between two other currencies, than to trade directly between those currencies. This is the purpose the US Dollar fulfills for almost 97% of all the trades involving Asian currencies. (Wooldridge & Tsuyuguchi, 2008)
For example, quanto derivatives have become particularly popular in Asia where investors would like to obtain exposure to major G10 currencies but prefer to have profit in the local currency. Using these derivatives they can for example invest in the exchange rate between the euro and US dollar, without being exposed to the fluctuations of the exchange rate between either foreign currency and their local currency by contracting a fixed quanto rate. This also meets the interest of western banks, who prefer to have major exposure in their own liquid currencies and a limited exposure in less liquid Asian currencies. By selling quanto derivatives to Asian investors these banks can increase their exposure to their own currencies by tapping the Asian market for these currencies, while having only limited exposure to the Asian currencies due to the quanto rate.
Quanto barrier derivatives are particularly popular in structured products in China as the government has been regulating the deposit rates for a long time, leading to deposit rates that are not very favorable and are often barely higher than the inflation. While the government is currently in the process of a slow liberation of these interest rates, these regulations have spurred a popularity of these derivatives as an alternative investment. (Porter & Xu, 2013) By creating an investment that is part derivative, the limitations put on the interest rates for deposits and other products can be avoided and in this manner it is possible to offer higher expected returns to potential investors. Since barrier options are less expensive than vanilla options using them in these structured products makes it is possible for investors to profit from the advantages with a lower initial investment cost than using vanilla quanto options would entail.
more prevalent and barrier options are once again one of the most traded derivatives. (Wystup, 2007)
2 Background & Literature Review
To fully comprehend the question at hand and its context, this section will explain the basic principles of option pricing, the properties of quanto barrier options and the properties of the different volatility models tested. After these are properly explained, both the regular pricing method for quanto barrier options and its approximation will be discussed.
2.1 Option Pricing
Since the subject of this thesis concerns only options, other types of derivatives such as forwards and swaps, will be disregarded for the sake of brevity. As explained shortly in the introduction, derivatives in general are assets who payoff depends on the value of other, underlying assets. Options are a type of derivative contract where the holder obtains the right, but not the obligation, to buy or sell this underlying asset at a pre-specified price at some pre-determined moment in the future. This allows the holder to speculate on the value of the underlying assets without having to invest in this asset itself. The fact that the holder is not obligated to exercise his right also limits the risk he is exposed to when his speculations are false; the most he stands to lose is the premium paid for acquiring the option and the rights that come with it. The pre-specified price for which the holder can buy or sell the underlying asset is called the “strike price” and the pre-specified moment in the future is called “at maturity”. Depending on whether the option gives the holder the right to buy or sell the underlying asset at maturity the option is either called a “call option” or a “put option” respectively. (Hull, 2015)
The payoff of such a vanilla option is defined as the difference between the strike price and the value of the underlying asset at maturity, T, if it exceeds zero and zero payoff otherwise. This payoff can be expressed as follows, where 𝑆𝑇 is the value of the underlying asset at time 𝑇 and 𝐾 is the strike price:
Call option: (𝑆𝑇− 𝐾)+ (2.1)
Put option: (𝐾 − 𝑆𝑇)+ (2.2)
it replicates, otherwise there would be arbitrage opportunities. If arbitrage opportunities exist, it is possible for investors to make a profit by short selling the more expensive of the two identical product while simultaneously investing in the less expensive product. This enables such an investor to profit without having to expose himself to any risk or having to invest any money. When this happens, the market dynamics of an efficient market will quickly compensate the difference in the prices of these product, thus eliminating the arbitrage and making the arbitrage pricing theory a reliable model for pricing assets.
According to the arbitrage pricing theory, the present value of a derivative is equal to the expected payoff at maturity discounted at the risk-free rate. (Bjork, 2009) This is the basis for many options pricing models, the most well known of which is the Black-Scholes (1973) model. This model is very popular because it enables investors to calculate options prices analytically for many cases. In order to facilitate this ease of calculation a few assumptions have to be made, most importantly the assumption that the returns on the underlying are log-normally distributed and that the risk-free rate and volatility are known, constant and deterministic. The assumption that the volatility is constant is, however, unrealistic and this can lead to inaccurate prices when using the Black-Scholes model. This is why more sophisticated models have been developed that more accurately cover the dynamics that the market exhibits. These models will be discussed in more detail later on in this chapter.
implement but often used by professionals in their pricing algorithms because of the relatively high calculation accuracy it has to offer.
Pricing quanto barrier options using stochastic volatility takes a rather large amount of computational power, since this requires the calibration of the volatilities to those implied from the market prices and the numerical solving of four-dimensional partial differential equations. The foreign exchange market is the world’s largest financial market and is also very liquid, trading up to $5.3 trillion per day, this is about $725 for every person living on earth! (Rime & Schrimpf, 2013) The derivatives traded on this market, such as quanto barrier options, are also extremely liquid, which requires them to be very accurately priced, as at these volumes of trades even an error of a fraction of a percent concerns a dazzling amount of money. To aid the parties selling quanto barrier options with keeping their prices accurate and up to date with the latest developments in the market in near real time using an approximation is a solution. The simplification created by Chuang (1996) will be tested to see how suitable it is for this application.
2.2 Quanto Barrier Options
A quanto barrier option is an option that combines the properties of both barrier and quanto options. In order to properly understand this construction, let’s start by examining the properties of barrier and quanto options separately.
2.2.1 Barrier Options
A barrier option is a type of exotic option that is similar to European call or put options, but has a payoff that is contingent on whether the underlying asset’s price has reached a certain level (the barrier) during the time to maturity of the option. Barrier options are attractive to some market participants since they are less expensive than a comparable vanilla option, which is due to the reduced chance of a positive payoff. Barrier options can be divided into two subtypes; knock-in and knock-out options. A knock-in option only exists after the underlying asset’s price has reached the specified barrier at any time during the lifetime of the option, while a knock-out option ceases to exist if the barrier is reached at any time before maturity.
variety, where a knock-in and knock-out options constitute down-and-in and down-and-out barrier options respectively. The name format for these varieties is of course based on the direction the price of the underlying needs to take to reach the barrier combined with the moving in or out of existence that is caused by reaching this barrier. (Hull, 2015)
Due to the properties of these barrier options the payoff they offer can be discontinuous, which means that there exists a sudden jump in the option value that occurs when the barrier is reached. This can happen for options that can hit the barrier when the option is in the money, the up-and-out call or the down-and-out put option. These kind of barrier options are often called reverse barrier options. This phenomenon is illustrated very clearly in the figure 2.1 on the right, which displays a
payoff diagram for an up-and-out call option that loses its value when the barrier is hit. Another property of inherent to the barrier option’s properties is path-dependency, which means that the eventual value of the options depends not only on the price of the underlying asset at maturity but also on the prices of the underlying asset during the entire time to maturity.
For example, consider an investor holding an up-and-out call barrier option on the AEX index with a strike price of €420 and a barrier of €450. If during the lifetime of this option the value of the AEX index hits or transcends the barrier of €450, the option will cease to exist and the payoff will be zero, even if the value of the AEX index is higher than the strike price at maturity. If the barrier is not hit during the option’s lifetime, the payoff will be that of a regular call option. Let’s say the value of the AEX index is €435 at maturity, this would payoff €15 when the barrier has not been hit and €0 if the barrier has been hit at any time during the lifetime of this option.
2.2.2 Quanto Options
A quantity-adjusting (quanto) option is mainly useful for investors looking to invest in assets that trade in a foreign currency. Quanto options are in the simplest form very similar to a vanilla call or put option where the underlying asset value and strike price are denominated in a foreign currency. But with a quanto option the payoff at maturity occurs in the investor’s domestic currency. When the option is exercised its intrinsic value will be calculated in the foreign currency, which is then converted to the domestic currency. The exchange rate (the quanto rate) with which the value is converted is fixed and is part of the option contract. Holding a quanto option is therefore
comparable to holding a vanilla option combined with a currency forward that has the option’s value as a variable notational amount. (Hull, 2015)
For example, consider an investor from the United States looking to speculate on the Nikkei 225 index, which is traded in the Japanese Yen (JPY). He can invest directly in an option on the index, but that would leave him exposed to the changes in the JPY-USD exchange rate, potentially harming his actual payoff. Now using a quanto option on this exchange instead would pay him a fixed amount of US dollars for every Japanese Yen that he has earned in the form of the option’s intrinsic value, removing exchange rate risk from the equation.
Using quanto options therefore allows investors to invest in assets that are traded in a different currency without having to be exposed to the fluctuations that occur in the currency exchange rate. This way an investor is only exposed to the fluctuations in the underlying asset and does not have to worry about the whims of the currency market. This makes quanto options a very attractive proposition for investors looking to invest in assets that are traded in different currencies than their own without being exposed to exchange rate risk.
2.2.3 Bringing Both Together
Now to form the quanto barrier option it is easiest to see the barrier option as replacing the regular option part of the quanto option above and having its payoff at maturity converted using the quanto rate. This makes the quanto barrier option an option where the barrier, strike price and underlying asset are denoted in the foreign currency and the value of that option is settled in the domestic rate at a fixed exchange rate. Therefore, the quanto barrier option allows investors to utilize the properties of a barrier option when looking to invest in assets traded in a foreign currency without exposure to exchange rate risk. Often the quanto barrier option is used not with a regular stock as underlying asset, but with a currency exchange rate as underlying asset so that one can speculate one the changes in a foreign currency pair without being exposed to the exchange rate risk to the between the foreign and domestic currencies.
study for efficiency, since the complexity also affects the speed with which these models can be calculated.
2.3 Volatility Models
Several different volatility models will be used to test the performance of the approximation in a representative and informative manner. These models will be explained below.
2.3.1 Black-Scholes
The most straightforward of all volatility models is the Black-Scholes model, where the volatility is assumed to be constant. This means that during the complete lifetime of the option the volatility remains the exact same number, independent from the other developments concerning the underlying asset or the option. By assuming that the volatility is constant it is possible to analytically solve the partial differential equations that describe the option prices. This gives formulas that consistently provide the same price for the same parameters.
2.3.2 Term Structure Volatility
Another simple volatility model is the term structure volatility model, where the volatility is not constant but depends only on the time to maturity. It is often observed in options that the volatility level implied from the Black-Scholes model is higher for options that have a shorter maturity. This can be explained by the probability of events that will seriously affect the underlying asset's value occurring in the near future. According to mean reversion theory this value will eventually move back towards its mean, so if the maturity is longer, the chance that these events will cancel out will be larger, leading to lower volatilities. While with a short maturity these events can have drastic consequences for the value of an option. This scenario is mainly observed during high volatility periods, as the short term volatility mimics the recently realized volatility. During low volatility periods the relationship might be the other way around, where the short term volatility mimics the low recently realized volatility and the long term volatility follows the higher, long term average volatility. (Xu & Taylor, 1994)
2.3.3 Local Volatility
are calibrated to match the volatility levels that the asset exhibits in the real world by using the volatility level implied from the Black-Scholes model. While this is a very attractive proposition in theory, in practice research has shown that using local volatility gives no better predictive or hedging performance. (Dumas, Fleming, & Whaley, 1998) This is particularly unfortunate for barrier options since these are very sensitive to the dynamics of the volatility smile and to the forward transition probability it influences. A single jump in the stock price, however short, after all, can knock-out the barrier option and completely destroy its value.
2.3.4 Stochastic Volatility
Stochastic volatility is a model that uses a random process for the volatility of the asset underlying the option. The simplest variety is using a geometric Brownian motion, a diffusion process, other more complex versions using for example jump-diffusion processes also exist but will not be discussed in this thesis. One of the most well-known and popular stochastic volatility models is the Heston (1993) model, which derives its popularity from the closed-form solution it offers for European call options. Stochastic volatility models provide more realistic dynamics for the option pricing process than Black-Scholes or local volatility models, but they are usually not able to fit the complete volatility smile at every maturity.
2.3.5 Stochastic Local Volatility
To overcome the shortcomings of the previously mentioned models the stochastic local volatility model has been developed. This model combines the strengths of both local and stochastic volatility models by letting the volatility be determined by function of both. While some initial simple approaches suggested combining both by taking a linear average, more sophisticated approaches have since been developed. An example of a very popular more sophisticated approach is the model used by Bloomberg (Taturu & Fisher, 2010) where the volatility process is structured as a stochastic multiplier (of around 1) that is imposed on local volatility. Taturu and Fisher (2010) also suggested in their paper that a term-structure model with a log-normal process for volatility realistically describes the market dynamics. Generally speaking, a stochastic local volatility model has a local volatility part represented by the so-called leverage function and a stochastic component represented by a random process. The leverage function, simply put, indicates the ratio between the local volatility and the conditional expectation of the stochastic volatility and is rather non-trivial to calibrate to the observed market dynamics.
Therefore, this volatility model will be used as the most realistic volatility model for the simulations done in this thesis.
2.4 Approximating the Price
The payoff of a quanto barrier option can be written as seen in the equation below.
Λ = (𝑆2𝑇− 𝐾)+𝑄 (2.3)
d
Here 𝑆2𝑇 is the value of the underlying asset at maturity 𝑇, 𝐾 is the strike price, and 𝑄 is the quanto fixed exchange rate . This equation cannot be easily used in calculation because the growth rate of 𝑆2𝑡 is only known in the currency of asset 𝑆2. To avoid this complication we convert the payoff from the quanto currency to the domestic currency of the asset by multiplying with the exchange rate 𝑆1𝑇.
Λ = (𝑆2𝑇− 𝐾)+𝑄𝑆1𝑇 (2.4)
D
In order to determine the correct price for a quanto barrier option one takes the expected value of this pay off and multiplies it with the probability that the option survives, that is; does not touch the barrier during the maturity, and discounts this value with the appropriate risk free rate.
The price of a vanilla call option the price can be expressed with the use of the probability density function for 𝑆𝑇, 𝑓(𝑆𝑇), risk-free rate 𝑟𝑑 and payoff(𝑆𝑇− 𝐾)+:
𝑉 = 𝑒−𝑟𝑑𝑇∫ (𝑆𝑇− 𝐾)+
∞ 0
∙ 𝑓(𝑆𝑇) 𝑑𝑆𝑇 (2.5)
D
Analogously the price of a barrier option can be expressed using the joint probability density function 𝑓(𝑆𝑇, 𝑀𝑇), where 𝐻 is the barrier level, 𝑀𝑇 = max(𝑆𝑇) and 𝟏(𝑀𝑇 < 𝐻) is a binary function conditional on 𝑀𝑇 being lower than the barrier. (Privault, 2013)
𝐵 = 𝑒−𝑟𝑑𝑇∬ (𝑆𝑇− 𝐾)+
∞ 0
∙ 𝟏(𝑀𝑇 < 𝐻) ∙ 𝑓(𝑆𝑇, 𝑀𝑇) 𝑑𝑆𝑇𝑑𝑀𝑇 (2.6)
D
Which can be also be expressed, somewhat simplified, as:
𝐵 = 𝑒−𝑟𝑑𝑇∫ (𝑆𝑇− 𝐾)+
∞ 0
∙ 𝑓(𝑆𝑇, 𝑆𝑡 < 𝐻) 𝑑𝑆𝑇 (2.7)
D
adjustments made in the payoff, which include adding a second asset value, the exchange rate 𝑆1𝑇 that also has to be included in the probability density function and should be integrated over. Note that the underlying asset of the previous equations is now referred to as 𝑆2𝑇. This gives the following equation for pricing a quanto barrier option:
𝑄𝐵 = 𝑒−𝑟𝑑𝑇∬ (𝑆
2𝑇 − 𝐾)+𝑄𝑆1𝑇∙ 𝑓(𝑆1𝑇, 𝑆2𝑇, 𝑆2𝑡 < 𝐻) 𝑑𝑆1𝑇𝑑𝑆2𝑇 ∞
0
(2.8)
Solving this equation using numerical methods will give the exact price for the quanto barrier option, assuming that the calibration of the probability density function is perfect. To obtain this calibration at a reasonable level of precision and to subsequently solve the equation is very complex and computationally intensive. Therefore, it is generally not ideal to use this approach and an approximation that simplifies this process is very welcome.
The approximation tested in this thesis manages to simplify the probability density function by splitting it into three different functions. This allow for a calibration that is much easier, as the probability density functions now rely only on a single stock each. The ease of solving this equation using finite difference methods also greatly improves, as these are limited in the amount of dimensions that can be solved and this approximation reduces this number. From Chuang (1996) it can be deduced that this approximation should be exact for the Black-Scholes model. This approximation will be derived below.
A standard principle for conditional probability functions is that they can be rewritten to a joint probability function of both variables divided by the probability function of the condition. This principle will be used to rewrite the probability density function to a conditional variant.
𝑓(𝑆1𝑇, 𝑆
2𝑇, 𝑆2𝑡< 𝐻) = 𝑓(𝑆1𝑇 | 𝑆2𝑇, 𝑆2𝑡 < 𝐻) ∙ 𝑓( 𝑆2𝑇, 𝑆2𝑡< 𝐻) (2.9)
D
Now an approximation of this conditional probability density function is made
𝑓(𝑆1𝑇 | 𝑆2𝑇, 𝑆2𝑡 < 𝐻) ≈ 𝑓(𝑆1𝑇 | 𝑆2𝑇) (2.10)
D
Once again, the same principle as above is applied but this time to revert the conditional probability density function back to a joint probability density function.
𝑓(𝑆1𝑇 | 𝑆2𝑇) =
𝑓(𝑆1𝑇, 𝑆2𝑇)
D
Substituting this into equation 2.9 gives us the following approximation.
𝑓(𝑆1𝑇, 𝑆2𝑇, 𝑆2𝑡 < 𝐻) ≈
𝑓(𝑆1𝑇, 𝑆2𝑇)
𝑓(𝑆2𝑇) 𝑓( 𝑆2𝑇, 𝑆2𝑡 < 𝐻) (2.12)
D
Which can then be substituted into equation 2.8 to form the approximation for the quanto barrier options price that will be tested in this thesis.
3 Methodology
This section explains the approach taken to testing the approximate formula for pricing quanto barrier options. Also it details some of the methods and techniques used to come to results that are as accurate and representative as possible.
3.1 Numerical Methods
While pricing options, even some more complex ones, it possible with analytical formulas in the Black-Scholes model, for pricing options under different volatility models numerical methods are required. The main approaches to numerical option pricing have already shortly been discussed in the option pricing section in the background and literature review chapter. This section brings these models into the context of pricing quanto barrier options for the purpose of this thesis.
The binomial tree method, while easy to implement in general, is notoriously hard to implement for simulations that require non-constant volatility, as the approach used here does. This makes the binomial tree method not very useful for testing the approximate pricing formula for quanto barrier options. The finite difference method is suitable for pricing options under different volatility models and does so quite accurately, but developing such a model is highly complex and somewhat limited in the amount of dimensions that can be solved. Therefore, the finite difference method is not very suitable for a simple testing of the approximation where computation time is less of an issue than in the real world. This is why in this thesis the prices for quanto barrier options will be simulated using the Monte-Carlo method.
on the amount of paths and the grid resolution, but can be made more accurate using variance reduction techniques.
3.2 Variance Reduction
While with finite computational time and power it will never be possible to calculate the price of a quanto barrier option to the point where it can be considered exact, it is important to keep the variance of the calculated prices as low as possible. This way the results obtained will be consistent and representative for the price of the option. To achieve this there are several available techniques for reducing the variance and these are discussed below.
3.2.1 Antithetic Variates
The antithetic variates method is a variance technique used for Monte-Carlo simulations. The main principle is that for every sample path obtained its antithetic path will also be used. Simply put, for every path that is generated we also take the negative of this path. This way whenever any unusually large or small output is calculated in a path it is balanced by an antithetic value of equal magnitude, resulting in a lower variance. (Glasserman, 2003) This technique not only has the advantage of reducing the variance and therefore improving the accuracy of the simulation, but also reduces the number of normal samples that need to be taken to obtain the desired number of sample paths. Antithetic variates will be used to generate the sample paths used in pricing the quanto barrier options with Monte-Carlo simulations in this thesis.
3.2.2 Control Variates
The control variates method is another variance reduction technique used for Monte-Carlo simulations. It is one of the most broadly applicable and effective methods for improving Monte-Carlo simulation accuracy. It relies on using the error of estimating a known quantity to reduce the error of the unknown quantity to be estimated. (Glasserman, 2003)
For example, take one known variable Y and an unknown variable X, that we wish to estimate. The control variate, Y, should be chosen on the expectation of being close to X. Now for variable Y we known the exact value it has, meaning that value has zero variance, this is the expected value of Y; 𝐸(𝑌). Also we can use a Monte-Carlo simulation to determine the value of Y with a non-zero variance. Now we can express our simulation for the value of X as a random variable Z:
𝑍 = 𝑋 + 𝐸(𝑌) − 𝑌 (3.1)
B
𝐸(𝑍) = 𝐸(𝑋) (3.2)
This means the Monte-Carlo simulation of Z will give us the value of 𝐸(𝑋) that we are looking for. Since adding the constant 𝐸(𝑌) to the Z equation does not influence the variance, the variance can be represented as follows:
𝑉𝑎𝑟(𝑍) = 𝑉𝑎𝑟(𝑋 − 𝑌) (3.3)
B
When the variance of X - Y is low, which is why the variable Y should be close to X, this gives a much lower variance on the simulation of Z, thus approximating 𝐸(𝑍) and therefore 𝐸(𝑋) with a higher degree of accuracy. For the simulating of quanto barrier options, the price as calculated with an analytical formula will be used as a control variate. This analytical formula is a combination made from an analytical formula for quanto options and a separate analytical formula for barrier options, both providing exact results under the assumptions of the Black-Scholes model. When simulating using Black-Scholes’ constant volatility model this should lead to an exact value, while serving to drastically reduce the variance when using other volatility models.
In order to achieve the maximum variance reduction when 𝑉𝑎𝑟(𝑋 − 𝑌) ≠ 0 it is possible to determine the coefficient 𝜃∗, which will lead to a minimized variance for 𝑍, from the following pair of equations:
𝑍 = 𝑋 + 𝜃∗(𝐸(𝑌) − 𝑌) (3.4)
𝜃∗=𝐶𝑜𝑣(𝑋, 𝑌)
𝑉𝑎𝑟(𝑌) (3.5)
3.2.3 Common Random Numbers
numbers. This way the difference in pricing accuracy of both approaches can be compared in a fair manner, without having different random number sets influence the results.
3.3 Brownian Bridge
Monte-Carlo simulations for path dependent options such as the quanto barrier option have a very important shortcoming; it can only simulate the path in discrete time. For barrier options this means that, for example, the check on whether the barrier has been hit only takes place once a day. For a barrier option to be priced accurately this check should occur continuously during that day and not just once, as it can hit the barrier at any time during that day. While it is of course possible to refine the time grid further, this would have drastic consequences for the computational power required and still leave a lot of inaccuracy. This is where the Brownian bridge comes in. A Brownian bridge is a continuous time stochastic process that not only has a fixed starting value but also a fixed end value, creating a stochastic bridge between both points. The Brownian bridge can be used as a continuous time bridge between the underlying asset’s value on the time grid points in the Monte-Carlo simulation. With the Brownian bridge process, it is then possible to determine the probability that the underlying asset has hit the barrier during the step in time taken. Since this probability comes from a continuous time process it is far more accurate in determining the probability of survival for a barrier option and therefore leads to more accurate pricing while being less computationally expensive. (Beaglehole, Dybvig, & Zhou, 1997)
3.4 Testing Approach
In order to test whether the approximate pricing process for the quanto barrier options also works for volatility models beside the Black-Scholes model, a Monte-Carlo simulation will be programmed in MathWorks MATLAB. MATLAB is used because it is a high level programming language and therefore quick to develop in and because it has a lot of useful mathematical functions and tools readily available, further contributing to the speed of developing the program. The simulation will be equipped to calculate the price of a quanto barrier option using the regular non-approximated approach and will also enable pricing of the options using the approximation. Also the volatility reduction techniques and Brownian bridge discussed above will be implemented in this program.
3.4.1 Underlying Asset Dynamics
𝑑𝑆𝑡 = 𝜇𝑆𝑡𝑑𝑡+ 𝜎𝑆𝑡𝑑𝑊𝑡 (3.6)
Where 𝑆𝑡 is the value of the asset at time 𝑡, 𝜇 is the percentage drift term, 𝜎 is the percentage volatility and 𝑊𝑡 is a Wiener process or Brownian motion. For the case of pricing quanto barrier options the drift term 𝜇 will be represented by the risk-free rate and the volatility 𝜎 will be implemented according to the volatility model that is simulated. Using Itō’s lemma the following analytical solution can be found for the above equation:
𝑆𝑡= 𝑆0𝑒
(𝜇−𝜎22)𝑡+𝜎𝑊𝑡 (3.7)
This is the equation that will be used to calculate the asset price at each time step 𝑡 on the grid of the Monte-Carlo simulation. However, because it is more efficient to calculate these prices at the log-scale than on a linear scale we will use the following equation that is easily derived from the one above:
ln(𝑆𝑡) = ln(𝑆0) + (𝜇 − 𝜎2
2) 𝑡 + 𝜎𝑊𝑡 (3.8)
3.4.2 Volatility Models
The simplest of the volatility models is naturally the Black-Scholes model, where the volatility 𝜎 is merely a constant that can be put into the formula above. For this constant volatility the quanto barrier option price from simulating the approximation should be about equal to the option price from simulating the full pricing process and both should be very close to the analytical price. Other volatility models are a bit more complex and will be implemented in a separate function in MATLAB, this way it is easy to switch between these models and it will also keep the code cleaner. Since it is very time consuming to calibrate these models to observed market dynamics, the program used for this thesis will define the volatility models based on some analytical formulas that closely replicate the behavior often observed in the market.
𝜎(𝑡, 𝜎𝑟𝑒𝑓) = 𝜎𝑟𝑒𝑓∙ (1 − 𝑥 + 𝑥𝑒−2𝑡) (3.9)
B
The local volatility model is described by the formula below, which also shares the term structure component above. This formula also depends on the spot price 𝑆𝑡, the initial value 𝑆0 and a factor 𝑑 that defines how much the volatility should increase for each standard deviation of 𝑆𝑡 from 𝑆0. (0.05 for these simulations for an increase of 5%)
𝜎(𝑡, 𝜎𝑟𝑒𝑓, 𝑆𝑡, S0) = 𝜎𝑟𝑒𝑓∙ (1 − 𝑥 + 𝑥𝑒−2𝑡) ∙ (1 + 𝑑 ∙ ( ln(𝑆𝑡) − ln(𝑆0) 𝜎𝑟𝑒𝑓∙ √𝑡 ) 2 ) (3.10) B
The stochastic local volatility model used is based on the local volatility model above, but adds a stochastic component responsible for stochastic variations in the volatility. This stochastic component is described by 𝑊𝑡2, a Wiener process that is correlated with the Wiener process 𝑊𝑡1 that drives the asset dynamics of 𝑆𝑡. It also contains a fractional coefficient 𝑘 that determines the impact of the stochastic process on the volatility relative to the local volatility component. (0.25 for these simulations for an impact of up to 25%)
𝜎(𝑡, 𝜎𝑟𝑒𝑓, 𝑆𝑡, S0, 𝑊𝑡2) = 𝜎𝑟𝑒𝑓∙ (1 − 𝑥 + 𝑥𝑒−2𝑡) ∙ (1 + 𝑑 ∙ ( ln(𝑆𝑡) − ln(𝑆0) 𝜎𝑟𝑒𝑓∙ √𝑡 ) 2 ) ∙ (1 +𝑘 5𝑊𝑡 2) (3.11)
3.4.3 Simulating the Approximation
In order to simulate the approximation, first the full Monte-Carlo quanto barrier option price will be simulated. This simulation will provide a set of data containing the outcomes for each path for both the underlying asset as well as the foreign exchange rate. To these outcomes it is possible to empirically fit the required probability density functions (PDFs) using kernel density estimation. Kernel density estimation is a non-parametric method to fit a PDF to a random variable, such as the asset values involved here. To execute the kernel density estimation in MATLAB, both for the 1-dimension and the 2-1-dimensional PDF, a third party package by Botev (2007) will be used.
These PDFs can then be used to calculate outcome of the following integral, where Λ is the payoff at maturity, 𝑓(𝑆2) is the PDF of the underlying asset 𝑆2 and 𝑓(𝑆1, 𝑆2) is the joint PDF of the underlying asset and the foreign exchange rate 𝑆1.
B
This integral gives us the expected payoff at maturity if there was no barrier involved, we multiply this with the survival probability of the option that is calculated using the Brownian bridge to obtain the intrinsic value of the option. This process is executed for each of the simulated outcomes for the underlying asset 𝑆2, then averaged and finally discounted at the risk-free rate to obtain the fair price for the option.
3.4.4 Robustness
While the simulating the prices of both the approximation and the full Monte-Carlo approach for a single configuration of the option parameters can give some insight into the performance of the approximation, it is important to see how robust this performance is across a range of possible combinations of parameters. This is why the simulations will be executed for a range of different strike prices, different barrier level, different maturities and different base volatility levels. With the extensive dataset that this provides it should be possible to infer a realistic picture of the performance of the approximation.
3.4.5 Interpreting the Numbers
When the dataset has been calculated it all comes down to the valid interpretation of the results it contains. Aside from the picture that the absolute and relative values of the difference between the two pricing approaches sketch, the performance is best compared using a standard statistical test. Seeing as the mean of the payoffs associated with all the sample paths from the Monte-Carlo simulation forms the expected value of the option, it is possible to compare the means from the approximated sample and the regular sample. This can be done with a 𝑡-test, which is the generally accepted approach for comparing the mean of two populations. The hypothesis of this test is that the two populations it is calculated for have the same mean. A regular 𝑡-test requires the assumptions that both populations have the same variance and that both are normally distributed. The assumption of equal variances might not always hold when comparing both of the approaches tested here, but the assumption of normality is no problem. This is why Welch’s 𝑡-test is the most suitable test for this scenario, as it does not rely on the assumption of equal variances and only on the assumption of normality. Therefore, this test will be used to judge the accuracy of the approximation compared to the full Monte-Carlo approach.
4 Results
The both pricing formulas have been successfully implemented in MATLAB and simulations have been run to generate a considerable dataset from which inferences about the performance can be made. The results calculated include the following:
Analytical quanto barrier price Full Monte-Carlo quanto barrier price Approximated quanto barrier price
Difference between full and approximated price
Relative difference between full and approximated price Standard deviation of the Monte-Carlo payoffs
Standard deviation of the approximated payoffs Results of the Welch t-test on these payoff (p-value) Full Monte-Carlo quanto price
Approximated quanto price
Relative difference between full and approximated quanto price
The full dataset containing these results can be found in the appendix of this thesis. In the next sections the most important results from this dataset will be presented on the basis of some excerpts from this dataset.
For reference, figure 4.1 below displays the probability density surface as estimated using the kernel density estimation function for 𝑓(𝑆1𝑇, 𝑆2𝑇).
Figure 4.1: Probability density surface for 𝑓(𝑆1𝑇, 𝑆2𝑇). Where the probability density is displayed for combinations of 𝑆1𝑇and
4.1 Black-Scholes Constant Volatility
When simulating using the Black-Scholes volatility model both the approximation and the full Monte-Carlo prices should be very close to the analytical prices. Also the approximation is known to be exact to the full Monte-Carlo price for this scenario. As can be seen in table 4.1, which shows the results of these pricing methods for different barrier levels, this is not entirely the case. The relative differences between the Monte-Carlo approach and the approximation are all lower than 1% and the Monte-Carlo approach is exact to the analytical approach. But the p-value from the Welch’s t-test (over the Monte-Carlo and the approximation) is low for most barrier levels, which means that the null-hypothesis, which states that both samples have the same mean is rejected
Table 4.1: Excerpt of simulation results for different barrier levels, showing the price from the analytical formula, the full Monte-Carlo approach (MC) and from the approximation. Also displays the relative difference between MC and Approximation and the p-Value from the Welch’s t-test.
Barrier (H) Analytical MC Approximation Relative Difference p-Value t-test
4.2 Term Structure Volatility
For the term structure volatility, the volatility curve presented in figure 4.2 on the right is used. The results for the quanto barrier options prices under term structure volatility show an increase in relative difference between the Monte-Carlo approach and the approximation, as can be seen in the results excerpt in table 4.2. This table shows the results of the price simulations for term structure volatility for different strike
prices. The increase in relative difference can also be found in all the robustness approaches used, so the difference increases when using term structure volatility.
Table 4.2: Excerpt of simulation results for different strike prices, showing the price from the analytical formula, the full Monte-Carlo approach (MC) and from the approximation. Also displays the relative difference between MC and Approximation and the p-Value from the Welch’s t-test.
Strike (K) Analytical MC Approximation Relative Difference p-Value t-test
€ 30 € 13,826 € 14,306 € 14,510 1,43% 0,000 € 35 € 11,131 € 11,494 € 11,674 1,56% 0,000 € 40 € 8,635 € 8,799 € 8,940 1,59% 0,000 € 45 € 6,424 € 6,391 € 6,499 1,69% 0,000 € 50 € 4,555 € 4,343 € 4,425 1,88% 0,000 € 55 € 3,053 € 2,730 € 2,794 2,31% 0,000 € 60 € 1,909 € 1,555 € 1,597 2,70% 0,000 € 65 € 1,090 € 0,772 € 0,801 3,78% 0,000 € 70 € 0,548 € 0,305 € 0,322 5,45% 0,000 € 75 € 0,226 € 0,078 € 0,086 10,12% 0,004 € 80 € 0,065 € 0,003 € 0,006 91,56% 0,091 € 85 € 0,008 € 0,003- € 0,002- -14,25% 0,537
4.3 Local Volatility
Now with local volatility the curve of the volatility does not only depend on the percentage of the option’s lifetime that has passed but also on the values of 𝑆0 and 𝑆𝑡. The surface that results from the volatility function used here is displayed in figure 4.3 below.
Figure 4.3: Local volatility surface, displaying volatility against the percentage of the option’s lifetime that has passed and the spot price, 𝑆𝑡
The results for the quanto barrier options prices under the local volatility model are shown in table 4.3 below.
Table 4.3: Excerpt of simulation results for different time to maturities, showing the price from the analytical formula, the full Monte-Carlo approach (MC) and from the approximation. Also displays the relative difference between MC and Approximation and the p-Value from the Welch’s t-test.
TTM (T) Analytical MC Approximation Relative Difference p-Value t-test
4.4 Stochastic Local Volatility
When using the stochastic local volatility, a stochastic component is added on top of the local volatility surface displayed in figure 4.3. This leads to a random walk added at each point, making the volatility surface move randomly away from the local volatility surface within a certain range. This can be seen in the local volatility plot in figure 4.4 on the right, where the jumps in volatility against the asset price 𝑆𝑡 is displayed.
The results for the quanto barrier options prices under the stochastic local volatility model are shown in table 4.4 below.
Table 4.4: Excerpt of simulation results for different reference volatilities, showing the price from the analytical formula, the full Monte-Carlo approach (MC) and from the approximation. Also displays the relative difference between MC and Approximation and the p-Value from the Welch’s t-test.
Sig_S2 Sig_S1 Analytical MC Approximation Relative Difference p-Value t-test
10% 5% € 2,319 € 2,553 € 2,554 0,04% 0,931 15% 10% € 2,477 € 2,839 € 2,846 0,24% 0,510 20% 15% € 1,909 € 2,412 € 2,423 0,45% 0,386 25% 20% € 1,345 € 1,926 € 1,954 1,42% 0,039 30% 25% € 0,938 € 1,537 € 1,561 1,58% 0,103 35% 30% € 0,664 € 1,279 € 1,280 0,03% 0,979 40% 35% € 0,481 € 1,069 € 1,032 -3,38% 0,057 45% 40% € 0,357 € 0,933 € 0,855 -8,36% 0,000 50% 45% € 0,271 € 0,799 € 0,682 -14,69% 0,000 55% 50% € 0,210 € 0,686 € 0,566 -17,49% 0,000 10% 5% € 2,319 € 2,553 € 2,554 0,04% 0,931 15% 10% € 2,477 € 2,839 € 2,846 0,24% 0,510
5 Conclusion
For determining whether a simplified equation for pricing quanto barrier options gives an accurate approximation to the exact equation for pricing these options, a set of simulations has been executed. These simulations have been performed using Monte-Carlo combined with several variance reduction techniques. This way a dataset has been established containing, among other parameters, the prices for both approaches, the difference between them and the results of Welch’s t-test on both samples. All results have been calculated for several strike prices, barrier level, time to maturities and reference volatility levels, providing an extensive dataset for robustness checks. It was found that while the relative difference between the approximated price and the full Monte-Carlo price is generally lower than 5%, the p-values generally indicate that both approaches have a different mean. The simulations run with a volatility from the Black-Scholes model show the highest p-values, but still these often fall below 0.05. This means that while the approximation is most accurate for Black-Scholes constant volatility, with relative deviations mostly below 1%, there are still large differences present between both samples according to the t-test. Under this volatility the approximation should be exact, so this makes a case for more extensive testing that should be performed at a higher level of accuracy.
When looking at the results for the other volatility levels the p-values are generally very close to 0. This indicates that the means of both approaches are highly unlikely to be equal. In a financial context this would mean that the performance of the approximation should be considered to be inadequate when looking only at this statistic. But the relative deviations for these models, especially with the stochastic local volatility, remain quite low, and this offers some hopeful perspective. Perhaps with more accurate simulation a more definitive conclusion can be drawn on the performance of this approximation, since a certain inaccuracy from the simulation obviously influences these results. The relative deviations however are small enough to suggest that this pricing approximation can be reasonable for a lot of applications, when executed with enough accuracy, but further research will have to prove so.
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7 Appendix
For the results please refer to the enclosed Results.xlsx file
