Amsterdam School of Economics
Faculty of Economics and Business
M.Sc. Financial Econometrics
Master Thesis
How American option exercise strategies are
affected by negative interest rates
Author1: Supervisor UvA2:
Thomas Christiaan Blokland prof. dr. ir. M.H. Vellekoop Second reader UvA3:
prof. dr. H.P. Boswijk
February 10, 2017
Abstract
This thesis investigates exercise strategies for American options on equity in a negative interest rate environment. Common assumptions about European and American options do not hold when negative interest rates are introduced. The paper investigates what happens to European and American options after the entrance in negative interest rate territory, for different maturities and strike prices. The options are priced using the so-called Longstaff-Schwarz algorithm. The underlying will be assumed to follow the SABR model; since this model is known to give a good fit in practice. Rules such as; “Never early-exercise an American call when there are no dividends” are shown not to hold when interest rates are negative. This thesis in fact shows that optimal exercise regions are mirrored when the interest rate drops below zero. Keywords: negative interest rates, option pricing, American options, European options, Black and Scholes model, SABR model, stochastic volatility, Longstaff-Schwarz, Monte Carlo, calibration.
1University of Amsterdam, student no. 6162460, Thomas.Blokland@student.uva.nl
2University of Amsterdam, Faculty of Economics and Business, Section Quantitative Economics 3University of Amsterdam, Faculty of Economics and Business, Section Quantitative Economics
Statement of Originality
This document is written by Thomas Blokland, who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those 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.
Contents
1 Introduction 3
2 Option Pricing Models 5
2.1 Introduction to basic options . . . 5
2.1.1 Mathematical framework . . . 5
2.1.2 Put-Call parity . . . 8
2.2 Pricing-models for European put and call options . . . 9
2.2.1 Black-Scholes model . . . 9
2.2.2 Local volatility models . . . 10
2.2.3 Constant elasticity of variance model (CEV) . . . 12
2.2.4 Stochastic Alpha Beta Rho model (SABR) . . . 12
3 Calibration Methods 15 3.1 Calibrating European options . . . 15
3.2 Calibrating American options . . . 16
3.2.1 Binomial tree model . . . 17
3.2.2 Longstaff-Schwarz Regression Method . . . 17
4 Empirical analysis 21 4.1 Data description . . . 21
4.2 Calibrating SABR . . . 21
4.2.1 Calibrating SABR separately . . . 21
4.2.2 Calibrating SABR simultaneously . . . 23
4.3 Exercise Boundary . . . 25
4.3.1 Exercise Boundary for an American Put when r > 0 . . . 27
4.3.2 Exercise Boundary for an American Call when r > 0 . . . 28
4.3.3 Exercise Boundary for an American Put when r < 0 . . . 28
4.3.4 Exercise Boundary for an American Call when r < 0 . . . 29
5 Conclusion 30
Bibliography 31
Appendices 33
1
Introduction
On the tenth of March 2016, the European Central Bank (ECB) made two major policy changes. It increased its monthly bond purchases from 60 billion Euros to 80 billion Euros, and it cut its deposit rate, charging banks 0.4 percent to hold cash overnight. This pushed interest rates deeper into negative territory. This news was not surprising since there were a couple of banks that preceded the ECB, in this policy change, in 2009, Sweden’s Riksbank led the journey into negative interest rate territory, and since then the ECB, the Danish National Bank, the Swiss National Bank and the Bank of Japan followed its track, causing considerable uncertainty and commotion in the financial sector. Issuing debt with negative interest rates may be part of an attempt to spur on commercial banks to lend more money to private parties instead of holding money and maintaining too large balance sheets, in order to raise the purchasing power of consumers and companies. It could also be a tool to create higher inflation, which is currently the primary goal of the Bank of Japan, but since this side effect is undesirable in Europe, the ECB combined adjusting the interest rate with purchasing more monthly bonds. Also, the fall in the price of oil and the worldwide slowing of growth keeps the fear of inflation to a minimum. Critics of the ECB’s policy say that this is a clear signal that traditional policies do not work and new policies have to be explored. Many European banks (especially the German banks) have protested to the present level of interest rates, since the financial health of banks is highly dependent on the interest rate.
Figure 1.1: Time series of interest rates of several central banks4.
Commercial banks have two options concerning this interest rate policy. They could, for instance, pass on the cost of holding money to their consumers, but this could inflict a bank run because if banks asked more money to hold savings, customers may want to withdraw their money. If uncontrolled, this would lead to the flow of money in the real economy, but it may not stimulate economic growth and purchasing power if the money would not be invested but kept under the mattress.
The second option is that the banks take the costs themselves, but due to the massive amount of money they loan, a small shift in percentages could have a major effect on their profits. Moreover, as has been seen in the market data, increasing negative rates may have a direct impact on bank profits. This may cause bank stock to fall and put downward pressure on all markets. The similarity of the movements of the spot price of Deutsche Bank and the interest rate is striking and very clear when comparing figure 1.1 to figure 1.2.
Figure 1.2: Time series of the spot price of Deutsche Bank in Euros (2001 - 2016)5.
As stated above,the ECB policy has had a great impact on the interest rate and equity prices, so it must have had an impact on interest rate-linked and equity-linked derivatives. Those derivatives have a significant market share: from $72 trillion in 1998 it has risen to $523 trillion in 2015, with a majority of the derivatives being interest rate derivatives. However, these derivatives may have been priced on the assumption that the interest rate is always positive. These fundamental beliefs may cause problems concerning the reliability of option pricing models, when this is no longer the case.
It is thus a natural question if options can be priced in the same way as before the entry into negative interest rate territory. Trading- and early exercise rules have been developed under the assumption of positive interest rates, but how will an option exercise strategy exactly be affected by negative interest rates? This the question that we will investigate in this thesis.
Structure of the thesis
Section 2 gives an overview of option pricing models, and an introduction to basic options is given. The characteristics of these options determine how their price is derived. The second section presents different option pricing models and formulas. To some extent an explanation will be given on how the models were obtained and we will clarify pros and cons about the usage of these models. Section 3 provides a guideline on how to calibrate the SABR model for basic European and American options. After these preliminary sections, section 4 will provide an empirical analysis of the SABR model in a negative interest rate environment. Paragraph 4.1 will give a motivation and explanation of the data that has been used, and paragraph 4.2 presents the results of the calibration of the SABR model to European and American options for different maturities at various strike prices. After calibration, the exercise boundary for American options is visualized in paragraph 4.3 for the call and put options in both negative and positive interest rate environments. The last section, section 5, contains concluding remarks and a critical analysis of the pricing model and strategies. The appendix at the end of this thesis gives Itˆo’s formula.
2
Option Pricing Models
This section will give an introduction on option pricing. The first section discusses different types of options and their characteristics. The second section will introduce several pricing models for options.
2.1
Introduction to basic options
Options are financial derivatives, a special class of financial instruments. They derive their price from the market price of another asset, ‘the underlying’. Options are contracts that give the owner the right, but not the obligation, to buy or sell a financial asset at a predetermined price on a particular date or during a certain period. An option contract involves two parties, the party that has sold the option, the option writer, and the party that has bought the option, the option holder. The seller has a so-called ‘short’ position, the buyer a ‘long’ position. There are two kinds of purchasers of options, speculators and hedgers. Speculators use options to speculate on market movements. This practice tends to be precarious because the speculator has to predict changes at the right time and for the right magnitude, but it can be highly profitable since a contract may contain a ‘leverage’-effect. On the other hand, hedgers use options to protect their positions against sudden market changes. For example, a put option protects the buyer on his long position in the underlying.
Other examples of commonly traded financial derivatives are forwards, futures, and swaps. Forwards and futures oblige the owner to buy or sell an asset at a predetermined time whereas futures have a predetermined period in which the underlying price changes are settled daily. Swaps are contracts that exchange future cash flows.
The call option (commonly referred to as ‘call’), gives the owner the right to buy a financial asset for a specified price. The put option, or just ‘put’, gives the owner the right to sell a financial asset at a specified price, the ‘strike price’. The options in these two classes can have two main styles, European-style options and American-style options. An option that is only exercisable at expiration, the final day that the option is ‘alive’, is called a European option. An American option can be exercised at any time till expiration. A put or call option that has a European-style or American-style is called a ‘vanilla’ or ‘non-exotic’ option. Options that have special exercise dates or have for example non-conventional rules are called ‘exotic’ options.
Most options mature at the third Friday of the month, but there are also options that are written to expire in a day or week or other set times. A common misconception is that a long position in a call is comparable to a short position in a put and vice versa. That would be the case if the contracts were forwards, where the owner is obliged to buy or sell an asset at a predetermined time for a predetermined price, but as stated before, exercising options is a right, not an obligation.
A trader with a long position in a call option would want the price to go up and surpass the strike price, the price at which the contract will be exercised. This is because if the spot price (the current market price of the underlying) is higher than the strike, the holder will make a profit. Conversely, the owner of a put option would want the spot price of the asset to drop, as he will make a profit when he can sell the asset for a higher price (the strike price) than the current market price.
An option price consists of two key elements. The first element is the intrinsic value, which is calculated by taking the difference between the price of the asset on the market, S, and the strike price, K. For a call, the intrinsic value is max(S − K, 0) for a put it is max(K − S, 0). The second element is the time value; it represents the value due to the fact that the intrinsic value at expiration may be larger than today.
2.1.1 Mathematical framework
Because this paper investigates the pricing of European and American options in a negative interest rate environment, we start with the fact that American options cost more than European options. This price difference is caused by the fact that the holder of an American option has the right to exercise the contract early. These extra opportunities can potentially give a higher return. Consequently, the contracts cost more.
C (price American-style call option) ≥ c (price European-style call option) P (price American-style put option) ≥ p (price European-style put option)
Why an American-style call option is never early-exercised when there are no dividends and interest rates are positive
If an agent on a financial market has a long position in an American call, the option would never be early exercised, given that the stock pays no dividend, if interest rates are positive. We give some intuition by approaching the statement in two ways.
The following lemma and proof is based on by Etheridge (2002) and shows why it is never optimal to exercise an American call option on non-dividend-paying stock before expiry. The intuition is to compare the value of the call when exercised, with the value of the call when it is not exercised. Consider the following portfolios;
• Portfolio A: One American call option plus an amount of cash equal to Ke−r(T −t) at time t.
• Portfolio B: One share.
We write St for the price of the underlying at time t. If the call option is exercised at time t < T the
value of portfolio A is St− K + Ke−r(T −t)< St. (obviously, the option will only be exercised if St > K.)
The value of portfolio B at time t is St. On the other hand, at time T , if the option is exercised then the
value of portfolio A is max(ST, K) which is at least that of portfolio B.
We have shown that exercising before maturity gives a portfolio whose value is less than that of portfolio B whereas exercising at maturity gives a portfolio whose value is greater than or equal to that of B. Therefore, it cannot be optimal to exercise early.
The following proof finds its arguments in the assumption of absence of arbitrage opportunities. An arbitrage opportunity is the possibility of a risk-free profit, it can occur when there is a price difference on two different markets between identical financial instruments. Arbitrage opportunities are present in real markets and can be a result of inefficiency of the market. The profit is locked in by purchasing and selling the financial instrument at the same time, causing the market price to balance. When the market price balances, arbitrage opportunities disappear very quickly. It is, therefore, reasonable to assume that there are no arbitrage opportunities.
This proof ultimately compares the same two scenarios as the previous proof: a scenario where the American call is exercised and when it is not exercised. We start by stating the following;
C(S, 0) ≥ S0− K. (2.1)
Here, C(S, 0) is the price of an American option at time 0, S0 is the spot price of the underlying at
time 0 and K is the strike price. This statement must be true since the price of an option is always positive and it cannot be the case that C(S, 0) < S0− K, because there would be an arbitrage opportunity.
A strategy where you would buy the option and exercise immediately would guarantee a risk-less profit (−C(S, 0) + (S0− K) > 0). This cannot be the case when there are no arbitrage opportunities, so equation
(2.1) has to be true. In the same way we can proof the following statement:
S0≥ C(S, 0). (2.2)
This equation is also proved by contradiction. Let S0 < C(S, 0), a strategy where the stock is bought
and the option is sold would guarantee a profit. The profit at time t = 0 would be −S0+ C(S, 0) > 0 and
we could deliver at any time since we hold the option, giving a risk-less profit. This cannot be the case in an arbitrage free environment, so equation (2.2) has to hold.
Hull (2014) proves that a lower bound for a European call is c ≥ S0− Ke−rT, with St, the spot price at
time t, K, the strike price, T , the time of expiration, and r, the risk-free interest rate. It must be that C ≥ c because of the extra exercise opportunities of the American option which means that C ≥ S0− Ke−rT. So
we have;
C ≥ c ≥ S0− Ke−rT when holding the American call option (2.3)
Assuming that the nominal rate of interest6r is positive 7 makes K < Ke−rT for positive T . So holding the American option call option rather then exercising gives a higher profit.
If at some time St > K and you believe the stock will drop below K it is still not optimal to exercise
the call option. The optimal strategy here is inspired by equation (2.1) where it is optimal to sell the option instead of exercising the call, because you believe the option is over-priced. Also, as shown by equation (2.1), you may get a higher profit.
Reasoning for an American put option in a negative interest rate environment
When the same reasoning for the American call in a positive interest rate environment is applied to an American put in a negative interest rate environment, the result is the same. Never exercise prior to maturity. The intuition to never exercise the American call is ultimately confirmed by stating that Ke−rT < K which can also be seen as paying K to get a stock now or pay K to get the stock later which is a difference of Kert− K. This difference is negative in a negative interest rate environment which means that it costs
money to wait till maturity. Therefore, the put is never early exercised. We conclude from these statements that calls are never exercised in a positive interest rate environment and that puts are never exercised in a negative interest rate environment. This is because of the opposite characteristics of the option.
Exercise strategies when the stock pays dividend
The above results only hold for non-dividend-paying stock. However, if the stock pays a continuous dividend8, value of the call at any time becomes;
C(S, t) = Se−qtKe−rt (2.5)
The continuous dividend yield can be seen as a discount for the option price, it devalues the asset price. When interest rates exceed the dividend rate, the same arguments can be used to never exercise the option. but when the dividend yield exceeds the interest rate, you would want to buy the underlying since the stock gives a higher yield than holding cash and earning the risk-free interest rate.
However, the result can be extended to show that if the underlying stock pays discrete dividends9, it can
only be optimal to exercise at the final time T or at one of the dividend times. More generally, the decision whether to exercise early depends on the ‘cost’ regarding lost dividend income.
Only when there are cash dividends, it is sometimes optimal to exercise the American call early, before the ex-dividend date and at no other time. The following proof makes use of a call option on an underlying that pays a dividend on a date between the settlement date t = 0 and the expiration date t = T . The time after the dividend payment can be compared to the situation described above, where there are no dividends paid. Since it is proven that it is not optimal to exercise the call option in this period we focus on the the time between the settlement date and the ex-dividend date. The time between t = 0 and the ex-dividend date can be compared to the case when there no dividends, so the option is not exercised. But at the ex-dividend date (the peak), it can be profitable to exercise the option. To give some intuition, assume the extreme case that the option is in the money at the ex-dividend date and the company chooses to pay all its assets as cash payments. All options that are in the money are exercised, because the option would have no value after the cash payment. When the dividend payment is large enough, the value of the call option can drop making it profitable to exercise prior to maturity.
A call option is only exercised when the value of the exercised call option is smaller than the intrinsic value. The two times that is can be possible to exercise is at the ex-dividend date or at expiration, but at no other time.
exercising at the ex-dividend date earns the holder of the option; Stlow+ Dt− K
6The nominal interest rate is the interest rate without taking inflation into account. The real interest rate is the nominal
interest rate without the effects of inflation.
7The common assumption is based on the fact that if r < 0, a risk-free investment would not provide advantages over holding
the cash. The market participants would then want to keep their money rather than spending it on a risk-free investment.
8Dividends are paid as a percentage of the stock price
The value of the option when not exercised;
C(t) = Stlow+ Ke−r(T −t)+ T.V.(t)
Here, T.V.(t) is the time value at time t, which is part of the premium of an option that is driven by the time between t and the expiration of the contract. The option is exercised if the value of exercising is higher than the value of holding;
Stlow+ Dt− K > Stlow− Ke
−r(T −t)+ T.V.(t)
⇒
Dt> K(1−e−r(T −t)) + T.V.(t) > 0
Figure 2.1: simplified graph of a stock when one dividend is paid
So the option is exercised if and only if the dividend is higher than the loss on the strike price plus the time value of the call. If Dt= 0 the call is of course never exercised before maturity.
Furthermore, the interest rate dynamics are critical when we derive prices of options. As interest rates increase, the expected return required by the investors from the stock tends to increase, and the present value of future cash flows decreases, causing an increase in the stock price. When stock prices increase, the value of the call can increase because the expected payoff increases. As said before, the holder of a call wants the price of the asset to go up. The price of the put will decrease because the demand for puts “decreases”. Following the same reasoning, decreasing interest rates may cause the value of the stock to decrease. As a result, the price of put options rise and the price of call options fall.
2.1.2 Put-Call parity
The Put-Call parity is derived in a few steps that start with the assumption of the absence of arbitrage. Consider a portfolio of a put option and the corresponding underlying. At maturity (t = T ) the payoff value of the portfolio is max(K − ST, 0) + ST = max(K, ST). A portfolio that consists of a call option and
a discounted cash amount K, will have a value of max(ST − K, 0) + K = max(ST, K) at maturity. Since
the two portfolios have the same payoff at T , they must have the same price at any time t before maturity. Otherwise, there would be an arbitrage opportunity. Therefore:
P (S, t) + S = C(S, t) + Ke−rt (2.6) P (S, t) − C(S, t) = Ke−rt− S (2.7) If the stock pays continuous dividend at rate q the equation becomes;
P (S, t) − C(S, t) = Ke−rt− Se−qt (2.8) If the stock pays discrete dividends Di, i = 1, . . . , n at times 0 ≤ t1, . . . , ti, . . . , tn ≤ T , the equation
becomes; P (S, t) − C(S, t) = Ke−rt+ n X i=1 Die−rti− S (2.9)
This parity has to be true for every pricing model at any time, that is why this equation is very helpful to derive risk metrics for option pricing models or to determine unreported interest rates or dividend rates.
2.2
Pricing-models for European put and call options
This section describes the historical development path of some popular models. Along the way, most models are rejected based on empirical findings from scientific papers and by looking at the characteristics of the data that is used for this article. The section ends with a description of the SABR model, the model that we will use for further analysis.
2.2.1 Black-Scholes model
In 1973 Fischer Black and Myron Scholes published an option pricing model which forms the basis of many option pricing models. The main idea is that by accurately modeling the underlying of an option it is theoretically possible to derive option values. It is for example possible to simulate the underlying and determine the price of an option. The Black-Scholes model can be used to price European-style options on equity.
The model is defined as follows;
dSt= µStdt + σStdWt, S0= s
This means that the spot price at time t, St, is modeled using a stochastic differential equation (SDE)
that classifies as a geometric Brownian motion. In this equation, Wt is a standard Brownian motion or
Wiener process, for further information about the Wiener process and the geometric Brownian motion look at Etheridge (2002) and Appendix A. A very important principle in the pricing of options is called the risk-neutral valuation. It means that we assume the investors to be risk-neutral. As said by Hull (2014); “Investors do not increase the expected return they require from an investment to compensate for increased risk.” Although investors in the real market are not risk-neutral, pricing options with this assumption gives the right option price. More about rsik-neutral valuation can be found in Hull (2012), Hull (2014) chapter 12 and Etheridge (2002). The risk-neutral world has three main features;
1. The expected return of the asset is the risk-free interest rate r.
2. Discounting the expected payoff of options is done with the risk-free interest rate r. 3. Assuming absence of arbitrage.
In the Black-Scholes model, in a risk-neutral world, µ equals the risk-free interest rate r. The σ is the volatility parameter, which is taken as a constant.
∂V (S, t) ∂t + 1 2 ∂2V (S, t) ∂S2 σ 2S2− rV (S, t) + r∂V (S, t) ∂S S = 0 (2.10)
Formula (2.10) is the Black-Scholes partial differential equation (PDE), and it can be derived using Itˆo’s stochastic calculus and the assumption that the underlying follows the geometric Brownian motion as described above. Here V (S, t) is the price of a derivative as a function of the spot price of the underlying and time. The function value at the final time V (S, T ) can be seen as a boundary function that fits the characteristics of the option. The PDE shows the relation between the function V (S, t) and its first- and second-order partial derivatives. The solution to this PDE is unique if and only if there is a boundary
condition. For the call this boundary function equals the payoff function max(S − K, 0). Solving the Black-Scholes PDE with boundary condition V (S, T ) = max(S − K, 0) gives the price formula of a call option;
C(St, t) = N (d1)St− N (d2)Ke−r(T −t) (2.11)
With N (x) the standard normal cumulative distribution function, and T the expiration date. Here T − t is the time till expiration, and d1,2 are defined as;
d1= 1 σ√T − t ln St K + (T − t) r + σ 2 2 , (2.12) d2= d1− σ √ T − t. (2.13)
The price of a put option can either be calculated by the put-call parity (section 2.1.2), but it is also possible to calculate it using the boundary condition for the put, V (S, T ) = max(K − S, 0).
Looking at the Black-Scholes model in a negative interest rate environment, it has little issues dealing with negative rates. The risk-free rate is introduced in the model by e−rT, in a negative interest rate environment, this part of the formula will add a premium, instead of a discount, in the value of the option. In the same year, Merton (1973) presented an extension of the model, that also accounts for continuously paid dividends by introducing a dividend rate parameter, q, in the model;
C(St, t) = St e−qt N (d1) − K e−rt N (d2) (2.14) P (St, t) = K e−rtN (−d2) − Ste−qtN (−d1) (2.15) with d1,2; d1= ln KS + (T − t)r − q +σ22 σ√t (2.16) d2= ln S K + (T − t) r − q −σ2 2 σ√t (2.17)
However, the assumption of constant volatility does not match most real market situations. In practice, the volatility is the only parameter that cannot be directly observed from the market, however, it is implied by the pricing function. At the time the option is priced all other parameters are known and given the fact the the price is an increasing function of σ, the implied Black-Scholes volatility, σB can be calculated via an
iterative process. The implied volatility can be described as a measurement of the expected price changes of the underlying. But the main problem with the Black-Scholes model is the fact that the implied volatility differs for different strike prices and maturity times. This variety in volatility values causes us to believe that another model, which allows an implied volatility which depends on the strike price and the maturity date is preferable.
2.2.2 Local volatility models
Local volatility models were mainly developed by Dupire (1994) and Derman et al. (1996). They show that a unique diffusion process, under risk-neutrality, is consistent with observed distributions of the market prices of European options. Derman and Kani explored discrete time and stock price steps, but the real breakthrough came from Bruno Dupire (1994) as he presented continuous-time equations. The previously described Black-Scholes model assumed constant volatility over time. However, there are clear signs that volatility may not be constant. Dumas et al. (1996) proves in an empirical analysis that the implied volatility surface does not match the assumption of a constant volatility.
As an example, the st.dev. of the Deutsche Bank returns is shown as a time series. The volatility graph clearly illustrates the presence of volatility clustering. As described by Mandelbrot (1967);“Large changes tend to be followed by large changes, of either sign and small changes tend to be followed by small changes.” The volatility clustering implies that the observations have autocorrelation10. This autocorrelation may be
caused by mean reversion of the volatility. Also, comparing the volatility distribution curve with the Normal Distribution curve shows that the volatility curve has a higher peak and fatter tails. On top of the first three observation it is clear that by looking at the returns of the underlying and the standard deviation, that volatility clustering is present. These four characteristics, motivate the use of a model with stochastic volatility as is further explained by Cont (2007). Dupire (1994) presented a new class of models which are called ‘Local volatility pricing models’. The use of such models is needed in this paper. This means that there is reason to believe that a stochastic volatility model is preferred over an IID model (independent and identically distributed model).
(a) Daily Returns (b) StDev of the Daily Returns
Figure 2.2: Informational graphs, Deutsche Bank (2001 - 2016).
The major difference between local volatility models and the Black-Scholes model is the fact that a stochastic process models the volatility. Dupire (1994) argues that the volatility should be obtained by calibrating the volatility model to observed market prices of standard traded European options. The option price will be expressed as a function of the forward price;
Ft= Ste RT
t r(s)ds= St/D(t, T ) (2.18)
Ftis the forward value of the spot price Stat time t, with D(t, T ) the discount function. r(s) is the interest
rate term structure which can be a constant or a function of time. The main reason to price option using the forward value of the underlying is to ‘eliminate’ the drift (µ) which are the effects of interest rate changes in a risk-neutral world. This is done by setting the initial value of the forward contract to zero and discounting the forward price between the initial date and the maturity date.
∂C(Ft, K, T ) ∂T = 1 2σ 2 loc(Ft, K, T ) K2 ∂2C(F t, K, T ) ∂K2 (2.19)
Here, C is the price of a call option, K the strike price and T the time to maturity. Here σloc, the local
volatility, is now dependent on strike prices. The calibration of local volatility models is a process where the chosen local volatility function is the starting point. The volatility is a function of the forward price and time, but is held constant over each time step, making it ‘locally’ constant. With the use of that local volatility function, the theoretical market price for an option can be calculated. Then the local volatility function is varied until the theoretical price matches the observed market prices for different strikes K and maturities T . The calculated local volatility is held constant until the consecutive time point where it is recalculated. This results in a local volatility model that handles volatility smiles and could give (at each time point) more accurate hedges than the Black-Scholes model.
The reason that these local volatility models will not be used to model the underlying in this paper is the fact that Hagan et al. (2002) argued that there are severe dynamic faults. The dynamics that are mentioned here are that of the ‘volatility smile’ or also known as ‘volatility skew’. It is the graph from plotting the implied volatility against the strike price which yields a skew or smile. It turns out that the the volatility
skew of the local volatility model behaves in the exact opposite of the observed market movements. As written by Hagan et al. (2002); “When the price of the underlying decreases, local volatility models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile changes to lower prices”. As a consequence, the predicted Greeks11 of those models contain an estimation fault and
could give, over a period of time, worse hedges than proposed hedges of the traditional Black and Scholes (1973) model.
2.2.3 Constant elasticity of variance model (CEV)
The CEV model forms the basis of the SABR model and makes volatility a random function of time. Introduced by Cox (1975), it gave better approximations than the early discussed Black-Scholes model because the CEV model produced an implied volatility that depends on strike prices.
dSt= µStdt + σS β
tdWt (2.20)
The parameters, σ and β are both assumed to be positive and, σ represents the volatility and β controls correlation between the price and volatility. This accounts for changes in volatility when prices change. When β is negative, it can also be interpreted as the leverage effect, meaning that the volatility of the underlying increases as the asset price declines. For equity markets, a shock that causes a fall in stock prices is usually followed by an increase of the volatility. In commodity markets, volatility increases when prices rise, this is called the inverse leverage effect.
When β is minus one, the model equals the Bachelier (1900) model, when β is zero, it is the same as the Black and Scholes (1973) model.
2.2.4 Stochastic Alpha Beta Rho model (SABR)
As argued in section 2.2.2, the market behavior of the volatility skew is the exact opposite of the predicted dynamic behavior in local vol models. That is why Hagan et al. (2002) introduced the Stochastic Alpha Beta Rho model, where the volatility is stochastic and there is a correlation between the price of the underlying and the volatility. Hagan et al. (2002) chooses to solve the fault in local volatility models by the use of a two-factor model. The two-factor model models the volatility and the forward price (see formula 2.18) in two stochastic differential equations which are driven by two correlated Brownian motions.
dFt= αtFtβ dW 1
t, F0= f (2.21a)
dαt= ν αtdWt2, α0= α (2.21b)
dWt1 dWt2 = ρdt. (2.21c)
Here β is restricted between one and zero, ν is the volatility parameter of ˆα making it the volatility of the volatility and where the two processes W1 and W2are correlated with each other through parameter ρ.
The SABR model captures the volatility smile well via the parameter β (introduced in the CEV model) and has an explicit expression for Black’s implied volatility (Black (1976)). A more detailed explanation of this is given in section 3.1. This makes the pricing of European options very straightforward and easy to understand. For the results of this calibration technique see section 4.2.
The explicit expression of the SABR model for the implied volatility by the Black-Scholes model, also
called ‘Hagan’s formula’; σB(K, f ) = A · z x(z) · B (2.22a) A = α (f K)(1−β)/2n1 +(1−β)2 24 log 2 f /K +(1−β)19204log4f /K + · · ·o (2.22b) B = 1 + (1 − β) 2 24 α2 (f K)1−β + 1 4 ρβνα (f K)(1−β)/2 + 2 − 3ρ2 24 ν 2 (T − t) + · · · (2.22c) z = ν α(f K) (1−β)/2log f /K (2.22d) x(z) = log ( p 1 − 2ρz + z2+ z − ρ 1 − ρ ) (2.22e) Here is σB the implied volatility by the Black-Scholes model and f the forward price. α, β, ρ and ν are
the parameters of the SABR model.
Hagan’s formula can be simplified for an at-the-money option (ATM) when K = f ;
σAT MH = σB(f, f ) (2.23a) = α f1−β 1 + (1 − β) 2 24 α2 (f )2−2β + 1 4 ρβνα (f )1−β + 2 − 3ρ2 24 ν 2 (T − t) + · · · (2.23b) Here, σAT M
H is the volatility by Hagan’s formula for an at-the-money option. Like the local vol models,
the forward price is used instead of the spot price, therefore the formula can also be used, without making any adjustments, for an underlying that does not pay dividend as well as for an underlying that does pay dividend. Two main arguments to use this formula is that it gives the prices for European options with very little computational time, and it provides good starting values for calibration techniques that use simulation. Riskmetrics SABR
This section explains how the SABR model reacts to changes in the parameters. An understanding of risk metrics is important when the SABR model is used for hedging market positions. Although our thesis does not discuss the application of SABR to hedge market positions it is interesting to see how the characteristics of the model change when parameters are alternated. That is where the Greeks or risk measures come in. These are derivatives of the option price with respect to various parameters such as the volatility α, leverage effect β, the correlation parameter ρ or the volvol parameter ν. Further information about the risk metrics of the SABR model can be found in the paper by Hagan et al. (2002), and they were modified later by Bartlett (2006).
Figure 2.3 shows how the PDF of the SABR model is affected by parameter changes. It is clear that varying the ρ and the ν have an effect on the skewness12of the function while the α and the β influence the kurtosis13 of the SABR density function.
The impact of the parameters on the implied volatility is interesting as well since they influence the curve quite a bit, as is shown in figure 2.4. The main reason to use different parameter values for these figures is the fact that variations of the parametrization is more clear with this parameter set. Secondly, the same parametrization was used in Vlaming (2011) which provided an ideal way to check if the results were correct. Looking at the influence of the parameters on the SABR implied Black volatility the results are as presented by Hagan et al. (2002). Inserting a high α, which represents the volatility, gives a higher implied volatility thus, controlling the height. α seems to have little effect on the skew of the volatility curve. With larger β the implied volatility curve flattens and gives a higher implied volatility, inserting a low value for β lowers the implied volatility but it makes it volatility curve to ‘smile’ more. ρ seems to have a great effect on the place of the smile, moving to the right as ρ becomes smaller, indicating that it controls the
12The skewness is a measure for the asymmetry of the distribution function.
13The kurtosis is a measure for the probability distribution that gives information about the ‘tailedness’ of the distribution
(a) Varying the Volatility (α) (b) Varying the Leverage Effect (β)
(c) Varying the Correlation (ρ) (d) Varying the VolVol (ν)
Figure 2.3: Plots of the PDF of the SABR model, with the following base scenario (blue line); S(0) = 0.05, α = 0.075, β = .5, ρ = 0.1, ν = 0.1 with a time step T of 1.
skewness. Put and call prices increase as ν increases, giving the volatility a bigger smile. This can be seen by comparing the lines in figure d. The wide variety of roles of the parameters, Hagan et al. (2002) argues that the fitted parameters tend to be very stable. The selection of β is mostly done a priori (Hagan et al. (2002), Vlaming (2011)) and taken usually at β = 1. It determines the distribution of the market noise taking β = 1 mean the model assumes market noise is lognormal distributed, taking β = 0 treats market noise as normal distributed. Proponents of β = 12 suspect the market noise is stochastic CIR14. We choose to fix β a priori at 1 in this thesis. It is important to note that fixing β has an effect on the Greeks of the SABR model, as shown by Bartlett (2006), which is expected since β contains information about the distribution of the market noise, but this is of more importance when hedging strategies are involved which are not covered in this thesis.
(a) Varying the Volatility (α) (b) Varying the Leverage Effect (β)
(c) Varying the Correlation (ρ) (d) Varying the VolVol (ν)
Figure 2.4: Plots of the implied volatility of the SABR model for different parameter values. The base scenario (blue line); S(0) = 100, α = 0.4, β = .9, ρ = 0.3, ν = 0.4 with a time step T of 1.
3
Calibration Methods
Calibration is a process where, in this case, parameters of a model are varied until they match real market data, by minimizing the measurement error. The main reason for calibrating is the philosophy of modeling, if the market data can be replicated with the calibrated model, it is possible to generate (or even predict) data that is not quoted in the markets, like complex, exotic, options.
In the previous section, different formulas and ideas are given to model the underlying of an option. This section will explain how these models can be implemented for pricing European options (section 3.1) and American options (section 3.2).
3.1
Calibrating European options
The fundamental characteristic of European options is the fact that the holder has the right to exercise at maturity, and only at maturity. That significantly limits the number of choices for the owner but makes a European option much easier to price than an American option.
The European option is priced by inserting all parameters in the model. Most of these parameters are easy to determine such as the strike price, the maturity date, the spot price and the time that the option is written. However, dividend rates, risk-free rates, and the volatility σ can not be observed directly.
The dividend rate of the stock and the risk-free rate can be found by solving the put-call parity (section 2.1.2) for different strikes using option price data.
various strikes is obtained by matching the Black-scholes pricing formula for the European put and call options with the observed market value of the option by varying the volatility parameter. Because we know that the pricing formula is strictly increasing for σ, the implied volatility by the Black-Scholes formula can be found. Then, the implied volatility is used to calibrate the SABR model (e.q. finding the parameters of SABR to match the implied volatility as accurately as possible).
The reader may wonder why the SABR model is not calibrated by matching prices directly. A short answer would be that it is harder since the implied volatility by the Black-Scholes model (equation 2.10) is easy to obtain and the SABR model (Hagen’s formula, equation 2.22) has an explicit expression for the volatility of an option given all other parameters. Hagan et al. (2002) used singular perturbation techniques to obtain the price of European options. Next, the implied volatility of that option is obtained. The significant advantage of this method is the fact that under the SABR model, Black’s formula then gives prices of European option. The main reason why option prices can be transformed into the implied volatilities of the SABR model. That provides a relatively straightforward solution to the problem.
Calibrating SABR
Minimizing the sum of squared errors (SSE) is an obvious choice to calibrate the SABR model: (ˆν, ˆρ, ˆα) = arg min
ν,ρ,α
X
i
[σB,i− σH(Ki, fi, ν, ρ, α)]2. (3.1)
Here σB is the market implied volatility obtained by the Black-Scholes formula, and σH is the implied
volatility from Hagan’s formula (equation 2.22).
To speed up the optimisation process, the number of parameters can be reduced by expressing α as a function of ν and ρ, resulting in α = α(ν, ρ). This can be done be using the at-the-money volatility from equation 2.23;
log σAT MH ≈ log α − (1 − β) ln f. (3.2)
Using this approximation, the at-the-money volatility can be rewritten as a polynomial in α, as shown by Frankena (2016); (1 − β)2(T − t) 24f2−2β α3+ ρβν(T − t) 4f1−β α2+ 1 +2 − 3ρ 2 24 ν 2(T − t) α − f1−βσAT MH (3.3) Solving this equation results in many roots, according to West (2005), who wrote a paper about calibrating the SABR model. The best root is the smallest real root.
The SABR model can also be calibrated using simulated paths (Monte Carlo). This is the method that we use in the calibration of the SABR model for American options. For European options it is a good method to test if the simulation schemes are well programmed since we expect the same results from the two calibration methods. After simulating multiple SABR paths the option price if priced calculating the expected discounted payoff. The SABR model is calibrated by comparing the simulated price with the observed market price.
To test whether the programs in Matlab were correct, European options were priced in the same envi-ronment as the European options that were priced in section 2.4 Vlaming (2011), there was an exact match between the findings in her thesis en the results that were found here. This applied to the European options that were priced directly with Hagan’s formula as well as to the options that were priced using the Monte Carlo simulation. Testing with simulations was needed to gain confidence to use the simulator for pricing American options.
3.2
Calibrating American options
Pricing an American option is more complicated than the European option since the holder has more exercise possibilities. This increase will not have much effect on the price of the call option since it is never optimal to exercise the option early, only at times before the date that a cash dividend is issued (see section 2.1.1). So this section will treat the pricing of American put options.
3.2.1 Binomial tree model
When it comes to pricing an American put in the Black-Scholes model, with all parameters known, the most straightforward way is to make a binomial tree. The binomial pricing method is a multi-step process that determines the price of an option at time zero by working backwards. With the use of an example, the algorithm is explained. The first thing that is needed is a binomial branch itself. It is created by determining the probability of the price to go up and down and by certain amounts. These were introduced by Cox et al. (1979); q := P(St+∆t= u · St) = er∆t− d u − d = 1 − P(St+∆t= d · St) u = eσ √ t d = e−σ √ t= 1 u
Here σ is the volatility with r the risk-free interest rate and q the probability of the spot price to go up. Picture (3.1) shows a two-step binomial model. Next, the option in the terminal nodes is priced. Which is very straightforward since it is just max(ST− K, 0) for a call and max(K − ST, 0) for a put. The last step is
to price the option in each node by backward induction. This is done by comparing the immediate exercise value with the sum of the discounted future values when holding the option. We exercise immediately when the value of holding is lower. The value of keeping the option at the final nodes is calculated by determining the intrinsic value:
VN,i= max(SN− K, 0), for an American call
VN,i= max(K − SN, 0), for an American put
The expected value of holding the option in earlier nodes is:
Vt−∆t,i= e−r∆t(q · Vt,i+1+ (1 − q) · Vt,i−1)
The option is immediately exercised when the value of exercising exceeds than the value of holding.
S0 S1 1 = d ∗ S0 S11 2 = d2· S0 1 − q S10 2 = d · u · S0 q 1 − q S10 = u ∗ S0 S01 2 = u · d · S0 1 − q S00 2 = u2· S0 q q
Figure 3.1: Example of a binomial tree with two steps
3.2.2 Longstaff-Schwarz Regression Method
The next problem that has to be tackled is the one concerning the calibration of American options in the SABR model. Knowing how to find the price of American options and knowing the parameters of the distribution of the underlying it is possible to simulate lots of future paths (using Monte Carlo simulation)
to determine the prices for the option for each node in the tree and take the expected (average) value to determine the option price. This value is compared with the observed market value whereafter the parameters are adjusted until the model fits the observed values as accurate as possible. Although time-consuming, it is feasible to use a different but similar tree model for this purpose. However, for getting a quick(er) result, Longstaff and Schwarz introduced the Longstaff-Schwarz regression method. It offers a practically implementable solution for the calibration/pricing problem. With the help of Wetterau and Kienitzg (2010) a Matlab code for this method has been written and presented in Appendix D. What follows is an explanation of the method and how it was applied to the SABR model.
The problem with American options are the early exercise opportunities. The price of the option is not only dependent on payoffs at maturity but also the potential payoff before maturity. For each time point, it is necessary to evaluate the payoff of immediate exercise and the expected payoff of holding the option for at least one time interval. The holder only exercises when the payoff of immediate exercise is higher than the payoff of keeping the option.
The payoff when exercised at time t ( 0 ≤ t ≤ T ) is easy to determine for it is max(St−K, 0) for a call and
max(K − St, 0) for a put option. However, the conditional expected future discounted payoff (when holding
the option) is much more challenging to calculate. The key insight that Longstaff and Schwartz (2001) give is that it is not necessary to simulate different paths from each time point that is investigated, but that it is possible to estimate the conditional expected payoff by regressing cross-sectional information with the use of the least squares estimator. So rather than averaging in the end, it uses a regression method to combine different simulated paths to predict expected payoffs. Since these paths are Monte Carlo simulated and because the least squares approach is taken, the technique is called the least squares Monte Carlo (LSM) method. In addition to the fact that this algorithm is easy to implement it can also be applied to models with more than one stochastic factor (such as the SABR model). The relatively easy implementation, the speed and the fact that it takes stochastic factors into account make that the LSM technique is used in this analysis.
There are m paths that simulate the forward of the undelying F for multiple time points ti, i from 0 to N ,
where in this case the paths follow the SABR process and where ti= i · ∆t, ∆t = T /N , t0= 0 and tN = T .
This can be seen as a m × N matrix, which will be called P .
So at any given time point a choice is made to either continue holding the option or exercising at that point. Following the notation of Wetterau and Kienitzg (2010) where Vi is the value of the option, hi the
payoff value and ci the continuation value at time ti. The option value at time ti is:
Vi =
(
ci, if hi< ci
hi, if hi≥ ci
The continuation value ci is computed with the use of regression of simulated paths that are still
in-the-money on a set of basis functions fj, j = 0, . . . , G. This results in the following approximate equation:
ci= e− Rti+1 ti r(s)ds· E[Vi+1|Fi] ≈ G X j=0 ajfj(Fi) (3.4)
To find the coefficients aj, least squares is applied to the following equation:
ci≈ G
X
j=0
ajfj(Fi) (3.5)
The payoff value hi is calculated by transforming the simulated forward Ft to the spot price St by
multiplying the forward with the discount function D(t, T ) which is defined as e−R0Tr(s)ds. The payoff of an
American at time t is:
hi= max(Ft· D(t, T ) − K, 0) (3.6)
This is done for all time points via backward induction to eventually determine the price of the option at time t = 0.
Application to SABR
To apply the LSM technique to the SABR model two very significant adjustments have to be made. First, the underlying path simulator has to be based on the SABR model. Second, since the SABR model is a two-factor model, the basis functions (which are used in the regression) have to implement two factors instead of one.
The LSM technique needs multiple simulated paths. The path of the spot price is based on the funda-mental stochastic differential equations presented in section 2.2.4 which can be transformed into formulas that approximate the process using discrete time steps as shown by Hu (2013);
Sk+1= Sk+ rSk∆t + αker(β−1)TS β k∆W 1, S 0= S (3.7) αk+1= αk+ ναk∆W2, α0= α (3.8)
∆W1and ∆W2are simulated with use of the Cholesky decomposition. Because W1and W2are Brownian motions their increments are Normally distributed with variance ∆t and can be represented by a standard normal distributed variable as follows;
∆W1= 1·
√
∆t (3.9)
Where 1and 2are independent of each other and have N (0, 1) distributions. To form ∆W2the Cholesky
decomposition is applied with the correlation parameter ρ: ∆W2= (ρ · 1+ p 1 − ρ2· 2) · √ ∆t (3.10)
However, the used simulation/discretization scheme for the paper is the one presented by Chen et al. (2011), where an unbiased scheme for SABR simulation is introduced. The two most important arguments to use this scheme over other schemes like Euler or Milstein (1975) is the fact that it is unbiased and robust to interest rate valuation, which is of importance since we experiment with different interest rate term structures that can become negative.
It is important to note that with the use of a two-factor model, twice as many values have to be sampled to simulate the payoff with the same accuracy as a one-factor model like Black-Scholes. An example of a SABR simulation for the spot price is given in figure 3.2.
Secondly, the basis functions on which the future payoffs are regressed will have to be adjusted. Since the SABR model is a two-factor model, the basis functions also have to be constructed using two variables. Commonly used basis functions for the Longstaff Schwarz technique are Laguerre polynomials, Legendre polynomials or basic polynomials. The latter will be used in this paper in a bivariate form of the second degree. The formula for the general bivariate polynomial and the structure of the second-degree polynomial is given by:
ci= a1+ a2αi+ a3Si+ a4αi2+ a5Si2+ a6αiSi (3.11)
To see if the algorithm worked, American options were priced in the same environment as in section 3.3 of Vlaming (2011) an the prices were compared. This gave an exact match, meaning that the program worked well and was ready to use for empirical analysis.
4
Empirical analysis
This section describes the process of the calibration of the SABR model on real market data. The first section will give a description of the data that has been used to calibrate the SABR model. The second section will show the results of the calibrations. The final section will analyze the exercise boundary for American options for different interest rate term structures using the final calibrated version of the SABR model.
4.1
Data description
The data is acquired from Datastream (Thompson Reuters). An important property of the data of these basic American and European options is the fact that they have the same underlying, namely Deutsche Bank. This makes the comparison between the calibrated models a lot better and more straightforward than when the options had different assets as underlying. To test interest rate fluctuations around zero it would have been better to use interest rate derivatives, but since such data was not readily available, the decision fell on equity options on Deutsche Bank. The main reason to choose Deutsche Bank was the fact that the spot price of the bank is highly correlated with the interest rate (see figure 1.1 and figure 1.2). So any shifts in the interest rate would influence the spot price, although not as directly as with interest rate derivatives such as swaptions. Another reason was the fact that Deutsche Bank did not issue any dividends lately, so these did not have to be modeled. This meant that the research could focus entirely on interest rate shocks. Also, the volume of these contracts is very large, and another reason to concentrate on this area. BIS (Bank for International Settlements) investigated the total volumes of interest rate derivatives and derivatives on equity; they found that the entire national outstanding on the global foreign exchange rose from 72 trillion dollars in 1998 to 522.9 trillion dollars in 201515. The ratio of IRS and equity-linked derivatives at that time is roughly four to one.
In this thesis we investigate option prices at one date for different maturities and a range of strike prices. For investigative purposes a date after the ECB policy, namely January the fifth 2017), is chosen. Deutsche Bank closed with a spot price of 18.26 euro that day. The maturity dates are the third Fridays of January (15 days), February (43 days), March (71 days) and June (162 days). The risk-free interest rate is not given in the data but is determined by solving the put-call parity, equation (2.8). Deutsche does not pay dividends due to the crisis and due to the settlements16because of the miss-selling of mortgage-backed securities. So,
in this case, the dividend rate, does not need to be modeled.
4.2
Calibrating SABR
The main target of calibrating the SABR model is to match the option prices implied by the model with the observed market prices. First, the SABR model is fitted separately for each maturity and each option style (European put/call and American put/call). The second part shows that the SABR model can be fitted for all the options over different maturities and strikes using the same parameter values for all series.
4.2.1 Calibrating SABR separately
In the case of a European option, the SABR model can be calibrated in two ways, directly and via Monte Carlo simulation. First, the direct approach is evaluated.
Since the SABR model has a direct formula for the Black-Scholes implied volatility it is possible to match Black-Scholes implied volatility with the SABR version to calibrate the parameters. The key idea is to transform the market prices into the implied volatility and match the volatility directly after which the model can produce the prices.
For options that expire shortly after the option is written the fit is less good than the fit for options with longer maturities. This is clearly seen in figure 4.1 and 4.2 for put and call options on Deutsche Bank. The graphs for Maturities with 15 and 162 days were the same as the ones presented, because they contained no extra information the figures are left out of the thesis.
15Source: http://stats.bis.org/statx/toc/DER.html
Figure 4.1: Calibrating the SABR model for European options directly using the implied Black-Scholes volatility transformation (Maturity, 43 days). Red dots are observed market prices, the green and the blue lines are the separate and
the simultaneous calibrated SABR.
An alternative way to calibrate the SABR model is to simulate paths of the underlying, calculate the average payoff of the option and match this to the observed market price. This is done by implementing the simulation/discretization scheme introduced by Chen et al. (2011).
The fit is nearly perfect implying that the model can represent the data very accurately.
Calibrating the SABR model with the use of market prices of American options is also done by simulation, only this time the value is calculated with the Longstaff and Schwartz (2001) algorithm which is described in section 3.2. The SABR model fits the curves of the market prices of the American options also very well, it does this so well it is hard to distinguish between the blue line that represents the calibrated SABR curve and the red line (representing market prices). The SABR model gave such a good fit for the prices of the options that it is not necessary to show the price curves for all maturities.
Since the SABR model fits the market data well it is possible to compare the calibrated parameters of different options for various maturities to see if there are any similarities. Table 4.1 gives a clear image of the parameters found by the calibration programs that calibrate SABR directly and via simulation (20 timesteps, 20000 simulations).
We note that the SABR calibration may lead to multiple minima, i.e. multiple optimal parameter combinations. The starting values for the optimization of these parameters are inspired by the values found by Vlaming (2011). Beta has been fixed since β would be used to fit the market noise as is stated by Hagan et al. (2002) and shown by Vlaming (2011). The calibrated β would, therefore, stay equal to its starting value, and a fixed β would not change the quality of the fit. That is why β is fixed at 1.
Figure 4.2: Calibrating the SABR model for European options directly using the implied Black-Scholes volatility transformation (Maturity, 71 days). Red dots are observed market prices, the green and the blue lines are the separate and
the simultaneous calibrated SABR.
have an interesting dependence on the length of the maturity. For European options as well as for American options they seem to decrease when maturity increases. However, it looks like the rate of decline for American options is bigger. This can also be the effect of an unobserved and unaccounted interest rate effect, but due to the small differences in the maturity, this conclusion is discarded. The decline of ν is unexpected as we expected the volatility to rise when maturities become larger. The relation between ρ and ν is interestings as ν becomes bigger as ρ approaches zero. It is also good to see that the direct approach and the simulation approach produce the same outcome which suggests that the models are correctly programmed.
What can be concluded from these results? Since the SABR model tries to capture the movements of the underlying (the asset price) and produces the same parameters for different options, the underlying of the options should have the same characteristics which is true because they are all options on the same underlying, namely Deutsche Bank. It can be therefore argued that the SABR model should be calibrated on all options for all maturities and strikes simultaneously.
4.2.2 Calibrating SABR simultaneously
The findings in the previous section are quite promising as it supports the argument for an overall calibrated SABR model. This paragraph presents that general calibration. In the last part the SABR model is calibrated on one kind of option for every individual maturity, but now the SABR model has to be calibrated on 24 options at once which increases the time to find an ‘optimal’ calibration. So the general calibration is
(a) European options (Maturity, 15 days).
(b) American options (Maturity, 15 days).
Figure 4.3: Calibrating the SABR model for European and American options by simulating paths for the underlying. Red dots are observed market prices, the green and the blue lines are the separate and the simultaneous calibrated SABR.
obtained by taking steps, closing in on the real value of the parameters. First a SABR calibration is done for all the European options and maturities with 15 timesteps, secondly a calibration for all American options and maturities with 15 timesteps and finally a universal calibration. The starting values for the simultaneous calibration of European options are chosen by taking the average of all the parameters found for European options. This is done in the same way for American options. The starting values for the final calibration were chosen by averaging the parameters of the simultaneous calibration for American and European options. The final calibration is then calibrated by simulating 20000 paths with 20 timesteps.
The table (4.2) shows that the hypothesis, fueled by the previous paragraph, seems to be true. There is a general calibration possible, and the calibrated parameter values are very close tot the values we expected. The two reasons to execute this calibration firstly is to show that the proposed hypothesis is likely to be true and secondly to use the calibrated parameter values for the next paragraph (4.3) were exercise boundaries will be modeled. The calibration of parameter ν is still not stable. This may be caused by the fact that ν has little effect on the price, as seen in the risk metrics section, which makes it hard do determine parameter value. ν may be better modeled when the SABR model tries to match volatilities instead of prices. The total simultaneous calibration is also shown in the previous graphs with a green line. The simultaneous calibration is close to the average of all parameters giving a much better fit for options with maturities of 43 and 71 days, than for options with 15 and 162 days.
Maturity α β ρ ν European Call 15 days Direct 0.3704 1 -0.241 3.7113
Call 15 days MC 0.3994 1 -0.2536 3.7549 Put 15 days Direct 0.3538 1 -0.1341 4.2849 Put 15 days MC 0.3998 1 -0.3457 2.4484 Call 43 days Direct 0.4173 1 -0.3192 1.5136 Call 43 days MC 0.4107 1 -0.3202 1.8367 Put 43 days Direct 0.4118 1 -0.2659 1.8166 Put 43 days MC 0.4104 1 -0.3157 2.0141 Call 71 days Direct 0.4094 1 -0.4154 1.2695 Call 71 days MC 0.4052 1 -0.3596 1.3579 Put 71 days Direct 0.4023 1 -0.3523 1.5231 Put 71 days MC 0.4056 1 -0.3367 1.5351 Call 162 days Direct 0.4057 1 -0.4396 0.9774 Call 162 days MC 0.4114 1 -0.4376 1.0398 Put 162 days Direct 0.4065 1 -0.3857 1.0853 Put 162 days MC 0.4116 1 -0.4229 1.1275
Maturity α β ρ ν
American Call 15 days LS 0.3905 1 -0.3392 3.7763 Put 15 days LS 0.3829 1 -0.2871 2.7515 Call 43 days LS 0.4217 1 -0.356 1.9388 Put 43 days LS 0.4108 1 -0.4002 1.8197 Call 71 days LS 0.4163 1 -0.4144 1.5258 Put 71 days LS 0.4019 1 -0.4477 1.5618 Call 162 days LS 0.4075 1 -0.5496 1.118 Put 162 days LS 0.4062 1 -0.5389 1.2746
Table 4.1: Parameter values for the calibrated SABR model.
Starting Values First Calibration Starting Values Second Calibration Starting Values Third Calibration
α 0.404683 0.4155 0.410138 0.4117 0.4136 0.4048
ρ -0.33408 -0.377 -0.34375 -0.4851 -0.43105 -0.4701
ν 1.956006 1.832 1.648038 1.6236 1.7278 1.2736
Table 4.2: Results Simultaneous Calibration.
4.3
Exercise Boundary
This section will show the results that focus on answering the research question: “How are option exercise strategies affected by negative interest rates?”. Exercise strategies for European options are very straight-forward and limited since the holder only has to compare the value of exercising at the maturity date with zero and choose which one is higher. In the case of American options, the problem is less trivial. That is why this thesis focuses on strategies for American-style options.
As explained in section 3.2, the exercise strategies for American options are calculated using the Longstaff and Schwartz (2001) algorithm. These exercise strategies can be framed in terms of a boundary: the American call option is exercised at the first possible exercise opportunity when the spot price is higher than the boundary. American put options are exercised if the spotprice is lower than the exercise boundary.
The holder of the American option, who can exercise the option at any time before maturity, has to compare the value of immediate exercise with the value of holding. This introduces a boundary which specifies the spot price, where the holder is indifferent between exercising and holding. This boundary changes over time and is in the the Black-Scholes model only depending on the spot price. However, this is not the case when the SABR model is implemented. The SABR model, introduced in section 2.2.4, is a two-factor model, that means that the simulator (that tries to model the actual movement of the asset) generates a path for the spot price of the asset but also a path for the volatility. So two variables instead
(a) Calibrated Rho (b) Calibrated Nu Figure 4.4: Averaged values of the parameters for different maturities
Figure 4.5: An example of the one-dimensional exercise boundary for a put option in a two-dimensional setting (added time component). The exercise boundary for a call option is in fact the mirror image of the exercise boundary for the put (on the line of the strike price), making it a downward sloping line.
of one are implemented in the basis functions of the Longstaff and Schwartz (2001) algorithm. Since these basis functions form the basis of the regression the choice of exercising depends on these two factors, namely the spot price of the underlying at any time and the corresponding volatility of that option. This makes the exercise boundary at any time point a line which will become a 2-dimensional surface when a time component is added.
The following paragraphs show exercises regions for the put and call options in negative and positive interest rate environments. Each scenario will be visualized by three graphs; at time t = T4, at half time t = T2 and at t = 3T4 . It is trivial what the exercise strategies are at t = T and because of its independence of the volatility of the underlying that graph is not shown.
Each graph is a cross-section at one time-point with the spot price for the simulations on the x-axis and the corresponding volatility on the y-axis. When a spot price/volatility combination is exercised, the point is marked blue, when the algorithm chooses to continue holding the option, the point is marked red. The exercise boundary is the border between the red dotted field and the blue dotted field. In the first two subsections, the exercise boundaries of the American put and call are discussed with an interest rate that is positive at 0.05 percent. The last two subsections look at the exercise regions of the American put and call in a negative interest rate environment. The interest rate is −0.05 percent there. The base scenario for
