Performance of robust model-free hedging via Skorokhod embeddings of digital double barrier options

Transcription

1 Performance of robust model-free hedging via Skorokhod embeddings of digital double barrier options University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance April 3, 2

2 Acknowledgements I am heartily thankful to my supervisor, Dr. Jan Ob lój, for his guidance, support, and encouragement during this project. I would also like to express my deepest gratitude to my wife, Clare, and my daughter, Eva, for their patience, understanding, and invaluable support throughout.

3 Abstract We analyse the performance of robust model-free hedging via Skorokhod embeddings of digital double barrier options against that of traditional hedging methods such as delta and delta/vega hedging. Digital double barrier options are financial derivative contracts which are most commonly traded in the foreign exchange markets and which pay out a fixed amount on the condition that the underlying asset remains within or breaks into a range defined by two distinct barrier levels. We performed the analysis in hypothetical markets, where we examined the influence of a set of market parameters and assumptions, and in two real-life situations, where we considered digital double barrier options written on the EUR/USD and the AUD/USD spot exchange rates.our findings suggest that, although robust model-free pricing is in general not competitive with respect to traditional pricing methods, robust model-free hedging can substantially outperform traditional methods when applied to forward markets. In spot markets, we find that the strong performance of model-free hedging relative to traditional hedging methods becomes conditional on the level of risk-neutral drift, the maturity of the option, and the volatility of the underlying.

4 Contents Introduction 2 SEP method 4 2. The Skorokhod embedding problem Application to financial markets SEP optimal strategies Application to spot markets Models used 3 3. Black-Scholes Heston Bates Variance Gamma with CIR stochastic clock Calibration Data Minimization problem Weighting scheme Two-step calibration Calibration results Numerical methods Finite difference method Coordinate transformation Finite difference method with concentration Single barrier options Digital down-and-in single barrier option Digital up-and-in single barrier option Digital down-and-out single barrier option Implementation COS method Simulation of sample paths Heston Bates VGSV Hedging comparisons Dynamic hedging with transaction costs Set-up Hypothetical markets Real markets Model chosen: VGSV Underlying assets i

5 Hedging strategies Barriers Simulation of the underlying assets Transaction costs Computed statistics General sample statistics Value-at-Risk Conditional Value-at-Risk Expected utility Risk-indifference price Sharpe ratio Results Hypothetical markets Forward markets Spot markets Real markets Conclusions 67 A FX option market conventions 68 B Robust hedging strategies and pricing 69 B. Double touch barrier option B.. Superhedging B..2 Subhedging B..3 Pricing B.2 Double touch/no-touch barrier option B.2. Superhedging B.2.2 Subhedging B.2.3 Pricing B.3 Double no-touch barrier option B.3. Superhedging B.3.2 Subhedging B.3.3 Pricing C Description of the COS method 77 C. Theory C.2 Pricing European Options with the COS method C.3 Truncation range D Additional tables and figures 82 E Market Data 96 ii

6 List of Figures 2. EURUSD SEP optimal strategies AUDUSD SEP optimal strategies Sample paths for EURUSD and AUDUSD; BS Sample paths for EURUSD and AUDUSD; Heston Sample paths for EURUSD and AUDUSD; Bates Sample paths for EURUSD and AUDUSD; VGSV Calibration weighting scheme EURUSD and AUDUSD calibrated volatility surfaces Coordinate transformation example Prices and Greeks on the finite difference grid; DT Prices and Greeks on the finite difference grid; DTNT Prices and Greeks on the finite difference grid; DNT Estimate errors for finite difference price and Greeks Convergence of MC price estimates towards FD Pdf of EURUSD and AUDUSD Prices and Vegas on the finite difference grid; DT Influence of volatility on the SEP method in spot markets D. Distribution of hedging errors; DT; BS D.2 Distribution of hedging errors; DT; BS HV D.3 Distribution of hedging errors; DT; BS LV D.4 Distribution of hedging errors; DT; VGSV iii

7 List of Tables 2. Sample of SEP optimal strategies Forward to spot market trading for SEP method Classification of model parameters for calibration Calibrated model parameters for EURUSD and AUDUSD Prices and Greeks for double barrier option; MC vs FD Call option prices by COS method, FFT, and MC Model combinations used in the hypothetical markets Summary of SEP performance; Normal transaction costs Summary of SEP performance; High transaction costs D. Description of column headings in result tables D.2 Colour codes used in result tables D.3 Hyp. Mkt.; Short Dig. Double Touch; True: BS; Mkt: BS; r d = % D.4 Hyp. Mkt.; Short Dig. Double Touch; True: BS HV; Mkt: BS; r d = % D.5 Hyp. Mkt.; Short Dig. Double Touch; True: BS LV; Mkt: BS; r d = % D.6 Hyp. Mkt.; Short Dig. Double Touch; True: VGSV; Mkt: VGSV; r d = % D.7 Hyp. Mkt.; Long Dig. Double Touch; True: VGSV; Mkt: VGSV; r d = 5% D.8 Short EURUSD Digital Double Touch; High transaction costs D.9 Long EURUSD Digital Double Touch No-Touch; Normal transaction costs D. Long AUDUSD Digital Double Touch No-Touch; Normal transaction costs D. Short AUDUSD Digital Double No-Touch; Normal transaction costs E. Deposit rates E.2 Implied volatility market quotes iv

8 Chapter Introduction The standard approach to pricing and hedging of double barrier options, and of many other financial products, is to postulate a model for the behaviour of the underlying asset and then use this model to derive prices and associated hedging strategies. The prices and hedges thus found will be correct only if the model precisely describes the real world. In practice, this is quite unlikely. On the other hand, the model-free methodology examined in this report, henceforth referred to as the SEP method, uses market data to deduce bounds on the option prices. Those prices are consistent with the no-arbitrage assumption and the associated super- and sub-replicating strategies. In contrast with the standard approach to pricing and hedging, the prices and replicating strategies derived by application of the SEP method are completetely independent of the model chosen for the underlying asset. In other words, the SEP method is robust to model misspecification. The hedging strategies thus derived are certain to work whatever the behaviour of the underlying asset. The SEP method is based on solving the Skorokhod embedding problem (SEP). We provide an introduction to the Skorokhod embedding problem and the SEP method in chapter 2, based on the description made in Obloj (29). The no-arbitrage price bounds derived by the SEP method are generally too wide to be of any use by traders for practical pricing because they are likely to fall outside the bid/ask spreads typically observed in most markets. However, the super- and sub-hedging strategies of the SEP method can still be useful to practitioners. We shall see in this report that the distribution of hedging errors resulting from their application can prove very desirable in some cases, such as when model uncertainty and market frictions are high. Such deviations from the ideal conditions required by traditional hedging methods can make hedging the exotic option using the SEP method substantially more efficient on a risk-adjusted basis than traditional delta or delta/vega hedging by shielding the trader from risks inherent in traditional hedging methods such as those related to: Discrete time rebalacing: in practice, the presence of transaction costs precludes continuous hedging as it implies infinite costs. As a result, hedging must be performed on a discrete time basis, which leads to systematic hedging errors. Transaction costs: when performing discrete time rebalancing, the trader must choose the frequency of such rebalancing and faces a trade-off between hedging errors and transaction costs. In the foreign exchange markets, the transaction costs incurred when delta hedging are very small and this trade-off might not be a significant problem. However, it is an important issue with respect to vega hedging, as the transaction costs incurred when trading vanilla options can be quite large, especially in highly volatile markets, where vega hedging is all the more important. Model risk: when hedging with traditional methods, the trader must choose a model for the evolution of the risk factors underlying the pricing of the double barrier option. Any differences between the chosen model and the true model followed by the underlying risk factors will result in systematic hedging errors. Those differences may arise because the trader s chosen model

9 ignores important features of the true underlying factors, such as jumps for example, or because the trader s estimates of the model input parameters deviate from the true parameters. In chapter 2, we show how the SEP method is not, or little, exposed to the above risks when appropriatly applied. In what follows, we examine the performance of the SEP method when applied to the hedging of digital double barrier options with continuously monitored barriers. Given barriers b < S < b, where S is the asset price at time t =, a digital double barrier option is a financial derivative paying if the underlying asset price has crossed or not crossed the barriers over the life of the option and otherwise. For example, a digital double touch option pays if, over the life of the option, the underlying has crossed both barriers, and pays if one of the barriers has not been crossed. Its payoff can be mathematically expressed as {ST b,s T b}, where T is the maturity of the option, S t = sup u t S u, and S t = inf u t S u. There are in total eight different types of digital double barrier options and they come in pairs. For example, {ST b,s T b} = {S T b or S T b}. The study of digital double barrier options can thus be reduced to that of four such options. Furthermore, by symmetry, it is enough to examine only one of the two digital double touch/no-touch options with payoffs {ST b,s T b} and {S T b,s T b}. In consequence, we consider in this report only three types of digital double barrier options: The digital double touch option with payoff {ST b,s T b}. The digital double touch/no-touch option with payoff {ST b,s T b}. The digital double no-touch option with payoff {ST b,s T b}. We compare the performance of SEP hedging of digital double barrier options to that of traditional hedging methods such as delta and delta/vega hedging. To this effect, we perform Monte Carlo simulations of the evolution of the underlying asset price under the risk-neutral probability measure in a variety of market scenarios and determine the distributions of hedging errors achieved by the hedging strategies under study. We perform those simulations in hypothetical markets and in two markets we believe close to real-life. In the hypothetical markets, we simulate the performance of the hedging methods when applied to a digital double touch option in forward and spot markets, with and without transaction costs. We consider four different models for the evolution of the underlying asset price under the risk-neutral measure: Black-Scholes, Heston, Bates, and Variance Gamma with CIR stochastic clock. Those models are described in chapter 3 and were calibrated to real market data from the EUR/USD and AUD/USD option markets. The calibration methodology we used is presented in chapter 4, along with the calibration results. The market-observed vanilla options used in our simulations by the trader to calibrate the traditional pricing and hedging methods, to vega hedge, and to compute the SEP strategies, are priced under one of the aforementioned models, only not necessarily the same one as that used to generate the risk-neutral dynamics of the underlying asset price. This allows us to examine the sensitivity of the hedging methods to model misspecification. In the two markets believed to be close approximations to real-life, we consider digital double touch options, digital double touch/no-touch options, and digital double no-touch options with a variety of different barrier levels and referencing respectively the EUR/USD and AUD/USD spot exchange rates in the foreign exchange interbank markets. In those two markets, we assumed that the model driving the risk-neutral dynamics of the underlying asset price and used by the market to price vanilla options is the Variance Gamma with CIR stochastic clock model calibrated in chapter 4. In chapter 5, we describe the numerical methods that were used to simulate the evolution of the underlying asset price and to compute the prices and hedge parameters of digital double barrier options and European vanilla options under the relevant models. The reader familiar with such numerical methods may skip this chapter without loss of continuity and proceed directly to chapter 6 where we define the numerous market scenarios in which we performed the hedging simulations and analyse the results. Those results suggest that in forward markets, or equivalently in spot markets with zero cost of carry, the SEP method substantially outperforms traditional hedging methods on a risk-adjusted 2

10 basis, especially in the presence of transaction costs and stochastic volatility. They also highlight some of the limitations of the SEP method when applied to spot markets with non-zero cost of carry, which stem from the hybrid way in which we apply the SEP method in such markets. Despite those limitations, we find that strong benefits can still be derived from the SEP method in spot markets with non-zero cost of carry when applied to short-dated digital double barrier options referencing highly volatile assets. 3

11 Chapter 2 SEP method 2. The Skorokhod embedding problem Problem 2. (Skorokhod embedding problem (SEP)). Given a stochastic process {X t : t } and a probability measure µ, find a minimal stopping time τ such that X τ has the law µ: X τ µ. By minimal stopping time τ we mean that if a stopping time ρ satisfies ρ τ and X ρ X τ then ρ = τ. When E[B τ ] =, minimality of τ is equivalent to {B t τ : t } being a uniformly integrable martingale (see Cox and Hobson (26)) and consequently when E[Bτ] 2 < it is further equivalent to E[τ] <. Note that there can be infinitely many minimal stopping times that embed the same distribution µ. Note also that the SEP problem does not necessarily have a solution and the existence of a solution depends on the choice of X and µ. However, for any continuous local martingale {X t : t } such that X = a.s., there is always a solution to the SEP. Let us now consider the SEP 2. for X t = B t, where {B t : t } is a Brownian motion. Denoting ΦandF µ thecumulativedistributionfunctionsof, respectively, thestandardnormaldistributionand µ, then Fµ (Φ(B )) µ and a solution to the SEP seems to be τ = inf{t 2 : B t = Fµ (Φ(B ))}. However, we note that E[τ] = and this solution is thus unlikely to be minimal. Skorokhod (965) imposes E[τ] < and solves the problem explicitly for any centered target measure with finite variance. A random walk is therein represented as a Brownian motion stopped at an increasing sequence of stopping times and properties of the random walk are deduced from the behaviour of Brownian motion. Since Skorokhod (965), several other explicit constructions have been found (see Obloj (24)). 2.2 Application to financial markets Denote O(S) T the payoff of a path-dependent exotic option with maturity T. We assume we can observe the market prices of European call options for all strikes K > and maturity T. This assumption is only needed to simplify the exposition of the method and can be relaxed to a finite number of observed European options as we will see later. We define as market input the set of observed European call options: {C(K) : K }, (2.) where C(K) denotes the price of an European option with strike K and maturity T. We denote X the set of all calls, forward transactions, and cash balances in money market accounts and Lin(X) the space of their finite linear combinations, i.e. the set of portfolios consisting of calls, forward transactions, and cash balances. We assume that we can enter forward transactions at no cost. We now introduce a pricing operator P such that if X T is a F T -measurable random variable, such as the payoff of an option maturing at T, then its price X at time t = is given by X = P(X T ). For example, P((S T K) + ) = C(K). We assume that P is linear and thus the put-call parity relation holds, i.e.: P((K S T ) + ) = K S +C(K). (2.2) 4

12 Definition 2.. We say that a pricing operator P admits no model-free arbitrage on X if X Lin(X) : X P(X). (2.3) We now assume no arbitrage as per definition 2. and define a market model as a filtered probability space (Ω,F, (F t ),P) with a continuous P-martingale S := {S t : t } which matches the market input (2.). We consider the model under the risk-neutral measure and thus P = E. Saying that S matches the market input is then equivalent to saying that E[(S T K) + ] = C(K),K >. In practice, S could be a forward or the spot price of an asset with zero cost of carry (for example: an underlying asset in a market with zero interest rates or a currency pair from economies with equal interest rates). We will see later in section 2.4 how the SEP method can be applied, albeit in a hybrid manner, to a spot price with non-zero cost of carry. A theorem by Monroe (978) establishes that any semimartingale is equivalent to a time change of Brownian motion. As the process S is a continuous martingale, it follows from Monroe s theorem that S t = B ρt where B := {B t : t } is a Brownian motion with B = S and {ρ t : t T} is an increasing sequence of stopping times. Now, we have the following lemma from Breeden and Litzenberger (978): Lemma 2.. Let T (, ) and r be the continuously compounded risk free interest rate. Let us assume that call prices with maturity T on the underlying process X := {X t : t } are known for every strike K (, ) and are given under the risk-neutral measure Q by We then have that Provided C(K) is twice-differentiable in K we have that C(K) = E Q [e rt (X T K) + ]. (2.4) Q(X T > K) = e rt K C(K). (2.5) Q(X T dk) = e rt 2 K2C(K). (2.6) Using lemma 2., we can infer from the market input 2. the distribution µ of S T. The stopping time ρ T is thus a solution to the SEP for µ. Conversely, let τ be a solution to the SEP for µ, i.e. B τ µ and {B t τ : t } is a uniformly integrable martingale, then the process { S t = B τ t T t : t [,T)} is a model for the process of the stock price consistent with the market input 2.. We thus have a correspondence between market models and solutions to the SEP. We can use this result to find lower and upper bounds to the price E[O(S) T ] of the exotic option by bounding E[O(B) τ ] amongst all solutions τ to the SEP as follows: inf E[O(B) τ] E[O(S) T ] sup E[O(B) τ ], (2.7) τ:b τ µ τ:b τ µ where we assume that O(S) T = O(B) ρτ a.s. and all stopping times τ are such that {B t τ : t } is uniformly integrable. To illustrate how the method works, let us consider a one-touch barrier option as an example. For such a functional payoff, the bounds in (2.7) can be found by using results from Azéma and Yor (979) and from Perkins (985), as shown in Brown et al. (2). Azéma and Yor (979) proved that for a probability measure µ such that xµ(dx) = B, the stopping time where τ AY = inf{t : Ψ µ (B t ) B t }, (2.8) Ψ µ (x) = uµ(du), µ([x, )) [x, ) is minimal and B τay µ. τ AY is also optimal as it stochastically maximises the maximum such that we have P[B τ α) P[B τay α) for all α and any minimal τ with B τ B τay. Perkins (985) later developed a stopping time which minimises the maximum. Brown et al. (2) showed 5

13 how those two solutions induce upper and lower bounds on the price of a one-touch option and derived the associated super- and sub-replicating strategies. When the exotic option is a digital double barrier options, Cox and Obloj (28), Cox and Obloj (29), and Cox and Obloj (2) developed solutions to the SEP which maximise or minimise P[sup u τ B u b,inf u τ B u b] for digital double touch options, P[sup u τ B u < b,inf u τ B u > b] for digital double no-touch options, andp[sup u τ B u b,inf u τ B u > b] for digital double touch/notouch options, respectively. They showed that there are no corresponding unique inequality in (2.7) and that different strategies may be optimal depending on the market input and the chosen pair of barriers. They characterised all those strategies in the following form: G(S T )+M T O(S T ) F(S T )+N T, (2.9) where G(S T ) and F(S T ) are payoffs of finite portfolios of European puts and calls and M T and N T are gains from self-financing trading strategies such as forward transactions. The prices of G(S T ) and F(S T ) are then, respectively, the lower and upper bounds on the price of the exotic option, and the left-hand side and right-hand side of equation (2.9) are, respectively, sub- and super-replicating strategies at those prices. They also gave precise conditions to decide which strategy should be used. We describe those strategies in section B and refer the reader to Cox and Obloj (28), Cox and Obloj (29), and Cox and Obloj (2) for the proofs. The no-arbitrage price bounds thus derived are generally too wide to be of any use by traders for practical pricing because they are likely to fall outside the bid/ask spreads typically observed in most markets. However, the super- and sub-hedging strategies can still be useful to practitioners. Let us assume a trader sells for a premium p a double barrier option with maturity T and payoff O(S) T. This trader could then set up the superhedge suggested by the SEP method for an initial premium p > p. At maturity T, the trader holds Π = O(S) T + F(S) T + N T + p p. Now, E[Π] = and Π p p, which may be desirable properties for the trader. The former property means that the trader s expected balance is zero, and hence any amount charged above p should yield a positive profit on average. The latter property means that the trader cannot loose more than p p, however adversely the market moves against the trader. This latter property can be very desirable in some cases, such as when model uncertainty and market frictions are high, and can make hedging the exotic option using the SEP method more efficient on a risk-adjusted basis than traditional delta or delta/vega hedging by shielding the trader from risks inherent in traditional hedging methods, as described in the introduction. In comparison to traditional hedging methods, the SEP method does not require continuous rebalancing. The SEP method only requires a small number of transactions and the monitoring of the breaching of the barriers. Transaction costs are thus likely to have a much lesser impact on the performance of this method. Also, although the monitoring of the breaching of the barriers will be performed on a discrete time basis in practice, which will introduce systematic hedging errors, there is no trade-off to be made between transaction costs and hedging errors when it comes to deciding on the frequency of this monitoring. Deciding on this frequency is a much simpler problem to formulate for the SEP method than for traditional hedging methods: the less often the trader monitors the breaching of the barriers in the SEP method, the more the distribution of hedging errors departs from the theoretical one. As for model risk, the SEP method is simply not exposed to any, as it is a model-independent hedging method. A further benefit of the SEP method relative to the traditional hedging methods is illustrated by comparing the shapes of the distributions of hedging errors resulting from application of those methods. Under ideal conditions for the SEP method, that is when the underlying of the double barrier option is a martingale (e.g.: a forward price or a spot price with no drift in the risk-neutral measure) and the breaching of the barriers is continuously monitored, the hedging errors resulting from the SEP method will be bounded below. This is in contrast with traditional hedging methods for which there is a non-negligible probability mass in the left tail of the distribution of hedging On the assumption the SEP method is applied to a martingale process, such as the forward price. If applied to a non-martingale process such as the spot price, we will see later that the SEP method looses some of its benefits when the risk-neutral drift is high relative to the volatility of the underlying process. 6

14 errors. This mass may become large under extreme market conditions, where transaction costs, the choice of rebalancing frequency, and model risk become major issues. A trader worried about the likelihood of heavy losses would benefit from the strict cut-off in losses offered by the SEP method. Of course, how strict that cut-off is depends on how close the dynamics of the underlying is to that of a martingale and how often the breaching of the barriers is monitored. Extreme market conditions are typically accompanied by higher asset price volatility and higher transaction costs. One would thus expect the SEP method s outperformance relative to traditional methods to be at its strongest then. Such market conditions would generally call for much more frequent risk monitoring on the traders part, and the issue related to how frequently the breaching of the barriers is monitored in the SEP method should therefore be minimized. 2.3 SEP optimal strategies Given the market input 2. with a finite number of strikes and the barriers b and b, it is possible to find which of the super- or sub-hedging strategies defined in section B is optimal and how to build its associated replicating portfolio using one of two methods. One method, henceforth referred to as the grid search method, consists in finding the supremum (minimum for superhedging and maximum for subhedging) of the prices that were computed using all the relevant proposed hedging strategies and all possible strike combinations. The second method, henceforth referred to as the explicit method, is based on the fact that, for a market input where there are an infinite number of arbitrary strikes, the optimal hedging strategy and its corresponding strikes can be explicitly determined. The explicit method is described in details in Cox and Obloj (28), Cox and Obloj (29), and Cox and Obloj (2). When confronted with only a finite number of strikes, the grid search method is more straightforward to implement than the explicit method, provided that there is not an excessively large number of strikes for the computer power at hand, and it is the method we chose to implement in this project. Specifically, the grid search method consists in computing all possible solutions to the optimization problem and selecting the best one. This method is best explained by considering an example. The application of the method to other cases can be trivially inferred from the example and is not reported here. Let us consider a digital double touch option with barriers b and b for which we want to find the optimal superhedging strategy and its associated strikes given the following market input: {C(x i ) : x i X}, (2.) where, denoting A := {,...,N}, X := {x i : x i,i A,x i < x j i j} is a finite set of N strikes in ascending order. From proposition B. we know that the optimal superhedging strategy is one of four possible strategies: H I (K), H II (K ), H III (K,K 2,K 3,K 4 ), or H IV (K,K 4 ). Given that the market input 2. has a finite number of strikes, there is also a finite number of possible prices for all four strategies. By computing all those prices and selecting the minimum one we can find the corresponding optimal strategy and its strikes. Some of the super- and sub-replicating strategies described in section B require the use of digital callorputoptions. Adigitalcall(put)optionisanoptionthatpaysoneatmaturityiftheunderlying price process S := {S t : t } is above (below) a given strike level at maturity T. Denote DC(K) the price of a digital call option with strike K. The payoff at maturity T of DC(K) is {ST K}. In our implementation, we assumed digital call or put options are not available for trading and we approximated those options by using a portfolio of European call or put options as follows. It can be shown (see Bowie and Carr (994) for example) that DC(K) is given by n DC(K) = lim n + 2 ( C ( K n ) C ( K + n )), (2.) where C(K) is the price of an European call option on S struck at K with maturity T. Similarly, denoting DP(K) the price of a digital put option with strike K and payoff at maturity {ST K}, 7

15 Table 2.: Sample of optimal SEP super- and sub-replicating strategies obtained for a digital double barrier option written on EURUSD with different lower and upper barrier levels (LB and UB). The market model is assumed to be the VGSV model calibrated in section 4.5. Position LB UB Opt. Strat. K K2 K3 K4 Price Double touch Short.5.46 H I Short.3.6 H III Long.4.62 H IV Long.35.5 H III Double touch/no-touch Short.4.6 G II.3946 Short G I Long.5.5 G II Long.4.78 G I Double no-touch Short.3.6 F I.6286 Short.4.78 F II Long F I Long.25.6 F II DP(K) is given by ( ( n DP(K) = lim P K + ) ( P K )), (2.2) n + 2 n n where P(K) is the price of an European put option on S struck at K with maturity T. In the foreign exchange interbank markets that we consider in this project, call and put options are generally available for trading at any strike price in increments of pip. For both spot foreign exchange rates considered in this project, namely the EUR/USD and the AUD/USD, pip equals 4. In light of this, we chose to set n = 4 in both (2.) and (2.2). In our implementation, we assumed that the market input consists of European call option prices computed using the relevant calibrated volatility surface from section 4.5 at strike prices ranging from max ( K ask 5 P,Kbid 5 P ) to min ( K ask 5 C,Kbid 5 C), where K z x is the strike price corresponding to the z := {bid,ask} quoted implied volatility at delta strike x := {5 P,5 C}. We chose those strike price ranges because options at strike prices falling outside those ranges would likely be too illiquid to be considered realistically tradeable in any market condition. Table 2. shows a sample of digital double barrier options with different barrier levels and the corresponding optimal SEP super- or sub-replicating strategies, the strike prices used to build the associated replicating portfolios, and the prices of the SEP super- or sub-replicating portfolios. Figures 2. and 2.2 show which of the super- or -sub-replicating strategies is optimal in function of the barrier levels for the digital double barrier options described later in section Note that those figures are only approximative as we used a coarse discretisation of the grid formed by the two barrier levels to reduce computation time. Using a fine grid should result in the borders between optimal strategy zones to be smooth. 8

16 (a) Double Touch, Super-replication (b) Double Touch, Sub-replication (c) Double Touch No-Touch, Super-replication (d) Double Touch No-Touch, Sub-replication (e) Double No-Touch, Super-replication (f) Double No-Touch, Sub-replication Figure 2.: Optimal hedging strategies in function of lower and upper barrier levels (LB and UB) for the EURUSD digital double barrier options described in section

17 (a) Double Touch, Super-replication (b) Double Touch, Sub-replication (c) Double Touch No-Touch, Super-replication (d) Double Touch No-Touch, Sub-replication (e) Double No-Touch, Super-replication (f) Double No-Touch, Sub-replication Figure 2.2: Optimal hedging strategies in function of lower and upper barrier levels (LB and UB) for the AUDUSD digital double barrier options described in section

18 2.4 Application to spot markets We consider here how the SEP method can be adapted to work when the underlying is a spot price with non-zero cost of carry. To directly apply the SEP method as described in the previous sections, we require that the underlying process be a martingale. Let us assume that the underlying process S := {S t : t } is a foreign exchange spot rate and that the domestic and foreign interest rates, r d and r f respectively, are constant and continuously compounded. The process S := {S t = S t e (r d r f )t : t } is then a martingale, and we could directly apply the SEP method to S 2. However, if we were to do so for a double barrier option, the constant barriers used for the spot process S would become timedependent when considering S. To illustrate the problem with the digital double touch option, we note that the option payoff, which we denote Π, is Π = {ST b,s T b} = { (u,v) T : S v be (r d r f )u,s v be (r d r f )v }. (2.3) We do not know how to apply the SEP method to such an option with time-dependent barriers. To apply the said method, we need the barriers to be constant. One way to circumvent the problem is to decompose Π into two parts, where one part has a payoff close to that of Π on which we could directly apply the SEP method, and the other is a residual part on which we would apply a model-dependent pricing and hedging method of our choice. Mathematically, denoting Π the following payoff: Π = { (u,v) T : S v b,sv b}, (2.4) and Π 2 = Π Π, we decompose Π as follows: Π = Π +Π 2. (2.5) We apply the SEP method to Π and standard model-dependent pricing and delta/vega hedging to Π 2. The performance of this method is expected to be close to that of the SEP method applied to forward markets when the residual term Π 2 is small relative to Π and when the pricing and hedging errors associated to Π 2 have a small influence on the overall pricing and hedging errors of Π. In our implementation, we priced and hedged Π 2 in the Black-Scholes framework. To apply the SEP method to Π, we need to estimate the risk-neutral distribution of the martingale process S := {S t = S t e (r d r f )t : t } at time t = T. Denoting C (K ) the price of a call option struck at K on the underlying process S with maturity T, we can estimate the distribution of S (T) by using lemma 2.. However, we assumed we can only observe the call option prices C(K), which are options with maturity T struck at K on the underlying process S. To infer C (K ) from C(K), we first note that the risk-neutral measure Q under which the discounted price process S with re-invested foreign interest payments is a martingale is also the probability measure under which the process S is a martingale. We then have that C(K) = e r dt E Q[ (S T K) +] = e r dt E Q[ (S Te (r d r f )T K) +] = e r ft E Q[ (S T Ke (r d r f )T ) +]. (2.6) By choosing K = Ke (r d r f )T we have C(K) = e r ft E Q[ (S T K ) +] = e r ft C (K ), (2.7) or, equivalently, C (K ) = e r ft C(K e (r d r f )T ). (2.8) 2 we could equivalently choose the forward price S t = Ste(r d r f )(T t)

19 Table 2.2: Hedging strategies in spot market: correspondance between the asset that would theoretically havetobetradedinthemarketwheretheunderlyingisthes processandtheassetthathastobepractically traded in the real market where the underlying is the spot process Product type S market Spot market Call option C (K ) e (r d r f )T C(K e (r d r f )T ) Put option P (K ) e (r d r f )T P(K e (r d r f )T ) Digital call option {S T b} {S T be (r d r f )T } Digital put option {S T b} {S T be (r d r f )T } Cash α e rdt α Underlying asset St e (r d r f )t S t We can use (2.8) to compute the call option prices C (K ) from the observed market option prices C(K). In practice, computing the prices C (K ) this way will only provide accurate estimations as long as there are observable market traded call options C(K) with K equal, or very close, to K e (r d r f )T. The lower the number of call options C(K) traded on the market, the less likely this will be. For the two assets we applied the method to, we had a large enough number of observable market traded options that the additional error on Π introduced by the error made when estimating C (K ) is negligible relative to the errors made in pricing Π 2. To complete the derivation of the price of the exotic option under the SEP method, we also require knowledge of the prices of put options and single digital options on the underlying S. To find the price of a put option P (K ) on S struck at K with maturity T, we use the same method as that used for the call option C (K ) and have that P (K ) = e qt P(K e (r d r f )T ), (2.9) where P(K) ] is the price of a put option on S struck at K with maturity T. To find the price P [ {S T >b} of a digital call option on S paying one at maturity T when ST > b, we note that P [ ] {S T >b} = E Q[ ] {S T >b} = E Q[ {ST>be (r d r f )T } ] = e rdt e rdt E Q[ {ST>be (r d r f )T } [ ] = e rdt P {ST>be (r d r f. (2.2) )T } P [ {S T >b}] can thus be inferred from the price of digital call option on S struck at be (r d r f )T. We make use of equations (2.) and (2.2) to approximate P[ {ST>be (r d r f ], assuming once again )T } that the strikes of the observable market traded options are not so far from be (r d r f )T ±/n, with n large, that they may introduce non-negligible errors relative to those made when pricing Π 2. The price of digital put options is found in the same way as in (2.2). To derive the robust model-free hedging strategies for Π, we adapt the hedging strategies described in section B by using table 2.2. In the middle column of table 2.2 we read the number of units (and the strike for options) of an asset that section B proposes to trade in the market where the underlying of the exotic option S is a martingale under the risk-neutral measure, and in the column on the right-hand side we read the translated number of units (and the strike for options) of the asset we need to trade in the market where the underlying of the exotic option S is not a martingale under the risk-neutral measure. ] 2

20 Chapter 3 Models used We consider in the following sections the models used in this project for the risk-neutral dynamics of the risky asset. All descriptions of the models in this chapter assume that the risky asset is a foreign exchange spot rate and that r d and r f are respectively the constant and continuously compounded annualized domestic and foreign risk-free interest rates. The descriptions can obviously apply to other types of risky asset by appropriately adapting the equations. For stock prices, for example, the equations are to be modified by substituting r and q, respectively the constant continuously compounded annualized risk-free interest rate and dividend yield, for r d and r f. 3. Black-Scholes The risk-neutral dynamics of the risky asset process S := {S t : t } in the BlackScholes model (Black and Scholes (973)) are given by ds t = (r d r f )S t dt+σs t dw t, (3.) where σ is the constant square root of the quadratic variation of the risky asset s log price process, henceforth referred to as the volatility of the risky asset, and W := {W t : t } is a Brownian motion under the risk-neutral probability measure. Proposition 3.. Denote S := {S t : t } a price process whose dynamics follow the Black-Scholes model (3.) and X := {X t = log(s t ) : t } the corresponding log price process. The conditional characteristic function φ(u,t,t) = E Q [e iuxt X t ] of X T given X t under the risk-neutral measure Q is given by { [ φ(u,t,t) = exp iu X t +(r d r f 2 σ2 )(T t) ] 2 } σ2 (T t)u 2. (3.2) Proof. Denoting W := {W t : t } a Brownian motion, we have φ(u,t,t) = E Q [exp(iulog(s T ) log(s t )] = E Q [exp(iu(log(s t )+(r d r f 2 σ2 )(T t)+σ(w T W t )) log(s t )] { [ = exp iu log(s t )+(r d r f ]} 2 σ2 )(T t) E Q [exp(iuσ(w T W t )) log(s t )] { [ = exp iu log(s t )+(r d r f ]} 2 σ2 )(T t) exp { 2 } σ2 (T t)u 2. 3

21 S.55 S t (a) EURUSD t (b) AUDUSD Figure 3.: Sample paths for EURUSD and AUDUSD driven by the Black-Scholes models as calibrated in section Heston The Heston model (Heston (993)) is a stochastic volatility diffusion model. The risky asset price process S := {S t : t } is assumed to be driven by the following diffusion ds t = (r d r f )S t dt+ v t S t dw () t, (3.3) where v t is the variance process, and W () := {W () t : t } is a Brownian motion under the risk-neutral probability measure. The volatility of the risky asset is assumed to follow an Orsnstein- Uhlenbeck process (see for example Stein and Stein (99)) d v t = β v t dt+δdw (2) t, (3.4) where W (2) := {W (2) t : t } is another Brownian motion under the risk-neutral probability measure, correlated with W (). We denote ρ the constant correlation coefficient between W () and W (2). It can then be shown by applying Ito s lemma to (3.4) that v t follows the process dv t = (δ 2 2βv t )dt+2δ v t δdw (2) t, (3.5) which can be written as the square-root process used by Cox et al. (985) dv t = κ(θ v t )dt+ζ v t dw (2) t, (3.6) where θ is the long run mean variance, κ is the mean reversion rate, and ζ is the volatility of v t, also known as the volatility of the volatility. In condensed form, the Heston model can be summarized as follows: ds t S = (r d r f )dt+ v t dw () t, t dv t = κ(θ v t )dt+ζ v t dw (2) t, (3.7) d W (),W (2) t = ρdt. Proposition 3.2. Denote S := {S t : t } a price process whose dynamics follow the Heston model (3.7) and X := {X t = log(s t ) : t } the corresponding log price process. The conditional characteristic function φ(u,t,t) = E Q [e iuxt X t ] of X T given X t under the risk-neutral measure Q is given by φ(u,t,t) = A(u,t,T)exp( B(u,t,T)), (3.8) 4

22 S S t (a) EURUSD t (b) AUDUSD Figure 3.2: Sample paths for EURUSD and AUDUSD driven by the Heston models as calibrated in section 4. where A(u,t,T) = { exp κθ(κ iρζu) [ cosh } ζ +iu[(r 2 d r f )(T t)+x t ] ) ( )] 2 κθ, + κ iρζu γ(u) sinh γ(u)(t t) ζ 2 2 ( γ(u)(t t) 2 B(u,t,T) = γ(u) coth (u 2 +iu)v t ( γ(u)(t t) 2 ) +κ iρζu γ(u) = ζ 2 (u 2 +iu)+(κ iρζu) 2. Proof. See for example Cont and Tankov (24)., 3.3 Bates Bates (996) proposed the following stochastic volatility jump-diffusion model, which is an extension of the Heston model: ds t S = (r d r f )dt+ t v t dw () t +J t dn t, dv t = κ(θ v t )dt+ζ v t dw (2) t, d W (),W (2) t = ρdt, (3.9) P(dN) = λdt, log(+j) N(log(+µ j ) 2 σ2 J,σ2 J ), where v t is the variance process, W () := {W () t : t } and W (2) := {W (2) t : t } are correlated Brownian motions under the risk-neutral probability measure with correlation coefficient ρ, θ is the long run mean variance, κ is the mean reversion rate, and ζ is the volatility of volatility, N := {N t : t } is an independent compound Poisson process with intensity λ > (i.e. E[N t ] = λt), and J t is the percentage jump size (conditional on a jump occurring). J t is assumed to be lognormally, identically, and independently distributed over time with unconditional mean µ j and where σj 2 is the variance of log(+j t ). Proposition 3.3. Denote S := {S t : t } a price process whose dynamics follow the Bates model (3.9) and X := {X t = log(s t ) : t } the corresponding log price process. The conditional 5

23 S.45.4 S t (a) EURUSD t (b) AUDUSD Figure 3.3: Sample paths for EURUSD and AUDUSD driven by the Bates models as calibrated in section 4. characteristic function φ(u,t,t) = E Q [e iuxt X t ] of X T given X t under the risk-neutral measure Q is given by φ(u,t,t) = A(u,t,T)exp(C(u,t,T) B(u,t,T)), (3.) where A(u,t,T) = { exp κθ(κ iρζu) [ cosh B(u,t,T) = γ(u) coth C(u,t,T) = λ(t t) } ζ +iu[(r 2 d r f λµ j )(T t)+x t ] ) ( + κ iρζu γ(u) sinh γ(u)(t t) 2 ( γ(u)(t t) 2 (u 2 +iu)v t ( γ(u)(t t) 2 [ exp ) +κ iρζu γ(u) = ζ 2 (u 2 +iu)+(κ iρζu) 2. Proof. See for example Cont and Tankov (24)., )] 2 κθ ζ 2, { 2 σ2 ju 2 +iu(log(+µ j ) 2 σ2 j) } ], 3.4 Variance Gamma with CIR stochastic clock As this model is the one we chose for the underlying assets in the real-life market scenarios, we describe it in details in this section. The Variance Gamma process with CIR stochastic clock (VGSV) is a pure jump Lévy process with infinite activity and stochastic volatility. It is based on the Variance Gamma (VG) model subordinated by the time integral of a CIR process. We begin by introducing the motivation behind suchamodel. WethendescribetheVGmodelandfinallyshowhowtheVGSVprocessisconstructed by subordinating the VG process to make its volatility stochastic. A theorem by Monroe (978) establishes that any semimartingale can be written as a timechanged Brownian motion. Since the assumption of no-arbitrage implies that there exists a probability measure under which the discounted prices of the risky asset are martingales, the prices of the risky asset must be semimartingales under the real-world probability measure. The same holds for the log-price of the risky asset. We thus have log(s(t)) = W(T(t)), (3.) 6

24 where W is a Brownian motion and T(t) is an almost surely increasing process which represents the time change. Economically, (3.) can be interpreted as how the price of the risky asset is function of the arrival rate of information. On days with little news, trading is slow and prices vary little. On days with influential news, trading is fast and prices vary wildly. T(t) is often referred to as operational time. It may be seen as a cumulative measure of economic activity, as suggested by Clark (973). The continuity of the process S(t) is equivalent to the continuity of the process T(t). Now, following the argument made by Geman and Street (22), if T(t) is continuous it can be written in the following form: T(t) = t a(u)du + t o b(u)dz(u). (3.2) Now, because T(t) must be increasing, we necessarily have b(u) = and the time change is thus locally deterministic, which is not something we desire as we believe that T(t) depends on random market activity such as the arrival of information or transactions. Therefore T(t) cannot be continuous, hence the choice of jump processes for T(t). We now briefly introduce the concept of subordination in the sense of Bochner (see Bochner (949)) and shall refer to it when we simply mention subordination in what follows. We already referred to a case of subordination when we described the time-changed Brownian motion above. By subordination in the sense of Bochner, we mean that if X is a Lévy process with characteristic exponent η X and T is an independent subordinator with Laplace exponent λ then Y(t) = X(T(t)) is a new Lévy process with characteristic exponent η Y = ( λ) ( η X ). The VG model (Madan and Seneta (99)) is a pure jump Lévy process with infinite activity. The VG process X VG := {X VG (t) : t } is defined by subordination of a Brownian motion W with drift θ and volatility σ by a gamma process G ν : X VG (t) = θg ν t +σw(g ν t), (3.3) where G ν := {G ν t : t } is a gamma process with unit mean rate and variance rate ν. A risky asset that follows a Variance Gamma process can then be considered to follow the Black-Scholes model in operational time G ν t, not in real time t. Denoting Γ(x) the gamma function, the probability density function at time t of the gamma process γ := {γ(t;µ γ,ν γ ) : t } with mean rate µ γ and variance rate ν γ is whilst its characteristic function Φ γ (u,t) is f γ (x) = xµ2 γ /νγ e xµγ/νγ (ν γ /µ γ ) µ2 γ /νγ Γ(µ 2 /ν γ ), (3.4) Φ γ (u,t) = E[e iuγ(t) ] = ( iuν γ /µ γ ) µ2 γ /νγ. (3.5) The probability density function at time t of the gamma process G := {G(t;t,νt) : t } with mean rate t and variance rate νt is thus whilst its characteristic function Φ G (u,t) is f G (x) = x( t ν ) e x ν ν t νγ( t ν ), (3.6) Φ G (u,t) = E[e iug(t) ] = ( iνu) t/ν. (3.7) The characteristic function of the VG process Φ VG (u,t) is obtained from (3.7) by conditioning on the gamma time and noting that the conditioned variable is Gaussian. Φ VG (u,t) is given by Φ VG (u,t) = E[e iuxv G(t) ] = ( iθνu+σ 2 νu 2 /2) t/ν. (3.8) Following Madan et al. (998), the VG process may also be expressed as the difference of two independent gamma processes, with one describing the up moves and the other the down moves, as the following proposition shows. 7