whereZ1−α2 denotes the 1−αquantile of the standard normal distribution. For further details, the reader is referred to Glasserman (2004).
2.5 Discretization
The computational implementation for simulation purposes requires a discretization of the continuous-time processes with the dynamics described in equation (9). The finite continuous-time horizon is partitioned into ntime steps of equal distance dtsuch that T ={0, dt,2dt, . . . , ndt=T} . Belaygorod and Olin (2005) state that it is difficult to distinguish whether many small jumps or one larger jump occurred in a short time interval. The discrete partition of the finite time horizonT into n time steps of size dtis not sufficiently granular to enable a clear distinction between the frequency of jump arrivals and the amplitude of jump sizes. For simulation, Belaygorod and Olin (2005) propose to model arrivals of jumps by Bernoulli random variablesJdts =Jdtv ∼Ber(λ). The Euler-Maruyama method is applied to discretize equation (9). The Euler discretization applied in this paper is
S(t+dt)−S(t)
where X1 and X2 are two standard normal variables correlated with coefficient ρ. µ is the drift.
In equation (9),µ=r+γ(t). Equation 18 presents only one possibility to discretize the solution of equation (9). The procedure with the solution (18) is a less precise method to discretize the solution of equation (9). Another more precise method is to solve equation (9) with Ito calculus for jump processes (Belaygorod and Olin, 2005). For further details, the reader is referred to section 4 in Belaygorod and Olin (2005). Broadie and Kaya (2006) describe an even more exact method for simulating affine jump diffusion processes. Since the scope of this paper is hedging, it is sufficient to apply the representation in equation (18). For different discretization methods, the reader is recommended to look into the book of Kienitz and Wetterau (2013). Hou et al. (2019) also apply the Euler-Maruyama discretization described in Johannes and Polson (2009). They calibrate the drift component with one parameterµ.
The calibrated parameters are reviewed in section 2.5.1.
2.5.1 Calibration
According to the concepts in section 2.2, options are priced under the risk-neutral measure PQ. Therefore, proper calibration requires the estimation of parameters under PQ. This is not feasible in the case cryptocurrencies, because options are not officially exchange-traded (Hou et al., 2019).
Therefore, Hou et al. (2019) calibrate the parameters Θ = {µ, µs, σs, λ, α, β, σv, ρ, ρj, µv}, where α = κθ and µs is the average jump size in returns, from returns under the physical measure P.
They apply Monte Carlo Markov Chain (MCMC), a Bayesian calibration method initially applied by Eraker et al. (2003), to estimate the parameters of the SVCJ for BTC and CRIX in a time horizon from 31.03.2014 to 29.09.2017. Perez (2018) extends the estimation to more cryptocurrencies until 30.09.2018.The purpose of this paper is to evaluate the hedge performance of options priced by Hou et al. (2019). Therefore, the parameters calibrated by Hou et al. (2019) presented in table 1 are assumed as given.
µ µy σy λ α β ρ σv ρj µv
mean 0.042 -0.049 2.061 0.051 0.010 -0.190 0.275 0.007 -0.210 0.709 p-value 0.006 0.371 0.432 0.007 0.001 0.009 0.069 0.001 0.364 0.089 Table 1: SVCJ calibrated parameters of the CRIX from Hou et al. (2019). Red means strong positive significance (below 0.001 %) and blue strong means negative significance (below 0.001 %)
SVCJ_CRIX
In table 1, λis small. The interpretation is that jumps are rare. As excepted, ρj is negative but statistically insignificant. This is in line with the conventions in Broadie et al. (2007) and the findings of Eraker et al. (2003), Eraker (2004) and Chernov et al. (2003). Broadie et al. (2007) outline the difficulties in estimatingρj by reason of the fact that jumps occur seldomly. Broadie et al. (2007) and
Branger et al. (2012) recommend to setρj= 0. We accept this suggestion and setρj = 0. Under this assumption, equation (9) distinguishes itself from the ’Stochastic Volatility with Jumps’ (SVJ) model by Bates (1996) by jumps in volatility. In other words, the model includes the additional parameter µv. Hou et al. (2019) outline thatρis positive and significant. This is contrary to what is expected.
It is expected that when prices increase, volatility decreases (Broadie et al., 2007). The interpretation of Hou et al. (2019) is an inverse leverage effect. They refer to the finding of Schwartz and Trolled (2009) on commodity markets.
In summary, this paper tries to mimic the behavior of the CRIX in a simulation study where the dynamics of the underlying are described by the SDE in equation (9). Equation (9) is discretized by equation (18) with the above-formulated parameter conventions, where ρj = 0. We assume that the values of the parameters are the ones listed in table 1. Figure 2 illustrates 25 trajectories of our data-generating processS(t).
Figure 2: 25 Euler discretized trajectories ofS(t) for a time horizon of 1 year
SVCJ_MC
The trajectories of figure 2 illustrate different scenarios. In equation (9), jumps sizes in returns are
Zs(t)|Zv(t)∼N µs+ρjZv(t), σs2
distributed. Accordingly, jumps sizes can be very large. This is illustrated by the blue and pink trajectory in figure 2. Sinceλis fairly small, jumps are rare. For the hedging procedure it is important to evaluate the relevance of these jumps. Figure 3a and 3b illustrate 4000 asset paths of the SVCJ.
The purpose of these illustrations is to evaluate the main drivers of our underlyingS(t). In figure 3a, jumps are present andλ6= 0. For comparison, in figure 3b the parameterλis set toλ= 0. In other words, the presence of jumps is excluded. In figure 3a large jumps amplitudes lead to prices rises up to 600 % within 1 year. However, in the majority of the cases paths range roughly up toS(t) = 250.
This price range corresponds to what is observed in figure 3b. This indicated that the main driver
of the price dynamics are not jumps but stochastic volatility. The discussion about adding jumps is summarized in Broadie et al. (2007). For example, Eraker (2004) finds that adding jumps may lead to a better model fit, yet has a small impact on option pricing. For further information, the reader is referred to Broadie et al. (2007). Hou et al. (2019) report that most studies on cryptocurrencies do not account for jumps. They choose to add jumps to pricing in reference to Scaillet et al. (2018).
Scaillet et al. (2018) report that in comparison to ”traditional markets”, jumps on Cryptocurrency markets tend to occur more frequently. Judging on the movements of the CRIX illustrated in figure 1, this paper believes that jumps should be included to pricing and that the presence of jumps should not be disregarded. Single cases can have extreme amplitudes in jump sizes. The question we ask ourselves is how this impacts the P&L.
(a) SVCJ trajectories (b) SVCJ trajectories without jumps
Figure 3: 4000 simulated trajectories ofS(t) with jumps in returns and volatility in 3a and without jumps in 3b
SVCJ_MC
3 Hedging
We have outlined the importance of hedging in terms of mitigation of risk and a deeper understanding of the market. In section 2.4.1 we address market incompleteness. There is no perfect replication (6). Thus, we face the trade-off between selecting a simpler and preferably complete market model or choosing alternative hedging strategies (Packham and Detering, 2016). A ”conservative approach”
(Cont and Tankov, 2003) is to almost surely hedge all risk associated with the contingent claim P
Vφ(t) =V(0) + Z t
0
φdS≥H
= 1 (19)
We review the explanation of superhedging in Cont and Tankov (2003). If the strategy is self-financing, the costs of superhedging correspond toVφ(0). Superhedging is neither desirable nor a very efficient strategy that hedges against large price movements resulting from, for example, jump amplitudes. To
demonstrate the inadequacy of this strategy in our investigation, it is sufficient to state that investors and financial institutions enter markets willing to bear some risk because the risk is rewarded. It is hardly the case that someone would participate in a market like a cryptocurrency market and to hedge all risk associated with this market.
A third way to go is utility-based hedging. We choose a hybrid method. We hedge under model misspecification, yet two of our models are incomplete market models such that we also apply quadratic hedging.
3.1 Hedging in discrete time
For the empirical implementation in discrete time, the finite time horizonT is partitioned intoT = {0, dt,2dt, . . . , ndt= T}. The hedging instruments are the underlying S(t) and the money market accountM(t), precisely, a risk-free assetB(t), t≥0 withB(0) = 1. Our strategyφis a self-financed strategy as defined in equation (2).
At timet0= 0 andB(0) = 1, the value of the portfolio is
Vφ(0) =CSV CJ(0, S(0)) =φ(0)S(0) + (CSV CJ(0, S(0))−φ(0)S(0))B(0) M(0) =CSV CJ(0, S(0))−φ(0)S(0)
(20) At timet≥0, the value of the portfolio is
M(t) =M(t−dt) + (φ(t−dt)−φ(t))S(t) B(t)
V(t) =φ(t−dt)S(t) +M(t−dt)B(t−dt)erdt=φ(t)S(t) +V(t)−φ(t)S(t) B(t)
| {z }
=M(t)
B(t) (21)
At maturityT, the final position is
V(T) =φ(T−dt)S(t) +M(T−dt)B(t) (22)
At maturity, the profit is VT −max{(S(t)−K)+,0}. We will implement this strategy in the delta hedge.