Advanced model calibration on bitcoin options

Abstract

In this paper, we investigate the dynamics of the bitcoin (BTC) price through the vanilla options available on the market. We calibrate a series of Markov models on the option surface. In particular, we consider the Black–Scholes model, Laplace model, five variance gamma-related models and the Heston model. We examine their pricing performance and the optimal risk-neutral model parameters over a period of 2 months. We conclude with a study of the implied liquidity of BTC call options, based on conic finance theory.

Appendices

Appendix A: The Laplace market model

A brief summary of the Laplace market model is given below. More details can be found in Madan (2016), and Madan and Wang (2017).

1.1 Laplace distribution

The density function of a Laplace distributed random variable L with mean $\mu$ and variance $\sigma ^2$ is given by

$$\begin{aligned} f_L(x) = \frac{1}{\sqrt{2}\sigma }\exp \left( -\sqrt{2} \ \frac{|x-\mu |}{\sigma }\right), \end{aligned}$$

(32)

and we denote $L \sim {\mathcal {L}}(\mu , \sigma ^2)$. We refer to the Laplace distribution with zero mean and unit variance as the standard Laplace distribution $L^*$. The characteristic function of L is given by

$$\begin{aligned} \phi (u) = \exp (iu \mu ) \left( 1 + \frac{\sigma ^2u^2}{2}\right) ^{-1}. \end{aligned}$$

(33)

We refer to Kotz et al. (2001) for a broad introduction to Laplace distributions and its extensions.

1.2 Market model

Assume that the log-returns of an asset S are modelled via the Laplace distribution:

$$\begin{aligned} \log (S_{t+s}) - \log (S_t) \sim {\mathcal {L}}\left( \mu s , \ \sigma ^2s\right), \end{aligned}$$

(34)

and hence

$$\begin{aligned} S_t \sim S_0\exp \left( \mu t + \sigma \sqrt{t}L^*\right) . \end{aligned}$$

(35)

We apply a mean-correcting measure change, assuming zero interest rates. The distribution is shifted to

$$\begin{aligned} S_t \sim S_0\exp \left( \log \left( 1-\frac{\sigma ^2t}{2}\right) + \sigma \sqrt{t}L^*\right) . \end{aligned}$$

(36)

Note that this equation is only valid for $\sigma ^2t < 2$. The characteristic function of the log-price process $\log (S_t)$ at time t equals:

$$\begin{aligned} \phi _t(u) = E[\exp (iu\log (S_t))] = \frac{\exp \left( iu \left( \log S_0 + \log \left( 1-\frac{\sigma ^2t}{2}\right) \right) \right) }{1 + \frac{\sigma ^2t}{2}u^2 }. \end{aligned}$$

(37)

Remark

This model is not identical to the variance gamma model with $\theta = 0$ and $\nu = 1$. However, the VG($\sigma$, 1, 0) distribution equals the Laplace distribution${\mathcal {L}}(0,\sigma ^2)$. On the other hand, in the VG($\sigma$, 1, 0) model the log-returns are distributed as

$$\begin{aligned} \log (S_{t+s}) - \log (S_t) \sim {\mathrm{VG}}\left( \sigma \sqrt{s}, \ 1/s, \ 0\right) . \end{aligned}$$

(38)

The two models hence only give the same results on a maturity of 1 year.

1.3 Vanilla option pricing

The price of a call option is given by

$$\begin{aligned} C(K,T;\sigma ) = S_0 SL\left( d, \frac{\sigma \sqrt{T}}{\sqrt{2}}\right) - K SL(d,0) \end{aligned}$$

(39)

with

$$\begin{aligned} SL(x,s)&= {\left\{ \begin{array}{ll} \frac{1}{2}(1+s)\exp ((1-s)x) &{} \quad {\text {for}} \,x \le 0 \\ 1 - \frac{1}{2}(1-s)\exp (-(1+s)x) &{} \quad {\text {for}} \,x > 0 \end{array}\right. }, \end{aligned}$$

(40)

$$\begin{aligned} d&= \frac{\log (S_0/K) + \log \left( 1-\frac{\sigma ^2T}{2}\right) }{\frac{\sigma \sqrt{T}}{\sqrt{2}}}. \end{aligned}$$

(41)

Appendix B: Option data of 29 June 2018

Underlying	Strike	Maturity (days)	Is call	Bid price (USD)	Ask price (USD)
5906	5500	6.92	1	434.06	472.44
5906	5750	6.92	1	256.88	289.36
5906	6000	6.92	1	132.87	147.63
5906	6250	6.92	1	64.96	91.53
5906	6500	6.92	1	29.53	50.2
5906	6750	6.92	1	8.86	26.57
5906	5500	6.92	0	70.86	91.53
5906	5750	6.92	0	132.87	168.3
5906	6000	6.92	0	256.89	301.18
5906	6250	6.92	0	431.09	484.24
5906	6500	6.92	0	634.84	699.8
5906	6750	6.92	0	862.21	930.12
5906	7000	6.92	0	1098.43	1169.29
5872	5500	27.92	1	607.7	648.8
5872	6000	27.92	1	352.29	393.39
5872	6500	27.92	1	193.76	211.37
5872	7000	27.92	1	102.74	126.23
5872	7500	27.92	1	58.71	76.32
5872	8000	27.92	1	35.23	49.9
5872	8500	27.92	1	17.61	29.36
5872	9000	27.92	1	14.68	23.49
5872	10,000	27.92	1	2.94	14.68
5872	5500	27.92	0	240.73	275.96
5872	6000	27.92	0	478.53	528.43
5872	6500	27.92	0	810.27	874.85
5872	7000	27.92	0	1203.66	1288.79
5872	7500	27.92	0	1652.83	1743.84
5872	8000	27.92	0	2122.55	2222.36
5872	8500	27.92	0	2606.95	2712.63
5872	9000	27.92	0	3097.22	3202.9
5872	10,000	27.92	0	4089.5	4195.19
5872	11,000	27.92	0	5087.65	5190.41
5872	12,000	27.92	0	6085.29	6190.97
5867	5500	90.92	1	920.96	973.76
5867	6000	90.92	1	692.19	739.12
5867	6500	90.92	1	513.23	530.83
5867	7000	90.92	1	387.12	428.18
5867	7500	90.92	1	287.41	328.47
5867	8000	90.92	1	217.02	255.15
5867	8500	90.92	1	164.23	199.43
5867	9000	90.92	1	126.11	158.37
5867	10,000	90.92	1	79.18	105.58
5867	11,000	90.92	1	43.99	70.39
5867	12,000	90.92	1	29.33	52.79
5867	13,000	90.92	1	17.6	38.13
5867	14,000	90.92	1	8.8	29.33
5867	15,000	90.92	1	2.93	23.46
5867	20,000	90.92	1	2.93	8.8
5867	5500	90.92	0	557.27	610.06
5867	6000	90.92	0	824.17	882.83
5867	6500	90.92	0	1132.14	1196.66
5867	7000	90.92	0	1492.9	1572.09
5867	7500	90.92	0	1886.08	1965.28
5867	8000	90.92	0	2317.27	2428.73
5867	8500	90.92	0	2748.46	2877.52
5867	9000	90.92	0	3206.04	3340.97
5867	10,000	90.92	0	4147.62	4253.21
5867	11,000	90.92	0	5112.65	5259.32
5867	12,000	90.92	0	6132.9	6238.49
5867	13,000	90.92	0	7073.79	7226.3
5867	14,000	90.92	0	8065.06	8217.57
5867	15,000	90.92	0	9056.33	9211.77
5867	20,000	90.92	0	14,027.34	14,203.31
5867	25,000	90.92	0	19,010.09	19,203.65
5867	30,000	90.92	0	23,989.9	24,206.92
5867	35,000	90.92	0	28,972.64	29,207.26
5867	40,000	90.92	0	33,955.38	34,210.53
5906	6250	181.88	1	1012.77	1086.58
5906	7500	181.88	1	658.45	729.31
5906	8750	181.88	1	439.95	504.9
5906	10,000	181.88	1	310.03	363.18
5906	12,500	181.88	1	159.44	203.73
5906	15,000	181.88	1	100.39	121.06
5906	20,000	181.88	1	41.34	59.05
5906	25,000	181.88	1	20.67	29.53
5906	30,000	181.88	1	11.81	20.67
5906	35,000	181.88	1	5.91	14.76
5906	40,000	181.88	1	2.95	11.81
5906	6250	181.88	0	1328.7	1417.28
5906	7500	181.88	0	2199.88	2350.47
5906	8750	181.88	0	3215.69	3398.77
5906	10,000	181.88	0	4328.8	4523.68
5906	12,500	181.88	0	6658.24	6900.35
5906	15,000	181.88	0	9064.7	9371.77
5906	20,000	181.88	0	13,966.13	14,391.31
5906	25,000	181.88	0	18,908.9	19,384.28
5906	30,000	181.88	0	23,863.48	24,442.2
5906	35,000	181.88	0	28,823.96	29,479.46
5906	40,000	181.88	0	33,787.4	34,519.66