With promises of services such as LedgerX to introduce derivatives such as swaps and options on digital currencies, partially reducing counter party risks with external clearing institutions, its time to dustoff some of the old strategies. Assuming it will be hard to exactly price the derivatives, there should be some arbitrage opportunities.
In my first blog post I’ll be revisiting the putcall parity equation. Blog describes how the arbitrage opportunity works with the American options on stocks. Hopefully we’ll soon be able to get our hands on cryptocurrency derivatives. At that point I’ll be able to amend the blog with the cryptocurrency version.
Putcall parity overview
 Let C_{K,T}(t) be the price of a call option with strike K and maturity T, at time t.
 Let P_{K,T}(t) be the price of a put option with strike K and maturity T, at time t.
 Let Z_{r,T}(t) be the price of a zero coupon bond with maturity at time T and rate r, at time t.
 Let S_{t} be the underlying stock price at time t.
 Portfolio F (fiduciary call) of a call option and zero coupon bond, that will have a value K (strike) at the time of maturity
 Portfolio P (protected put) of a put option and the underlying stock
C_{K,T}(t) + KZ_{r,T}(t) = P_{K,T}(t) + S_{t}
 C_{K,T}(t) = 5.90$ (bid: 5.75$, ask: 6.05$)
 P_{K,T}(t) = 0.90$ (bid: 0.78$, ask: 1.01$)
 K*Z_{r,T}(t) = K*e^{rT} = 24.58$ (with the risk free rate of 2%)
 S_{t} = 30.45$
 Portfolio F. C_{K,T}(t) + KZ_{r,T}(t) = 5.90$ + 24.58$ = 30.48$
 Portfolio P. P_{K,T}(t) + S_{t} = 0.90$ + 30.45$ = 31.35$
Theory and practice
European vs. american options
No dividends

With dividends



Before the exdividend date(s)

After exdividend and before expiration


American Call

Never optimal to exercise assuming r > 0
(Cai, 2005; Hull, 2015)

Can be optimal to exercise immediately before the exdividend date if
D > K(1 – e^{r(Tt)})
(Hull, 2015)

Never optimal to exercise
(Hull, 2015)

American Put

Exercise early if time value is smaller than interest loss
(Cai, 2005)

Never optimal to exercise before exdividend date
(Cai, 2005)

May be optimal to exercise early if time value is smaller than interest loss
(Cai, 2005)

S_{t}  D  K =< C_{K,T}(t)  P_{K,T}(t) =< S_{t}  KZ_{r,T}(t)
G_{T}(K,t) <= K + D  S_{t} G_{T}(K,t) >= KZ_{r,T}(t)  S_{t}
Early exercise

At time T if
option was exercised early


At time T if option was not exercised early



Starting position

Exercise at time t’

S_{T} <= K

S_{T} > K

S_{T} <= K

S_{T} > K

Sell stock= +S_{0}
Sell put: +P
Buy bond: (K+D)
Buy call: C

Stock: S_{t’}
Put: (KS_{t’})
Bond: +Ke^{r(t’t0)}

Stock: De^{rT}
Put: 0
Bond: +De^{rT}
Call: 0

Stock: De^{rT}
Put: 0
Bond: +De^{rT}
Call: S^{T}K

Stock: S_{T}
Put: S_{T }– K
Bond: K
Call: 0

Stock: S_{T}
Put: 0
Bond: K
Call: S_{T }– K

Totaling
+K(e^{r(t’t0)}1) > 0

Totaling
0

Totaling
S^{T}K > 0

Totaling
0

Totaling
0


Sell bond: +K
Sell call: +C
Buy stock: S_{0}
Buy put: P

Call: KS_{t’}
Stock: +S_{t’}

Bond: K
Call: +Ke^{r(Tt’)}
Stock: 0
Put: K – S_{T}

Bond: K
Call: +Ke^{r(Tt’)}
Stock: 0
Put: 0

Bond: K
Call: 0
Stock: S_{T}
Put: K – S_{T}

Bond: K
Call: S_{T}K
Stock: S_{T}
Put: 0

Totaling
K

Totaling
+ Ke^{r(Tt’)} – S_{T} > 0

Totaling
+K(e^{r(Tt’)}1) > 0

Totaling
0

Totaling
0

Transaction costs
Next we need to introduce transaction costs which occur explicitly or implicitly at various times of the trade.
There are a plethora of online brokers nowadays and in such a competitive environment fee structures can vary greatly. A typical online broker however will charge (1) a fee for opening and closing a position and (2) an assignment fee, which will occur in the event the option we have written gets exercised. Usually the part of the fee will be fixed and another part variable, relative to the value, number of shares, etc.
Then, there is an implicit fee in bidask spread, which can sometimes be very high relative to the price of an instrument. Especially for options the spread could be high at times of high volatility, for options with long expiration times or far outofthe money strikes. It is customary to take the midway between the bid and the ask as the execution price at testing, but with big spreads it is unrealistic. I’ll intentionally exclude the bidask spread from the further fee consideration, and will limit to the universe of instruments with small bidask spread.
Let’s define a quantity F(s) as a sum of all explicit transaction costs, relative to an abstract notion of position size s. Updating the equations to account for transaction costs yields the following two equations.
G_{T}(K,t) <= K + D  S_{t}  F(s) G_{T}(K,t) >= KZ_{r,T}(t)  S_{t }+ F(s)
The bond leg
The bond leg in the transaction can be replaced by lending or borrowing money. Obviously as an individual you will lend at lower interest rate than the rate at which you can borrow money. Moreover if we replace the tbills with of example borrowing / lending funds to or from our broker, it can no longer be considered riskless borrowing, as we are assuming a counterparty risk.
Conclusions
When options prices are misaligned with the putcall parity equation, we might be able to take advantage of an arbitrage opportunity. With the exchange traded stock and vanilla options, such opportunities are rare, and mostly available for long term options, where the profits, that you are able to lock in, are meagre.
Cryptocurrencies look like a new frontier. Nowadays they are traded at fairly low volumes and are very volatile. LedgerX just started offering digital currency derivatives to the institutional investors in October 2017. The volumes during the first days are still quite low and the options spreads very high. Hopefully similar offering will soon be available for the retail investors as well. Then we can start the search for the “bitcoins in front of a steamroller”.
References
 Hull, John C. Options, Futures, And Other Derivatives. 9th ed. Upper Saddle River, NJ: Pearson Education, 2015. Print.
 Cai, David. Derivatives, Spring 2006, NYU, http://www.math.nyu.edu/~cai/Courses/Derivatives/lecture8.pdf.
 Pavel Cízek, Wolfgang Härdle, Rafal Weron, 2005, http://fedc.wiwi.huberlin.de/xplore/tutorials/sfehtmlnode40.html.