An Innovative way of making options
The Timer Option
Gianni Baroncini
Rocco Letterelli
Timer Option
Brief History
The first to study these options was Neuberger in 1990, also if he called them
“mileage options”. Bick, in 1995, extended the analysis of such options to a
continuous time setting.
In more recent times, 2009-2010, also Li, Bernard and Cui, have analyzed
these options.
In April 2007 Timer Options were introduced in the market for the first time by
Société Générale Corporate and Investment Banking.
Timer Option
Main Characteristic
The Investor is the decisional center
He chooses
The Maturity
The Volatility
Timer Option
The variance budget
The volatility and the maturity specified from the investor, are used
to calculate the “Variance Budget”
The product between target
volatility squared and target
maturity
Once that the Variance Budget is consumed, the option expires.
When realised volatility squared, multiplied by
the number of expired days divided by 252, is
greater than the Variance Budget
The timer option expires if
(σ^2) * (d/252) ≥ Variance Budget
Timer Option
Timer Option
Payoff
The payoff of a Timer Call or Timer Put, is similar to
that of a Call or Put
Timer Call (K) = max (Sτ-K ; 0)
Timer Put (K) = max (K- Sτ ; 0)
The difference with plain vanilla option is in τ, that it’s not known at t=0
Thus we can write that the price of a timer call option at time 0:
EQ[e-r τ max (Sτ - K ; 0)]
Timer Option pricing:
By definition τ depends on the variance process, and the payoff at time τ
depends on Sτ. That problem involves two different processes (St; Vt):
𝑑𝑆𝑡 = 𝑟𝑆𝑡 𝑑𝑡 + 𝑉𝑆𝑡 1 − ρ2 𝑑𝑊1𝑡 + ρ 𝑑𝑊2𝑡
𝑑𝑉𝑡 = α𝑡 𝑑𝑡 + 𝐵𝑡 𝑑𝑊2𝑡
W1 and W2 are indipendent Wiener processes
ρ 𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝑡ℎ𝑒 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑆𝑡𝑎𝑛𝑑 𝑉𝑡
The difference with a plain vanilla is about 𝑑𝑉𝑡
A well-known stochastic volatility model that satisfies the above conditions is the
Heston Model:
In that model the process Vt+Δt|V t is distributed as a non central chi-square:
Vt+Δt|V t= γ (1-e-kΔt) X
4k
Timer option pricing
MONTECARLO:

We can simulate the joint path of the correlated
processes (St, Vt) to obtain n simulations of (T,ST):

We simulate Vt+Δt and St+Δt for each time step and we
stop when Vj reaches V
Vj is the accumulated variance at time j
V is the variance budget

Finally we record (j,Sj) as an approximation of (T,ST)
Timer option pricing:
Remarks:

The smaller is Δt the better is the
approximation of Vt+Δt

The main disavantage: implementation
takes a long time

We suppose ρ,r to be constant
if r = 0
call0 timer = call0 B&S

put0 timer = put0 B&S
Timer Option
Greeks
The timer option, Greeks should be calculated in a numerical way.
For instances, using numerical approximation, we can express the first derivative of
our option as [f(x+Δx) – f(x – Δx)] / 2*Δx
However they can be expressed as expectations of the Greeks calculated in Black &
Scholes model, using the target volatility and the target maturity specified by the
investor in the model.
C0= Eq[Cbs(S0,K, r, (V/τ)^0.5, τ)]
Timer Option
Applications (Speculate)
Investors that have a strong expectation about the future stock price, and
want to exploit it, should buy a Call (Put) Timer option when they belive
that the implied volatility is greater respect to its effective volatility (in that
case the plain vanilla is over-valued).
A study that have analyzed all stocks in the Euro Stoxx 50 index
from 2000 to 2007, calculates that 80% of three-month calls that
have matured in-the-money were overpriced.
Timer Option
Applications (Stock Hedging)
A Timer Option could be used also to hedge a stock position, exploiting the
negative correlation that generally exists between stocks return and realized
volatility.
For instance, imagine that an investor wants to hedge himself against a collapse
of the market, and for this reason he buys a Timer Put.
If the market goes down, he is protected, and moreover this generally implies an
increase in the realised volatility (his Timer Option will expiry sooner)
On the contrary, in case of bullish market, that are usually markets with low
realised volatility, the expiry date for the Timer Put will be longer respect to a
simply Plain Vanilla Put.
This means that, the investors cost for hedging will be lower, the need to roll the
Timer Option is less frequent than for the Vanilla Option
Timer Option
Applications (Combination with a Plain Vanilla)
Timer Options could be used also to create combination with Plain Vanilla
Options:
If we think that the implied volatility of the market is too high:
• buy a Timer Call
• sell a Plain Vanilla Call
with same S and K (setting the Volatility Target below the implied
volatility)
If we are in a bullish market, the tenor of our Timer Option will be higher
respect to the Plain Vanilla, and so we will capture the value due to the
difference in time value of the two options.
At time t=0 The cost of the Timer Call is lower respect to the Plain Vanilla
If we are right, timer call will expire later than plain vanilla Call thus:
At time t=T we have plain Call= max(ST-k,0), timer Call= plain Call + time value