# An Innovative way of making options

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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, , 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 + 𝑉 𝑆𝑡 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