Forward-Start Options



Forward-Start Options

Mark Rubinstein

December 9, 1990

(published under the title: "Pay Now, Choose Later," in RISK 4 (February 1991), p. 13)

How much would you be willing to pay now for the following opportunity related to a prespecified underlying asset: after a known elapsed time t in the future (the "grant date"), you will receive at no extra cost a call option with time to expiration T-t, and with a strike price set so that the call will be at-the-money at the time the option is granted? An example of such a forward-start option can be found in corporate incentive stock option arrangements, where employees look forward to receiving options, which are at-the-money on the day of grant.

If we assume only that

(1) homogeneity: the call option value, when it is granted, will be homogeneous of degree one in the underlying asset price and the strike price

(2) state variable: all uncertainty in valuing the option after time t is resolved once the underlying asset price after time t is known

(3) date-invariance: the variables determining the value of the option are not date-dependent

(4) payout: the underlying asset through the grant date has a known constant payout rate d,

such an opportunity is surprisingly easy to value and to hedge.[1]

Let: S ( current value of underlying asset,

St ( (uncertain) value of underlying asset after time t,

C(X, Y, T–t) ( value of a call with concurrent underlying asset price X, strike price Y, and time to expiration.

From the homogeneity assumption, the value of the forward-starting at-the-money call on the grant date can be written as:

C(St, St, T–t) = StC(1, 1, T–t)

From the date-invariance assumption, no date subscript is required for the function C. From the state variable assumption, C(1, 1, T–t) is known in advance since it is none other than the current value of an at-the-money option with underlying asset price equal to 1 and with time to expiration T–t.

If we can somehow arrange to make an investment now which will for sure produce exactly the cash flow StC(1, 1, T–t) after time t, then the current cost of this investment must be the value of the forward-start option. Interpreting C(1, 1, T–t) as a number of shares, to replicate now the value of the option after time t we need to hold C(1, 1, T–t) shares of underlying asset. Using the payout assumption, correcting for the loss of dividends over time t,

Sd-tC(1, 1, T–t)

is then the current value of the forward-start option. Again, using the homogeneity assumption, we can rewrite this as

d-tC(S, S, T–t).

In other words, the value a forward-start option is simply the current value of d-t calls which are currently at-the-money, with time to expiration T–t.

Ignoring dividends, replicating such an option up to the grant date is quite simple: from the current time to the grant date, we simply need to hold C(1, 1, T–t) shares of stock. Since C(1, 1, T–t) is a constant, a buy-and-hold (in contrast to a dynamic) strategy is required. Such a strategy is, of course, trivially self-financing.

We can quickly generalize these results in a number of ways. Without further complicating the above proof, the terms of the forward-start option can easily be generalized to permit the granting of options which are "proportionally in- or out-of-the-money." That is, we can write the contract so that on the grant date the call is worth C(St, (St, T–t), where ( is a prespecified positive constant. Then, the current value of the option will be d-tC(S, (S, T–t).

In some cases, we may be uncertain about the grant date perhaps because an employer is unwilling to make a definite commitment. In that case, if there are to be no payouts prior to the grant date, our results continue to hold. Since the seller can perfectly hedge his sale by continuing to hold C(1, 1, T–t) shares until whenever the grant date occurs, the seller can afford to sell the forward-start option for the current cost of this portfolio, namely SC(1, 1, T–t). However, if payouts are positive, then the longer the grant date is postponed, the lower the current value of the option since the holder of a forward-start option gets no benefit of payouts prior to the grant date.

Finally, if receipt of the option requires that we continue to be employed, if the probability of continued employment is independent of the underlying asset price, and if we can assume risk-neutrality toward the risk of being fired, then the above formula for the value of a forward-start option needs to be adjusted downward by multiplying it by one minus the probability of being fired.

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    [1] These assumptions hold for many approaches to option pricing including the Black-Scholes option pricing formula and its generalization to binomial price movements as well as for the jump-diffusion formula developed by Robert Merton.

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