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I want to value the following contract for times $0<t<T$, i.e. determine $V(t,\cdot)$ where $\cdot$ refers to all other dependences (strike, spot, volatility, etc.). The contract is long and has a lot of non-standard features. Any suggestions would be helpful (feel free to exclude some of the features of the contract if you aren't able to offer up a solution that includes them; hopefully we will eventually get to a point where we can value the entire contract).

We have two parties, A and B, that have entered into the following premium-offsetting option contract on an underlying equity whose price is $S$.

Option 1

  • Type: Put
  • Exercise: European
  • Strike: $K$
  • Expiry: $T$
  • Buyer: Party A
  • Seller: Party B

Option 2

  • Type: Call
  • Exercise: American
  • Strike: $K$
  • Expiry: $T$
  • Buyer: Party B
  • Seller: Party A

Both options are written on a quantity $N$ of the underlying, and there is no up-front premium due from either party (the option premiums are designed to be 0 on a net basis).

Furthermore, there are a number of additional features written into the contract.

Simultaneous Exercise If Party B exercises $N'\leq N$ options prior to expiration, then $N'$ puts held by Party A will be deemed exercised.

Payoff Determined by Post-Exercise Averaging Period Upon exercise by Party B (or automatic exercise at maturity), Party A will determine a $k$-consecutive day averaging period, acting in good faith based on the number and value of the shares to be paid/delivered, as exercised by Party B. For this purpose, $k\leq15$, unless Party A is advised by counsel that $k>15$ is reasonably necessary or advisable in order to effect its hedging, hedge-unwind, or other settlement obligations to Party B.

The relevant price recorded on each averaging date is the dollar volume-weighted average price per share (Bloomberg "VWAP Price").

The final payoff is determined on the final averaging date as the volume weighted average of the recorded relevant prices.

Strike Price Reduction

If the equity pays an extraordinary dividend $D$, then the $K\mapsto K-D$ for each call and put option that remains unexercised (i.e. the strikes of both options are reduced by the amount of the extraordinary dividend).

Extraordinary dividend is defined as the aggregate amounts of (as determined by Party A) (a) any dividend declared on shares at a time $t'$ when the issuer had not previously paid dividends for $t\in[t'-1,t']$ (i.e., the past 4 quarters); (b) dividend amounts paid in excess of the regular dividend; (c) dividends expressly paid outside the normal course of operations or dividend schedule. In all cases, the ex-dividend date $\tilde{t}$ must satisfy $0<\tilde{t}<t$.

For (b), the regular dividend is specified as $0$ in the contract. Essentially this implies all dividends are extraordinary. I might as well state now that the contract was written on LAZ in 3/22/2012, and looking at the http://www.nasdaq.com/symbol/laz/dividend-history it seems they have had a pretty consistent dividend payout since 2005. Although, it seems they skipped their usual February dividend that year, so perhaps this is why the regular dividend is zero.

Additional Premium Due to Party A on Settlement Dates

For each exercise date (including the automatic exercise at maturity), there is a corresponding compounding period, the resulting value of which is due to Party A from Party B upon Party B's exercise.

For each day in the compounding period, commencing anew either from the beginning of the contract $t=0$ or from the most recent exercise date and up-to the next exercise date or final maturity date, Party B will pay Party A upon exercise the sum of the daily adjustments during the compounding period.

For each day $t_{j}$ in the $k$th compounding period, the daily adjustments are defined as $$D^{k}_{j}=N^{k}\cdot r_{j}\cdot(K^{k}-D^{k}_{j})$$ where $N^{k}$ are the number of call options remaining during the $k$th compounding period; $r_{j}$ is the daily rate as defined below; $K^{k}$ is the dividend adjusted strike price (as defined above) as of the beginning of the compounding period; $D^{k}_{j}$ is the aggregate dividend adjustment starting from the beginning of the $k$th compounding period and upto time $t_{j}$.

Thus,

$$P^{k}=\sum_{j}D^{k}_{j}.$$

The daily rate is defined as a per annum rate given by $$r_{j}=L(1)+\frac{\alpha}{360}$$

where $L(1)$ is the US 1 Month LIBOR rate and $s=35$ bps for $0\leq t\leq T/2$ and $s=50$ bps for $T/2\leq t\leq T$.

The contract seems to emphasize that $r_{j}$ should be expressed as a per annum rate, as opposed to a per diem rate as suggested by its purpose for computing the daily adjustment. This seems strange to me. Also, it is not clear from the contract whether the 1M LIBOR is adjusted for each day or per month. The contract states "...determined as if the first day of such compounding period were the reset date." To me, this suggests the rate is adjusted per month beginning with the compounding period (I also assume the actual rate is fixed 2 business days before the reset date as is standard for USD denominated swaps).

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  • $\begingroup$ I'll take a look at this tomorrow, seems like it shouldn't be too hard, as a lot of the "features" in there just seem to be to make things fairer (ie the averaging period, the dividend adjustment on the strike), and so they're not worth anything. $\endgroup$ – will Jul 4 '16 at 21:07
  • $\begingroup$ Could you clean it up a little bit? the formula for the daily adjustments reused the same variable ($D_j^k$), and then you mention $s$ later, but it's not in any formulae. For the averaging period, you should be able to ignore this, since the expected value is jsut the average of the fwd curve. Also, from what you've written, it seems that the additional premium payable to A cancels out the call options - and so really it's just an option to exercise A's puts (which look like calls to B), so it's just an american call with forced exercise at expiry. Does that sound reasonable? $\endgroup$ – will Jul 4 '16 at 22:20
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The first thing that occurs to me is that Party B controls the exercise of both Option 2 and Option 1, so it is equivalent to Party B owning 2* the notional of Option 2. Have I misunderstood something?

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  • $\begingroup$ This is how I see it too, often with these options it's just a case of working out what the strange wording of the term sheets mean. Likewise, the averaging period can be dealt with by adjusting the vol accordingly. $\endgroup$ – will Jun 5 '16 at 13:32

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