Exotic Pricing Example

Here are some exotic contract details quoted on a chat between a trader and an interbank exotics broker on July 30, 2013: 3mth 30 oct-1 nov 13 tky/0.8100 aud/usd ot/spot 0.9080/tv 5.35% vol 11.4/del A$2.1/–55/0.30/A$1-2 vh.

Translating, that is:

• Contract type: one-touch (ot): A one-touch contract pays out cash at maturity if the barrier level trades prior to maturity.
• Currency pair: AUD/USD (aud/usd).
• Expiry: 3mth, and the exact dates are specified for clarity: expiry date: October 30, 2013/delivery date: November 1, 2013 (3mth 30 oct-1 nov 13).
• Cut: Tokyo (tky).
• One-touch barrier level: 0.8100.
• Notional: AUD1m to AUD2m (A$1-2).

This is enough information to enter the contract details into a pricing tool as shown in Exhibit 18.2.

Reference AUD/USD market data is also given:

• Spot: 0.9080 (spot 0.9080)
• 3mth swap points: –55/USD deposit rate: 0.30% (–55/0.30)
• 3mth ATM volatility: 11.4% (vol 11.4)

This market data should be close to mid values observed in the market at the time of pricing. Exhibit 18.3 shows the market data in the pricing tool and Exhibit 18.4 shows the pricing tool outputs.

Under Black-Scholes, market data to a given expiry date is fully defined using spot, forward, one interest rate, and the volatility. Therefore, these exotic contract details could be priced by any trading desk in the market using the supplied market data and the same TV would be generated.

By entering the trade details and market data, the resultant TV can be checked against a reference. This TV matching helps confirm that the correct contract is being priced. Given the extra parameters within exotic contracts this additional safety is important. TV matching on exotic contracts occurs when trading in the interbank broker market plus some institutional clients also use it.

Going back to the example trade, the TV shown in the pricing tool is 5.35 AUD%, which matches the broker details (tv 5.35%). The delta in the pricing tool is 214AUD% which also matches the broker details (del A$2.1). Once the TV has been matched, the trader quotes a price in the form of a two-way TV adjustment. The key questions are therefore: What does the TV adjustment represent, and how is it calculated?

Pricing the Volatility Smile

The primary element within the TV adjustment is usually the volatility smile. Therefore, the first step in understanding exotic pricing is knowing how to estimate the impact of the volatility smile for a given contract.

Starting from first principles, exotic contracts with “good” features compared to the Black-Scholes model will cost more than TV to buy (a positive TV adjustment) and exotic contracts with “bad” features compared to the Black-Scholes model will cost less than TV to buy (a negative TV adjustment).

In the context of the volatility smile, long smile exposures (i.e., long-skew or long-wing exposures) are “good” features while short smile exposures (i.e., short-skew or short-wing exposures) are “bad” features. Therefore, the exotic contract is assessed to determine whether it has long smile exposures, in which case it will have a positive TV adjustment and will “trade over TV,” or it has short smile exposures, in which case it will have a negative TV adjustment and will “trade under TV.”

Just as the volatility smile can be split into the wings and the skew (as described in Chapter 12), exposures to the volatility smile can be split into the same two elements: exposure to the wings of the smile and exposure to the skew of the smile. A vega versus spot profile of an exotic contract is a simple method for assessing smile exposures and hence whether a contract will trade over or under TV.

Pricing the Skew

In a currency pair with a topside risk reversal, the vega profile of a long risk reversal position gets longer to the topside and shorter to the downside as shown in Exhibit 18.5.

The zeta on the long topside strike within this risk reversal will be positive and greater than the zeta of the short downside strike, which may well be negative. Therefore, net zeta on this risk reversal is positive. By definition, buying a risk reversal results in a long smile position.

A similar methodology can also be applied to exotic contracts. In a currency pair with the risk reversal for topside, if the vega exposure on an exotic contract gets longer to the topside and shorter to the downside, the exotic is said to be “long risk reversal” or “long skew.” In the same currency pair, if the vega gets longer to the downside and shorter to the topside, the exotic is said to be “short risk reversal” or “short skew.”

More generally, if the vega exposure on an exotic contract gets longer on the higher side of the volatility smile, the exotic is said to be “long risk reversal.” If the vega gets longer on the lower side of the volatility smile, the exotic is said to be “short risk reversal.”

Exhibit 18.6 shows the vega versus spot profile from a long 1yr 130.00 one-touch contract. As stated previously, a one-touch contract pays out a fixed amount of cash at maturity providing spot has touched the barrier level throughout the life of the option. When calculating a TV adjustment on a single exotic contract, a long position in the option is assumed.

Intuitively, the long one-touch option is long vega because higher spot volatility makes the barrier knock, and hence the payout, more likely. With spot at 90.00, the one-touch vega profile in Exhibit 18.6 is “long risk reversal” because vega gets longer to the topside and shorter to the downside, exactly like a long risk reversal position in this currency pair with topside skew.

Therefore, in this example, with spot at 90.00 and a risk reversal for topside, the one-touch will trade over TV (i.e., it will have a positive TV adjustment).

Buying this one-touch contract makes a trading position longer topside vega. To hedge this vega exposure, vanilla options with topside strikes must be sold. These topside vanilla options trade at a higher implied volatility than the ATM due to the risk reversal being for topside. Selling vanilla options at a higher volatility than the ATM earns zeta. Therefore, the exotic must cost more than TV: If it were possible to buy the one-touch under TV and then sell topside vanilla options on the hedge over TV, this would be a (very weak) form of arbitrage.

When hedging the one-touch contract with vanilla options, the smile value of the vanilla hedge should be approximately equal (but negative) to the smile value of the exotic option. For example, if the one-touch contract was bought in USD1m at a TV adjustment of +USD6.5%, that implies +USD65k of smile value has been purchased. On the vanilla hedge, a similar amount of smile value should be sold.

Pricing the Wings

The wing exposure of the exotic contract must also be considered within pricing. This can be done using the butterfly contract. The vega on a long butterfly contract gets longer to both the topside and downside, as shown in Exhibit 18.7.

Ignoring broker fly complications, volatility smiles have positive wings and the sum of the zetas on the topside and downside same-delta strikes within the butterfly will be positive.

By definition, buying a butterfly results in a long smile position that, if priced within the exotics framework, would have a positive TV adjustment.

Assuming the exotic contract is vega hedged (which a butterfly is by construction), if the vega gets longer in the wings (i.e., away from current spot), the exotic is said to be “long wings” or “long fly.” If the exotic vega gets shorter in the wings, the exotic is said to be “short wings” or “short fly.”

Exhibit 18.8 shows the vega versus spot profile from a long 1yr 80.00/100.00 double-no-touch (DNT). A double-no-touch contract pays out cash at maturity providing spot hasn't touched either of the two barrier levels throughout the life of the option. Therefore, the long double-no-touch contract is short vega; lower spot volatility makes it more likely that spot stays within the range to get the payout.

It is hard to judge from Exhibit 18.8 whether the double-no-touch contract is long wings or short wings. Hedging the vega at current spot with an ATM option gives more clarity. This is valid since hedging with the ATM does not affect the smile position of the exotic contract plus hedge.

Exhibit 18.9 shows the vega versus spot profiles from a long 1yr 80.00/100.00 double-no-touch and its ATM vega hedge separately while Exhibit 18.10 shows the aggregate vega profile from the double-no-touch plus the ATM vega hedge.

With spot at 90.00, the double-no-touch is said to be “long wings” since the vega hedged long double-no-touch gets longer vega in the wings, exactly like a long butterfly position.

Therefore, in this example, with spot at 90.00, the double-no-touch will trade over TV, i.e., it will have a positive TV adjustment.

Buying this double-no-touch contract with vega hedge makes the position longer vega to both the downside and topside. To hedge this wing vega exposure, topside and downside wing vanilla options must be sold. On average, wing vanilla options trade at a higher implied volatility than the ATM due to the shape of the volatility smile. Selling vanilla options at a higher volatility earns zeta. Therefore, the exotic must cost more than TV because buying it allows vanillas that trade higher than TV to be sold on the hedge. If it were possible to buy the double-no-touch under TV and then sell wing vanilla options on the hedge over TV, again, this would be a form of arbitrage.

As in the risk reversal case, when hedging the double-no-touch contract with vanilla options, the smile value of the vanilla hedge should be approximately equal (but negative) to the smile value of the exotic option.

VVV Pricing Example: Part 1

VVV (vega/volga/vanna) pricing formalizes this approach. Second-order vega Greeks are used to measure the exposure the exotic contract has to the skew and wings of the volatility smile:

• Vanna gives the exposure of the exotic contract to the skew of the volatility smile.
• Volga gives the exposure of the exotic contract to the wings of the volatility smile.

VVV is a heuristic rather than a model. One possible implementation replicates the vega, volga, and vanna exposures in the exotic contract using ATM, 25d call, and 25d put vanilla options to the same maturity as the exotic. The TV adjustment of the exotic is estimated by calculating the cost of the vanilla replication “on the smile” (i.e., its cumulative zeta) and weighting this cost by the stopping time (explained later in this chapter) of the exotic contract.

This methodology can be applied to the example broker AUD/USD 3mth one-touch with 0.8100 barrier contract that was introduced earlier in the chapter. In AUD/USD the risk reversal is for downside and the one-touch has a downside barrier also. Exhibit 18.11 shows the AUD/USD 3mth volatility smile on the deal horizon while Exhibit 18.12 shows the vega profile from the one-touch option.

Under Black-Scholes, and with all exposures quoted in AUD% terms, the long one-touch contract has:

• Vega: 1.99%
• Vanna: –47.1%
• Volga: 0.29%

The signs of these exposures should not be a surprise: Vega is long because higher spot volatility increases the chance of the barrier touching, vanna is negative because vega gets longer to the downside at current spot, and volga is positive because vega overall gets longer in the wings.

The AUD/USD 3mth 25d call vanilla has a 0.9360 strike and:

• Vega: 0.165%
• Vanna: 1.95%
• Volga: 0.006%

The AUD/USD 3mth 25d put vanilla has a 0.8660 strike and:

• Vega: 0.115%
• Vanna: –1.85%
• Volga: 0.007%

The vega component can be ignored since it does not impact the smile exposures. What remains is a system of two linear equations with two unknowns—the notionals of the topside and downside vanilla options respectively, n_call and n_put:

which solves to give:

Therefore, the vega profile from the one-touch can be approximately replicated by buying 31.6× the one-touch notional of 25d downside vanilla options and 5.8× the one-touch notional of 25d topside vanilla options. Therefore, if AUD2m of this one-touch contract were bought, approximately AUD60m of 25d put options and AUD10m of 25d call option could be sold to hedge the vega profile.

On the smile, the 3mth 25d topside vanilla is marked at 10.45% implied volatility, which equates to –0.14 AUD% zeta (recall that the ATM volatility is 11.4%), and the 25d downside vanilla is marked at 12.95% implied volatility, which equates to +0.25 AUD% zeta. Therefore, the replication has a cost on the smile equal to . The positive TV adjustment implies that the one-touch option has long smile exposures, as expected since vega gets longer to the downside and the risk reversal in AUD/USD is also for downside.

The final step in the VVV methodology is to weight this cost by the stopping time of the contract.

Stopping Time

Stopping time (also known as first exit time or expected life) is the expected length of time an exotic option will stay alive. This is useful information about the risk on the trade and it is an important measure within exotic option pricing and risk management.

Stopping time takes a value between 0% and 100%, expressed as a percent of the life of the option. Therefore, if the stopping time of an exotic option is 50%, the option is expected to live for half of its life. The stopping time of a European vanilla option is 100% since the option always lives right up to expiry.

The most common reason an exotic option might not live through to expiry is that it contains continuously monitored barriers (called American barriers). If spot is close to an American barrier, stopping time is low. If spot is far from the barrier, stopping time is high. Higher volatility reduces the stopping time since barriers are more likely to trigger earlier. Exhibit 18.13 shows how the stopping time reduces (expected barrier knock sooner) as volatility increases.

It is important to appreciate that stopping time doesn't depend on the option payoff, only the relative positioning of barriers within the contract.

Stopping time on American barrier options is conceptually similar to the valuation of a no-touch contract (see Chapter 23) that pays out a fixed amount of cash at maturity if barrier levels do not touch prior to expiry. However, the no-touch value will be lower than stopping time because if the barrier trades prior to expiry, the no-touch has zero value where realized stopping time is non-zero. Exhibit 18.14 compares stopping time with the TV of an equivalent no-touch contract.

Stopping time is also an important measure for target redemption options (see Chapter 28) where the option expires based on a target.

When stopping time is displayed in a pricing tool it is important to understand what methodology is used to calculate it: Is a single ATM volatility to expiry, the full ATM curve but no smile or the full volatility surface used? Are single interest rates used or is the full interest term structure used?

VVV Pricing Example: Part 2

Back to the VVV pricing example: The stopping time of the example AUD/USD 3mth 0.8100 one-touch contract is 98.6%—very close to 100% since the barrier is far from current spot (given its maturity). Applying this weighting to the cumulative zeta from the replication has minimal impact: TV Adjustment = +7.1 AUD% × 98.6% = +7.0 AUD%. This VVV TV adjustment is fairly close to the market TV adjustment of +5.35%, although given the market bid–offer spread for this contract would be around 2.0%, the VVV TV adjustment is not accurate enough to be used in practice.

The advantages of VVV are that it is very quick to calculate and it gives intuition about the risks on the trade: By analyzing vanna and volga exposures it is possible to judge whether an exotic is long or short skew and long or short wings at current spot. Also, VVV suggests an appropriate vanilla hedging strategy. In the AUD/USD one-touch example, the majority of the smile risk can be hedged by selling 30× the one-touch notional of 25d downside vanilla options.

However, VVV prices do not consistently match the market because only exposures at current spot are used within the price and the fact that exposures change over time or at different spot levels is ignored. Other problems include valuation jumps near barriers, and if the skew within the volatility smile is large, it is possible to generate VVV TV adjustments larger negatively than TV itself. Historically, much effort was put into applying fixes to the VVV methodology to adjust for its deficiencies. For example, different delta vanilla options were used within the replication, more advanced exposures (e.g., ) were added, or different weightings or floors and caps were used. However, for exotic option pricing, most FX derivative trading desks have now moved to using pricing models calibrated to the whole volatility surface rather than a VVV-based approach.

Finally, note that this is stylized analysis. Issues regarding differing vanna exposures from ATM contracts with different premium currencies and TV exposure versus smile exposure inconsistencies have been brushed under the carpet; for most traders these aren't important concerns.

Path Dependence

A key feature of exotic options is their path dependence. Path dependence means that the exotic option payoff is affected by the path that spot takes over the life of the option. Vanilla options are path independent because their payoff depends only on the spot level at the option expiry. Although note that when a vanilla option is delta hedged infrequently within a trading portfolio, the P&L from the option and delta hedges is highly path dependent. Within exotic contracts, the presence of barriers, averages, or targets makes the option path dependent.

For pricing and risk management the consequence of path dependence is that the full ATM curve and interest rate curve must be used to value options and these factors must be included within the TV adjustment. Alternatively, some bank trading desks use two TV values: an ATM TV and a term structure TV.

This is worth restating to be as clear as possible: Trading desks use forward curves, interest rate curves, and the volatility surface for valuing derivative contracts. For a vanilla option, only the forward, interest rate, and implied volatility for the specific strike and expiry date is used for pricing. For a path-dependent exotic option, the entire interest rate term structure and volatility surface (up to the option maturity) must be incorporated into the pricing, hence a different pricing methodology is required.

Traders learn which exotic option contracts have significant path dependence. For example, consider a window barrier option (see Chapter 26) with a knock-out barrier that is only active for the first month of the trade. If the ATM curve is upward sloping (short-date ATM volatility lower than long-date ATM volatility), the single volatility TV calculation will overestimate the chance of the window barrier knocking.