Exotics

Barrier options: the eight standard types

A barrier option is a vanilla whose existence is conditional on the underlying touching, or not touching, a stated level at any time during the life. Two option types times two barrier directions times two conditions gives eight contracts, and the whole family is spanned by two arithmetic relationships.

Field	Value
Formula	Knock-in payoff = VanillaPayoff * 1{barrier touched}; Knock-out payoff = VanillaPayoff * 1{barrier never touched}
The eight	Down-and-in call, down-and-out call, up-and-in call, up-and-out call, and the same four for a put
Monitoring	Continuously monitored barriers admit the closed forms below. Discretely monitored barriers are strictly less likely to knock and are therefore worth strictly more if knocking out, and less if knocking in
Rebate	A cash amount paid if a knock-out is triggered, or if a knock-in never triggers. Enters as the E and F terms and is zero for most listed structures
Reverse barrier	A knock-out whose barrier lies in the money, so the option is destroyed at the moment it is worth most. The up-and-out call with K below H is the standard case
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. With H = 105 on a K = 100 call, the up-and-out is worth 0.065637 against a vanilla of 4.485236 - it retains 1.46 percent of the vanilla value, because the barrier sits only 5 percent above spot and above the strike, so almost every path that would pay also knocks. The mirror up-and-in is 4.419599, and the two sum to the vanilla exactly. Moving the barrier out to H = 110 lifts the up-and-out to 0.659722, ten times the value for a 5-point move in the barrier

• The value of a reverse knock-out is dominated by the barrier and almost independent of the strike, which is the opposite of the intuition a vanilla builds. The up-and-out at H = 105 above is worth less than a hundredth of the underlying's daily range in premium terms.
• A discretely monitored barrier and a continuously monitored one can differ by more than the entire bid-ask spread. The monitoring frequency is a contract term, not a modelling detail, and it belongs in the term sheet.
• Every barrier is short or long a digital at the barrier, which is why the delta jumps there. That is the origin of every practical difficulty with the family.

Source: Merton 1973; Reiner and Rubinstein 1991

The Reiner-Rubinstein barrier closed form

All sixteen continuously monitored single-barrier prices are built from six terms. The construction uses the reflection principle: a barrier at H is handled by adding an image of the diffusion reflected in ln(H), scaled by a power of H over S that carries the drift.

Field	Value
Formula	With b = r - q, mu = (b - sigma^2/2)/sigma^2, lambda = sqrt(mu^2 + 2*r/sigma^2), v = sigma*sqrt(T), phi_sign = plus 1 for a call and minus 1 for a put, eta = plus 1 for a down barrier and minus 1 for an up barrier: x1 = ln(S/K)/v + (1+mu)*v; x2 = ln(S/H)/v + (1+mu)*v; y1 = ln(H^2/(S*K))/v + (1+mu)*v; y2 = ln(H/S)/v + (1+mu)*v; z = ln(H/S)/v + lambda*v
A	phi_sign*S*exp((b-r)*T)*N(phi_sign*x1) minus phi_sign*K*exp(-r*T)*N(phi_sign*x1 minus phi_sign*v). This is the vanilla
B	The same expression with x2 in place of x1
C	phi_sign*S*exp((b-r)*T)*(H/S)^(2*(mu+1))*N(eta*y1) minus phi_sign*K*exp(-r*T)*(H/S)^(2*mu)*N(eta*y1 minus eta*v). The reflected image
D	The same expression with y2 in place of y1
E	Rebate*exp(-r*T)*[N(eta*x2 minus eta*v) minus (H/S)^(2*mu)*N(eta*y2 minus eta*v)], the in-barrier rebate
F	Rebate*[(H/S)^(mu+lambda)*N(eta*z) plus (H/S)^(mu-lambda)*N(eta*z minus 2*eta*lambda*v)], the out-barrier rebate
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. Down-and-out call at H = 90 evaluates to 4.418182 and the down-and-in to 0.067055, summing to the vanilla 4.485236. Two independent checks were run rather than one. First, in plus out equals the vanilla across all 112 combinations of seven strikes from 85 to 115 and eight barriers from 80 to 120, with a worst deviation of 1.5e-14. Second, a Crank-Nicolson finite-difference solve of the Black-Scholes equation with a Dirichlet zero boundary at ln(90) returns 4.41824015 on a 400-by-400 grid, 4.41817188 on 1600-by-1600 and 4.41817923 on 3200-by-3200, against the closed-form 4.41818154 - agreement to 2.3e-06 at the finest grid. At H = 95 the closed form gives 3.62599049 and the same solver 3.62598931, agreeing to 1.2e-06

• The A term is the vanilla, so every out-barrier price is the vanilla minus a correction, and the correction is what the barrier costs. Reading the formula that way makes the sign of each term obvious.
• The (H/S) powers carry the drift through mu and lambda. At b = 0 they collapse and the formulas reduce to pure reflection, which is where the reflection-principle intuition is exact rather than adjusted.
• In-out parity is not a coincidence to be verified, it is an identity: a knock-in and a knock-out with the same barrier and strike together pay the vanilla on every path. Use it as a test of any implementation, including this one.

Source: Merton 1973; Reiner and Rubinstein 1991

In-out parity

A knock-in and a knock-out written on the same underlying with the same strike, expiration and barrier, and the same rebate structure of zero, together replicate the vanilla. It holds path by path, so it holds for any process and any monitoring convention.

Field	Value
Formula	KnockIn(K, H) + KnockOut(K, H) = Vanilla(K), for a zero rebate and identical monitoring
Why it is model-free	On any single path exactly one of the two options survives, and the survivor pays the vanilla payoff
Where it breaks	A rebate on either leg, or different monitoring conventions between the two, breaks it by exactly the value of the difference
Use	The primary implementation test for any barrier pricer, and the way a knock-in is normally quoted, as the vanilla minus the knock-out
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. At H = 105 on the call: in 4.419599 plus out 0.065637 equals 4.485236, the vanilla, to six decimals. At H = 90 on the put: in 2.695096 plus out 0.795124 equals 3.490220. Tested across 112 combinations of seven strikes and eight barriers, both option types, the worst absolute deviation was 1.5e-14, which is double-precision rounding and not method error

• Because parity is exact, quoting a knock-in and a knock-out that do not sum to the vanilla is an internal inconsistency in the quote, not a view.
• The identity says nothing about the delta of either leg. Two positions can sum correctly in value and both be badly hedged, and near a barrier they usually are.
• It also gives the cheapest correct way to price a knock-in: price the knock-out, subtract from the vanilla. That halves the code and removes a whole class of case-table errors.

The barrier hedging problem: a discontinuous delta at the barrier

A knock-out is worth its remaining value one tick above the barrier and zero at it, so the value function has a jump and the delta is unbounded as the barrier is approached. No finite position in the underlying hedges a jump, which is the entire practical difficulty with the family.

Field	Value
Formula	As S approaches H from above for a down-and-out, Value approaches 0 while Value at H plus epsilon is positive, so dValue/dS is unbounded
Reverse knock-out, the worst case	The barrier sits in the money, so the option is destroyed when its intrinsic value is largest. Delta near the barrier can exceed 1 in magnitude and flip sign
Standard static approach	Replicate with a strip of vanillas chosen so the portfolio is worth zero along the barrier, which converts a dynamic problem into a fixed position
Gap risk	A gap through the barrier realises the discontinuity in one print, and no rehedging frequency reduces it
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. Up-and-out call, K = 100, H = 105. Value 0.065637 with spot at 100. The contract is worth zero the instant spot touches 105 while the vanilla it tracks is worth 5.00 of intrinsic at that point, so 5.00 per share of value disappears at a single price. Compare the two barriers: at H = 105 the value is 0.065637 and at H = 110 it is 0.659722, so a 5-point barrier shift is worth 0.594085, nine times the entire premium of the tighter contract. The delta is also the wrong sign for a call: the same up-and-out is worth 0.095133 at S = 95, 0.065637 at S = 100, 0.041629 at S = 102, 0.014117 at S = 104 and exactly zero at S = 105, so it falls as the underlying rises. A hedge sized from a vanilla intuition would be long the underlying against a position that loses when the underlying gains

• The sensitivity to the barrier level dwarfs the sensitivity to every other input. On the worked pair, moving the barrier 5 points is worth nine times the premium, while a 5-point rise in spot destroys the contract entirely and a 5-point fall raises its value by 45 percent.
• Static replication with vanillas is the standard answer because it removes the need to trade through the barrier at all. Its cost is that the replicating strip is only exact under an assumption about the process, so it trades model risk for gap risk.
• A risk system that reports a finite delta near a barrier is reporting the delta of the grid it is using, not of the contract.

Digital and binary options in closed form

A digital pays a fixed amount, or the asset itself, conditional on finishing beyond a strike. Both variants are single terms of the Black-Scholes-Merton formula, which is why the vanilla decomposes into them exactly.

Field	Value
Formula	CashOrNothing call = R*exp(-r*T)*N(d2); CashOrNothing put = R*exp(-r*T)*N(-d2); AssetOrNothing call = S*exp(-q*T)*N(d1); AssetOrNothing put = S*exp(-q*T)*N(-d1)
Vanilla decomposition	Call = AssetOrNothing(K) minus K*CashOrNothing(K), exactly
Cash digital identity	Call plus put = R*exp(-r*T), the value of the certain payment, since exactly one of them pays
Asset digital identity	Call plus put = S*exp(-q*T), the value of the asset
Delta	CashOrNothing call delta = R*exp(-r*T)*phi(d2)/(S*sigma*sqrt(T)), which is unbounded as T goes to zero at S = K
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. with a 1.00 payout. Cash-or-nothing call = exp(-0.01)*0.519938810 = 0.99004983*0.51993881 = 0.51476533. Cash-or-nothing put = 0.47528451. The two sum to 0.99004983, exactly exp(-r*T), as required. Asset-or-nothing call = 55.96176924 and put = 44.03823076, summing to 100.00000000 which is S*exp(-q*T). Decomposition check: 55.96176924 minus 100*0.51476533 = 4.48523641, recovering the vanilla call to eight decimals

• The cash digital value is a discounted risk-neutral probability, so it is the cleanest market-implied probability available and it is not the delta. At the reference inputs it is 0.5148 against a delta of 0.5596.
• Digital delta and gamma blow up at the strike as expiry approaches, worse than any vanilla, because the payoff itself is a step. This is the same problem as pin risk with the smoothing removed.
• Because minus the dual delta of a vanilla equals the cash digital, a digital can always be priced off a strike curve without a separate model. If the two disagree, the strike curve is what to trust.

Source: Reiner and Rubinstein 1991; Cox and Rubinstein 1985

Static replication of a digital with a tight vertical

A digital is the limit of a call spread as the strike gap goes to zero with the notional scaled by one over the gap. That makes a vertical an exact hedge in the limit and a conservative over-hedge at any finite width, which is how digitals are actually risk-managed.

Field	Value
Formula	Digital(K, payout R) = limit as w goes to 0 of (R/w)*[Call(K - w/2) minus Call(K + w/2)]
Units	R/w contracts of the vertical. A 1.00 digital replicated with a 0.10-wide vertical needs 10 spreads per unit of payout
Direction of the finite-width error	The centred vertical is worth slightly more than the digital, so the replication over-hedges. The error is second order in w
Practical constraint	Listed strike increments set the minimum achievable w, and the notional scales as 1/w, so a tight replication is a large position
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. The 1.00 cash-or-nothing call at K = 100 is 0.51476533. Centred verticals: width 10.00, 0.10 units, value 0.51549939, error plus 0.00073407. Width 5.00, 0.20 units, 0.51496510, error plus 0.00019977. Width 2.00, 0.50 units, 0.51479805, error plus 0.00003273. Width 1.00, 1 unit, 0.51477354, error plus 0.00000821. Width 0.50, 2 units, 0.51476738, error plus 0.00000205. Width 0.10, 10 units, 0.51476541, error plus 0.00000008. The error falls by a factor of four each time the width halves, confirming second-order convergence

• Halving the width cuts the error by four and doubles the notional. That trade-off is the whole of digital replication and there is no width at which both are small.
• A 5.00-wide vertical on a 100 strike is already within 0.0002 of the digital, which on most desks is inside the spread. The exotic premium over a listed vertical has to be justified against that number.
• The over-hedge direction matters when selling: a short digital hedged with a short vertical of finite width is under-hedged, not over-hedged, because the sign flips.

Geometric-average Asian options have a closed form

The geometric average of a lognormal price path is itself lognormal, so a geometric-average Asian option prices in a Black-Scholes-style formula with a reduced volatility and an adjusted drift. Continuous averaging over the whole life divides the variance by three.

Field	Value
Formula	With m = ln(S) + (r - q - sigma^2/2)*T/2 and v = sigma^2*T/3: Price = exp(-r*T)*[ exp(m + v/2)*N(d1) - K*N(d2) ], d1 = (m - ln(K) + v)/sqrt(v), d2 = d1 - sqrt(v)
Effective volatility	sigma/sqrt(3), about 57.7 percent of the spot volatility, because averaging over [0,T] gives variance sigma^2*T/3
Effective forward	exp(m + v/2), which is below the spot forward because the average is taken over the whole path, not the endpoint
Relation to the arithmetic average	The geometric average never exceeds the arithmetic average, so the geometric-average option is a lower bound on the arithmetic-average option at the same strike
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. m = ln(100) + (0.04 - 0.02)*0.125 = 4.60767019. v = 0.04*0.25/3 = 0.00333333, so sqrt(v) = 0.05773503, which is sigma*sqrt(T)/sqrt(3) = 0.100000/1.732051 exactly. Effective forward exp(m + v/2) = 100.41753593. Price = 2.49745602 against a vanilla call of 4.485236 - the Asian is 55.7 percent of the vanilla. Independent check by Simpson quadrature over the terminal lognormal with 200,001 nodes: 2.49745602, agreeing to 2.5e-09. Monte Carlo cross-check with 250 averaging dates and 40,000 antithetic paths, seed 20260827: 2.494664 plus or minus 0.018014, within 0.16 standard errors of the closed form

• The volatility reduction to sigma over root three is the whole reason averaging structures are cheaper. It is a property of continuous averaging over the full life; averaging over a window inside the life gives a different and larger factor.
• The effective forward is below the spot forward. Comparing an Asian to a vanilla at the same strike therefore compares two options with different effective moneyness, not just different volatilities.
• Discrete averaging over n dates gives variance sigma^2*T*(n+1)*(2n+1)/(6*n^2), which converges to sigma^2*T/3 from above. Using the continuous formula on a monthly-averaged contract understates the volatility and therefore the price.

Source: Kemna and Vorst 1990

Arithmetic-average Asian options and moment matching

A sum of lognormals is not lognormal, so an arithmetic-average Asian has no closed form. The standard approximation matches the first two moments of the arithmetic average to a lognormal and then applies Black-76 to that surrogate.

Field	Value
Formula	M1 = S*(exp(b*T) - 1)/(b*T) with b = r - q; M2 = 2*S^2*exp((2*b + sigma^2)*T)/((b + sigma^2)*(2*b + sigma^2)*T^2) + (2*S^2/(b*T^2))*[1/(2*b + sigma^2) - exp(b*T)/(b + sigma^2)]; sigma_a = sqrt(ln(M2/M1^2)/T); Price = Black76(M1, K, T, r, sigma_a)
M1	The forward price of the arithmetic average, exact
M2	The second moment of the average, exact. Only the lognormal shape assumption is an approximation
Direction of error	The true distribution of an arithmetic average is less skewed than a lognormal, so the approximation is small and its sign depends on moneyness
Degenerate case	b = 0 makes M1 and M2 indeterminate as written; take the limits or perturb b
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. M1 = 100.50167084, M2 = 10134.42336710, so sigma_a = sqrt(ln(10134.42336710/10100.58579)/0.25) = 0.11566256. Black-76 on a forward of 100.50167084 at K = 100 with that volatility returns 2.54649178. Independent Monte Carlo with 250 averaging dates and 40,000 antithetic paths, seed 20260827: 2.541318 plus or minus 0.018307, so the approximation sits 0.28 standard errors above the simulation. Ordering check: the geometric-average price at the same inputs is 2.49745602, below the arithmetic 2.54649178 as required by the arithmetic-geometric mean inequality, and both are well below the vanilla 4.485236

• The moment-matched effective volatility of 0.1157 is above sigma over root three, 0.1155, by two ten-thousandths. That tiny gap is the entire difference between the arithmetic and geometric averaging problems at these inputs, and it is worth 0.049 of premium.
• The approximation degrades where the average is far from the strike and where the tenor is long, because the true average becomes visibly less lognormal. Check against a simulation before quoting it on anything long-dated.
• The geometric-average price is a strict lower bound, so it is a free sanity check that costs one closed form. Any arithmetic Asian price below it is wrong.

Source: Turnbull and Wakeman 1991

Lookback options

A lookback pays off against the extreme of the path rather than a fixed strike. A floating-strike lookback call pays the terminal price minus the running minimum, so it can never finish worthless, and a closed form exists because the joint law of the terminal value and the running minimum of a Brownian motion is known.

Field	Value
Formula	With b = r - q, a1 = [ln(S/m) + (b + sigma^2/2)*T]/(sigma*sqrt(T)), a2 = a1 - sigma*sqrt(T), a3 = [ln(S/m) + (-b + sigma^2/2)*T]/(sigma*sqrt(T)), y1 = -2*b*ln(S/m)/sigma^2: C = S*exp((b-r)*T)*N(a1) - S*exp((b-r)*T)*(sigma^2/(2*b))*N(-a1) - m*exp(-r*T)*[N(a2) - (sigma^2/(2*b))*exp(y1)*N(-a3)]
Payoff, floating strike call	S_T minus min over the life of S_t. Always non-negative, and strictly positive unless the terminal price is the minimum
Payoff, floating strike put	max over the life of S_t minus S_T
m	The running minimum observed so far. At inception m = S
Monitoring bias	Discrete monitoring cannot see the true continuous extreme, so a discretely monitored lookback is strictly worth less. The bias vanishes at order 1/sqrt(number of observations)
Worked	Reference inputs: S = 100, K = 100, T = 0.25, r = 0.04, q = 0, sigma = 0.20, at which the vanilla call is 4.485236 and the vanilla put is 3.490220. These are inputs chosen to make the arithmetic checkable, not observations of any market. with m = S = 100 at inception, so the option is at its floating strike. Closed form: 8.204387, which is 1.83 times the vanilla at-the-money call of 4.485236 - the premium for never being able to finish worthless. Monte Carlo with antithetic paths, seed 20260827, showing the monitoring bias explicitly: 250 monitoring dates and 40,000 paths gives 7.850105 plus or minus 0.032783; 1,000 dates gives 8.037098 plus or minus 0.033062; 4,000 dates and 20,000 paths gives 8.074975 plus or minus 0.045853. The estimates rise monotonically toward the continuous-monitoring value as the monitoring frequency increases, which is the signature of the known downward discretisation bias rather than of a formula error

• The Monte Carlo figures above do not equal the closed form and are not supposed to. Discrete monitoring is a different contract, worth less, and the gap at 250 dates is 0.354 - four percent of the price, and eleven standard errors. Anyone reconciling a lookback pricer against a simulation has to match the monitoring convention first.
• The sigma-squared over two-b factor is singular at b = 0. That is a removable singularity in the mathematics and a division by zero in code, and it is the most common implementation bug in this formula.
• A lookback cannot expire worthless, which is why its premium is roughly double an at-the-money vanilla. It is the extreme of the same trade-off every option makes and not a different kind of instrument.

Source: Goldman, Sosin and Gatto 1979

Cliquets as a strip of forward-starting options

A cliquet resets its strike to the prevailing price at each observation date, so it is a strip of forward-starting at-the-money options. When the periods are uncapped and unfloored, the strip values by homogeneity with no additional machinery, because a forward-starting at-the-money option is worth a fixed fraction of the spot.

Field	Value
Formula	Uncapped cliquet = sum over resets i of S*exp(-q*t_i) * BS(1, 1, tau_i, r, q, sigma), where tau_i is the length of period i and t_i its start
Homogeneity	BS is homogeneous of degree one in S and K, so a forward-starting at-the-money option is worth S*exp(-q*t_i) times the unit-strike value
Volatility exposure	Positive vega in every period, and no exposure to the level at all once the strike resets. It is a pure forward-volatility position
What caps and floors change	A cap on each period return breaks the homogeneity argument and reintroduces dependence on the forward-volatility skew, which no closed form covers
Worked	Four consecutive quarterly resets, S = 100, r = 0.04, q = 0, sigma = 0.20, each period 0.25 years. The unit at-the-money quarter call is BS(1, 1, 0.25, 0.04, 0, 0.20) = 0.04485236, so each period is worth 100*0.04485236 = 4.485236 and the strip is worth 17.940946. A single one-year at-the-money call at the same inputs is 9.925054. The cliquet is 1.8076 times the one-year option. Resetting the strike three times is what buys that: each quarter contributes the full value of a fresh at-the-money option rather than a share of one annual variance

• The uncapped cliquet is worth almost twice the single-period option of the same total length. Resetting the strike is what buys that, and it is why the structure exists at all.
• Every real cliquet is capped, floored, or both, and the moment it is, the value depends on the volatility of forward volatility. The clean arithmetic above is the boundary case, not the traded product.
• A cliquet has no delta at inception beyond the discounting, because each period's strike is unset. Risk reports that show a large delta on one are showing the first period only.

Source: Rubinstein 1991

Autocallable structure

A note that redeems early, paying accrued coupons, if the underlying is at or above an observation level on any scheduled date, and otherwise continues. At maturity, principal is repaid in full unless a downside barrier has been breached, in which case the investor takes the underlying performance. It has no closed form and its arithmetic is entirely in the schedule.

Field	Value
Formula	Payoff at the first observation date i with S_i at or above the call level: Principal + i * Coupon. If no call occurs: Principal if S_T is at or above the barrier, else Principal * S_T/S_0
Embedded positions	Long a coupon stream contingent on a sequence of digitals, and short a down-and-in put at the barrier struck at the initial level
Why no closed form	The payoff depends on the first passage across a sequence of discrete levels, which is a multi-dimensional first-crossing problem
Valuation method	Monte Carlo, or a lattice with the observation dates aligned to time steps
Worked	Stated terms chosen for checkability, not a real issue: principal 1,000.00, four semi-annual observations, call level 100 percent of the initial 100.00, coupon 30.00 per observation, downside barrier 70 percent observed at maturity only. If the first observation prints at 101, the note redeems for 1,000.00 plus 30.00 = 1,030.00 after six months, a 3.00 percent return over 0.5 years. If it survives to the fourth observation and prints at 100.50 there, it pays 1,000.00 plus 4*30.00 = 1,120.00 after two years. If it never calls and the final level is 85, above the 70 barrier, it pays 1,000.00 and the total return is zero. If the final level is 60, below the barrier, it pays 1,000.00*60/100 = 600.00, a loss of 400.00 against a maximum possible coupon stream of 120.00

• The maximum coupon and the maximum loss are wildly asymmetric by construction: 120.00 against 1,000.00 in the worked terms. That asymmetry is the structure, not a defect in it, and it is fully visible from the term sheet arithmetic.
• Early redemption is the outcome the structure is built to produce, which means the realised holding period is short in the cases that pay and long in the cases that do not. Any yield quoted to the final maturity describes the worst path.
• The embedded short put is a down-and-in, so its value is dominated by the barrier level and the skew there, not by the at-the-money volatility. Pricing one off a single volatility number misses most of the risk.

Compound options

An option on an option: at an intermediate date the holder may pay a second premium to acquire an underlying option that expires later. The Geske closed form expresses it through the bivariate normal, and the same value is obtained by one-dimensional quadrature over the intermediate price.

Field	Value
Formula	CallOnCall = exp(-r*t1) * E[ max( C(S_t1, K2, T2 - t1) - K1, 0 ) ], with the expectation over the risk-neutral law of S_t1
K1	The strike of the outer option, paid at t1 to acquire the inner option
K2 and T2	The strike and expiration of the inner option acquired
Closed form	Geske 1979, in terms of the bivariate normal distribution with correlation sqrt(t1/T2)
Degenerate check	K1 going to zero must return the value of the inner option today, since the outer option is then always exercised
Worked	S = 100, r = 0.04, q = 0, sigma = 0.20. Outer strike K1 = 1.50 payable at t1 = 0.10; inner call K2 = 100 expiring T2 = 0.35. Value by Simpson quadrature over the intermediate lognormal with 100,001 nodes: 3.981820. The inner call priced today is 5.412643, so the compound option costs 73.57 percent of the option it may buy while deferring 1.50 of the premium for 0.10 years. Degenerate check: setting K1 = 0 returns 5.412643, exactly the inner call, confirming the construction

• The compound option costs 73.6 percent of the underlying option in the worked case, and the total outlay if exercised is 3.981820 plus 1.50 = 5.481820, which is above the 5.412643 the option costs outright. The structure defers cash and pays for the deferral, exactly as the arithmetic requires.
• Because the payoff is a max against a value rather than against a price, the vega is second order: it is the vega of an option on a vega-bearing asset. Compound options are the standard example of positive volga.
• The one-dimensional quadrature is exact to quadrature error and needs no bivariate normal, which makes it the easier implementation to verify. Use Geske's form for speed and the quadrature to test it.

Source: Geske 1979

Chooser options

A simple chooser lets the holder decide at an intermediate date whether the contract is a call or a put, both struck at the same K and expiring at the same T2. It is worth less than a straddle, because the choice is made once at t1 rather than kept to expiry, and it has a closed form.

Field	Value
Formula	V = S*exp(-q*T2)*N(d) - K*exp(-r*T2)*N(d - sigma*sqrt(T2)) - S*exp(-q*T2)*N(-y) + K*exp(-r*T2)*N(-y + sigma*sqrt(t1)), d = [ln(S/K) + (r - q + sigma^2/2)*T2]/(sigma*sqrt(T2)), y = [ln(S/K) + (r - q)*T2 + sigma^2*t1/2]/(sigma*sqrt(t1))
Equivalent decomposition	A call struck K expiring T2, plus a put struck K*exp(-(r-q)*(T2 - t1)) expiring t1 on the forward
Bounds	At least max(call, put) today, and at most the straddle. Both limits are attained at t1 = 0 and t1 = T2
Worked	S = 100, K = 100, r = 0.04, q = 0, sigma = 0.20, choice date t1 = 0.10, expiration T2 = 0.35. Closed form: 7.28380200. Independent check by Simpson quadrature over the intermediate lognormal, valuing max(call, put) at t1 with 100,001 nodes: 7.28380201, agreeing to 3.9e-09. Context: the T2 call is 5.412643, the T2 put is 4.022397, the straddle is 9.435040, so the chooser is 77.20 percent of the straddle. Boundary checks: at t1 = 0.349999, effectively the expiration, the formula returns 9.435034, converging on the straddle; at t1 = 1e-08 it returns 5.412643, exactly max(call, put) today

• The chooser is 77 percent of the straddle for a choice made 29 percent of the way through the life. The value of optionality about direction decays much faster than the value of the direction itself.
• Both boundary checks are worth running on any implementation: the t1 to zero limit and the t1 to T2 limit are known exactly, and an implementation that misses either has a sign or a tenor wrong.
• A complex chooser, where the call and put have different strikes or expirations, has no such simple form and requires the bivariate normal or numerics. The word simple in the name is load-bearing.

Source: Rubinstein 1991

Quanto adjustment

A quanto pays a foreign-asset payoff in the domestic currency at a fixed exchange rate. Converting the payoff at a fixed rate changes the drift of the foreign asset under the domestic risk-neutral measure by the covariance between the asset and the exchange rate, and that correlation term is the entire adjustment.

Field	Value
Formula	Price a quanto with the domestic rate r_d for discounting and an effective dividend yield q* = q_f + r_d - r_f + rho*sigma_S*sigma_FX, then apply the standard Black-Scholes-Merton formula
rho	Correlation between the foreign asset return and the exchange rate, quoted in the convention where a rise means the domestic currency weakens
Sign of the effect	A positive rho raises q*, which lowers a call and raises a put. A negative rho does the reverse
Magnitude	rho*sigma_S*sigma_FX, a product of three numbers each below one, so it is a small adjustment to the drift and a visible one to the price
Worked	S = 100, K = 100, T = 0.25, sigma_S = 0.20, sigma_FX = 0.10, r_d = 0.04, r_f = 0.02, q_f = 0.01, all chosen for checkability. q* = 0.01 + 0.04 - 0.02 + rho*0.02. At rho = minus 0.50, q* = 0.020000 and the quanto call is 4.211052. At rho = minus 0.25, q* = 0.025000 and the call is 4.144260. At rho = 0, q* = 0.030000 and the call is 4.078169. At rho = plus 0.25, q* = 0.035000 and the call is 4.012777. At rho = plus 0.50, q* = 0.040000 and the call is 3.948082. Moving rho from minus 0.5 to plus 0.5 changes the call by 0.262970, which is 6.45 percent of the rho = 0 value of 4.078169. For comparison, the same option priced with the domestic rate and the raw foreign yield, ignoring the quanto adjustment entirely, is 4.346739 - an error of 0.268570 at rho = 0, larger than the entire correlation range

• The rate differential term r_d minus r_f is usually larger than the correlation term and is the part most often left out. In the worked case it is worth 0.02 of yield against 0.01 for a rho of 0.5.
• The correlation input has no market quote in most cases, so a quanto price carries an unhedgeable parameter. The honest presentation is a price range across a plausible rho, not a single number.
• Vega on a quanto is with respect to sigma_S, but sigma_FX enters through q*, so the position has an exchange-rate volatility exposure with no exchange-rate delta. That is the defining feature of the structure.

Source: Reiner 1992