inverse-options-pricing

Inverse (Coin-Settled) Options Pricing

Deribit BTC/ETH options are inverse-settled: 1 contract has a notional of 1 BTC (or 1 ETH), premium is quoted in coin, payoff at expiry is in coin. This changes the Greeks.

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Step 1: Understand the Payoff

For a Deribit BTC call with strike K, at expiry with index price S_T:

payoff_in_BTC = max(S_T − K, 0) / S_T       (call)
payoff_in_BTC = max(K − S_T, 0) / S_T       (put)

The / S_T divisor is the inverse-settlement twist. As S_T rises, the call payoff in BTC grows slower than a USD call (and the BTC-denominated value is bounded by 1 BTC even as S_T → ∞).

USD payoff (for comparison): payoff_in_USD = max(S_T − K, 0) exactly as a vanilla call.

The two are related by:

coin_payoff = USD_payoff / S_T

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Step 2: Pricing Formula

Two equivalent ways to think about it:

Method A — Price in USD, convert to coin

1. Compute the vanilla USD BS price C_usd(S, K, T, r, σ).
2. Divide by spot: C_coin = C_usd / S.

This is what Deribit's mark_price field returns.

Method B — Risk-neutral coin-numeraire

Under the coin-as-numeraire measure, the drift of S becomes r − q + σ² (the additional σ² is the inverse-numeraire correction). The pricing formula has different d1/d2 inputs.

For practical work, Method A is simpler and gives identical results for European inverse options. Use it.

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Step 3: Inputs

Field	Notes	Default
S	Spot (BTC index = btc_usd)	required
K	Strike (USD)	required
T	Time to expiry in years (DTE/365)	required
r	Risk-free; Deribit historically uses 0	0
q	Dividend / staking yield (BTC = 0; ETH staking ≈ 3-4%)	0
σ	IV (decimal)	from Deribit mark_iv
type	call or put	required

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Step 4: Compute (Python)

from math import log, sqrt, exp, pi
from scipy.stats import norm

def usd_bs_price(S, K, T, r, sigma, q=0, opt='call'):
    if T <= 0 or sigma <= 0:
        return max(S - K, 0) if opt == 'call' else max(K - S, 0)
    d1 = (log(S/K) + (r - q + 0.5*sigma**2)*T) / (sigma*sqrt(T))
    d2 = d1 - sigma*sqrt(T)
    if opt == 'call':
        return S*exp(-q*T)*norm.cdf(d1) - K*exp(-r*T)*norm.cdf(d2)
    return K*exp(-r*T)*norm.cdf(-d2) - S*exp(-q*T)*norm.cdf(-d1)

def deribit_inverse_price(S, K, T, r, sigma, q=0, opt='call'):
    """Returns premium in coin units (BTC for BTC options)."""
    return usd_bs_price(S, K, T, r, sigma, q, opt) / S

To compare against Deribit's mark_price, this should match within rounding.

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Step 5: Inverse Greeks (the trick)

Deribit returns Greeks in coin-per-coin-of-underlying-move units. Conversions:

delta_coin = delta_usd / S − usd_price / S²
           ≈ delta_usd / S      (when usd_price ≪ S, deep OTM or near expiry)

More precisely, with V_coin = V_usd / S:

delta_coin = ∂V_coin / ∂S = (S · ∂V_usd/∂S − V_usd) / S²
           = (delta_usd · S − V_usd) / S²
           = delta_usd / S − V_usd / S²

For practical use:

• USD-equivalent delta = delta_coin × S (this is "how many USD do I make per $1 of spot move")
• Coin notional: 1 contract = 1 BTC, so coin delta of 0.5 means a $1 spot move changes the position by 0.5 coin × (1 / S_change) ≈ 0.5/S BTC. Confused yet? Always reconvert to USD before reasoning about hedge ratios.

Vega

Coin vega: vega_coin = vega_usd / S. Convert with × 1/100 for "per vol point" the same way.

Theta

Coin theta: theta_coin = theta_usd / S (assuming S doesn't drift — small approximation).

Gamma

Coin gamma is a mess of cross-terms. Use the USD price's gamma and convert with gamma_coin ≈ gamma_usd / S − 2·delta_usd / S² + 2·V_usd / S³. Or just use the USD Greeks and remember that hedge ratios should be in USD-equivalent terms.

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Step 6: Sanity Checks

1. Put-call parity (inverse form): C_coin − P_coin = 1 − K·e^(−rT)/S. If r = 0: C − P = 1 − K/S. Verify within 1e-5.
2. Bound: C_coin ≤ 1 always (you can't make more than 1 coin from a 1-coin call). P_coin ≤ K·e^(−rT)/S.
3. Deep ITM call: C_coin → 1 − K·e^(−rT)/S as S → ∞.
4. Deep OTM call: C_coin → 0.

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Step 7: Respond to User

1. Coin-quoted price (matches Deribit's mark_price).
2. USD-equivalent price = coin_price × S.
3. Greeks: report BOTH coin-denominated and USD-equivalent, with labels.
4. Comparison against the live Deribit mark_iv / mark_price if the user gave a contract name.

For full surface fitting, recommend iv-surface and tell the user the SVI fit is for vanilla — inverse-option vols can be back-solved to USD vols using the same σ (the IV is the same under both numeraires for European options, by the volatility invariance of measure change).

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Reference Files

• references/inverse_math.md — Full derivation of inverse Greeks, USD numeraire vs coin numeraire, ETH staking yield correction