ブラック・ショールズ・ガンマ・スキャルピング戦略：オプションマーケットメーカーのための定量的知見

/*backtest
start: 2025-01-04 00:00:00
end: 2026-01-02 00:00:00
period: 1d
basePeriod: 1d
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
*/

//@version=6
strategy("Black-Scholes Gamma Scalping Strategy", 
         overlay=true,
         default_qty_type=strategy.percent_of_equity, 
         default_qty_value=10,
         pyramiding=5)

// ============================================================================
// STRATEGY CONCEPT:
// This strategy simulates gamma scalping - a volatility arbitrage technique
// used by options market makers. The core idea:
//
// 1. We model a synthetic long straddle position (long call + long put)
// 2. The straddle has positive gamma, meaning delta changes as price moves
// 3. We continuously delta-hedge by trading the underlying
// 4. If realized volatility > implied volatility, hedging profits exceed theta decay
// 5. Trading signals generated when price moves create hedging opportunities
//
// The Black-Scholes equation provides: Delta, Gamma, Theta, Vega
// ============================================================================

// === INPUT PARAMETERS ===
string grp_opt = "Option Parameters"
strike_offset_pct   = input.float(0.0, "Strike Offset from Current Price %", group=grp_opt, step=0.5)
days_to_expiry      = input.int(30, "Days to Expiration", group=grp_opt, minval=1, maxval=365)
implied_vol         = input.float(0.25, "Implied Volatility (Annual)", group=grp_opt, minval=0.05, maxval=2.0, step=0.01)
risk_free_rate      = input.float(0.05, "Risk-Free Rate (Annual)", group=grp_opt, step=0.001)
dividend_yield      = input.float(0.0, "Dividend Yield (Annual)", group=grp_opt, step=0.001)

string grp_vol = "Volatility Analysis"
hist_vol_period     = input.int(20, "Historical Volatility Period", group=grp_vol, minval=5)
vol_ratio_threshold = input.float(1.2, "HV/IV Ratio for Long Vol Signal", group=grp_vol, minval=1.0, step=0.05)
vol_ratio_short     = input.float(0.8, "HV/IV Ratio for Short Vol Signal", group=grp_vol, minval=0.1, maxval=1.0, step=0.05)

string grp_trade = "Trading Parameters"
gamma_scalp_threshold = input.float(0.5, "Gamma Scalp Threshold (ATR mult)", group=grp_trade, minval=0.1, step=0.1)
hedge_band_pct        = input.float(2.0, "Delta Hedge Band %", group=grp_trade, minval=0.5, step=0.5)
use_vol_filter        = input.bool(true, "Use Volatility Regime Filter", group=grp_trade)
max_positions         = input.int(3, "Max Concurrent Positions", group=grp_trade, minval=1, maxval=10)

string grp_risk = "Risk Management"
stop_loss_atr_mult    = input.float(2.0, "Stop Loss (ATR Multiple)", group=grp_risk, minval=0.5, step=0.5)
take_profit_atr_mult  = input.float(3.0, "Take Profit (ATR Multiple)", group=grp_risk, minval=1.0, step=0.5)
max_drawdown_pct      = input.float(15.0, "Max Drawdown % to Pause Trading", group=grp_risk, minval=5.0, step=1.0)

// ============================================================================
// BLACK-SCHOLES MATHEMATICAL FUNCTIONS
// ============================================================================

// --- Standard Normal CDF (Abramowitz & Stegun approximation) ---
norm_cdf(float x) =>
    float result = 0.0
    if na(x)
        result := na
    else
        float ax = math.abs(x)
        if ax > 10
            result := x > 0 ? 1.0 : 0.0
        else
            float t = 1.0 / (1.0 + 0.2316419 * ax)
            float d = 0.3989422804 * math.exp(-0.5 * x * x)  // 1/sqrt(2*pi) * exp(-x²/2)
            float p = t * (0.31938153 + t * (-0.356563782 + t * (1.781477937 + t * (-1.821255978 + t * 1.330274429))))
            result := x >= 0 ? 1.0 - d * p : d * p
    result

// --- Standard Normal PDF ---
norm_pdf(float x) =>
    na(x) ? na : 0.3989422804 * math.exp(-0.5 * x * x)

// --- Black-Scholes d1 and d2 ---
bs_d1(float S, float K, float r, float q, float sigma, float T) =>
    T > 0 and sigma > 0 ? (math.log(S / K) + (r - q + 0.5 * sigma * sigma) * T) / (sigma * math.sqrt(T)) : na

bs_d2(float S, float K, float r, float q, float sigma, float T) =>
    float d1 = bs_d1(S, K, r, q, sigma, T)
    na(d1) ? na : d1 - sigma * math.sqrt(T)

// --- Greeks ---
// Call Delta
call_delta(float S, float K, float r, float q, float sigma, float T) =>
    T <= 0 ? (S >= K ? 1.0 : 0.0) : math.exp(-q * T) * norm_cdf(bs_d1(S, K, r, q, sigma, T))

// Put Delta
put_delta(float S, float K, float r, float q, float sigma, float T) =>
    T <= 0 ? (S <= K ? -1.0 : 0.0) : math.exp(-q * T) * (norm_cdf(bs_d1(S, K, r, q, sigma, T)) - 1)

// Gamma (same for call and put)
option_gamma(float S, float K, float r, float q, float sigma, float T) =>
    if T <= 0 or sigma <= 0
        0.0
    else
        float d1 = bs_d1(S, K, r, q, sigma, T)
        math.exp(-q * T) * norm_pdf(d1) / (S * sigma * math.sqrt(T))

// ============================================================================
// HISTORICAL VOLATILITY CALCULATION
// ============================================================================

// Calculate annualized historical volatility
calc_historical_vol(int period) =>
    float log_return = math.log(close / close[1])
    float std_dev = ta.stdev(log_return, period)
    
    // Annualization factor based on timeframe
    float periods_per_year = switch
        timeframe.isdaily => 365.0
        timeframe.isweekly => 52.0
        timeframe.ismonthly => 12.0
        => 365.0 * 390.0 / (timeframe.isminutes ? timeframe.multiplier : 1440.0)
    
    nz(std_dev * math.sqrt(periods_per_year), 0.20)

hist_vol = calc_historical_vol(hist_vol_period)

// ============================================================================
// STRATEGY CALCULATIONS
// ============================================================================

// Dynamic strike price (ATM or offset)
float atm_strike = math.round(close / syminfo.mintick) * syminfo.mintick
float strike_price = atm_strike * (1 + strike_offset_pct / 100)

// Time to expiration in years
float tau = days_to_expiry / 365.0

// Calculate Greeks for synthetic straddle (1 call + 1 put at same strike)
float c_delta = call_delta(close, strike_price, risk_free_rate, dividend_yield, implied_vol, tau)
float p_delta = put_delta(close, strike_price, risk_free_rate, dividend_yield, implied_vol, tau)
float straddle_delta = c_delta + p_delta  // Net delta of straddle

float gamma = option_gamma(close, strike_price, risk_free_rate, dividend_yield, implied_vol, tau)
float straddle_gamma = 2 * gamma  // Straddle has 2x gamma

// Volatility ratio: Historical / Implied
float vol_ratio = hist_vol / implied_vol

// ATR for position sizing and stops
float atr = ta.atr(14)

// ============================================================================
// TRADING SIGNALS
// ============================================================================

// --- Signal 1: Volatility Regime ---
// Long volatility when realized > implied (gamma scalping profitable)
// Short volatility when realized < implied (collect theta)
bool long_vol_regime = vol_ratio > vol_ratio_threshold
bool short_vol_regime = vol_ratio < vol_ratio_short

// --- Signal 2: Gamma Scalp Trigger ---
// When price moves significantly, delta changes. We trade to capture this.
float price_move = close - close[1]
float move_threshold = atr * gamma_scalp_threshold

bool significant_up_move = price_move > move_threshold
bool significant_down_move = price_move < -move_threshold

// --- Signal 3: Delta Hedge Bands ---
// Trade when straddle delta deviates from neutral
float delta_band = hedge_band_pct / 100
bool delta_long_signal = straddle_delta < -delta_band  // Need to buy to hedge
bool delta_short_signal = straddle_delta > delta_band   // Need to sell to hedge

// --- Combined Entry Signals ---
bool vol_filter = use_vol_filter ? long_vol_regime : true

// Long entry: Price dropped significantly OR delta needs positive hedge
bool long_entry = vol_filter and (significant_down_move or delta_long_signal)

// Short entry: Price rose significantly OR delta needs negative hedge  
bool short_entry = vol_filter and (significant_up_move or delta_short_signal)

// ============================================================================
// RISK MANAGEMENT
// ============================================================================

// Track equity for drawdown protection
var float equity_peak = strategy.initial_capital
equity_peak := math.max(equity_peak, strategy.equity)
float current_drawdown = (equity_peak - strategy.equity) / equity_peak * 100
bool drawdown_exceeded = current_drawdown > max_drawdown_pct

// Position counting
int open_trades = strategy.opentrades

// ============================================================================
// STRATEGY EXECUTION
// ============================================================================

// Stop loss and take profit levels
float long_stop = close - atr * stop_loss_atr_mult
float long_target = close + atr * take_profit_atr_mult
float short_stop = close + atr * stop_loss_atr_mult
float short_target = close - atr * take_profit_atr_mult

// Entry conditions
bool can_trade = not drawdown_exceeded and open_trades < max_positions 

if can_trade
    // Long entries
    if long_entry and strategy.position_size <= 0
        strategy.entry("GammaLong", strategy.long, 
                       comment="Δ:" + str.tostring(straddle_delta, "#.##") + " Γ:" + str.tostring(straddle_gamma, "#.###"))
        strategy.exit("LongExit", "GammaLong", stop=long_stop, limit=long_target)
    
    // Short entries
    if short_entry and strategy.position_size >= 0
        strategy.entry("GammaShort", strategy.short,
                       comment="Δ:" + str.tostring(straddle_delta, "#.##") + " Γ:" + str.tostring(straddle_gamma, "#.###"))
        strategy.exit("ShortExit", "GammaShort", stop=short_stop, limit=short_target)

// Exit on extreme conditions
if drawdown_exceeded
    strategy.close_all(comment="Drawdown Protection")

// Exit if volatility regime shifts against us
if use_vol_filter and short_vol_regime and strategy.position_size != 0
    strategy.close_all(comment="Vol Regime Shift")

// ============================================================================
// VISUALIZATIONS
// ============================================================================

// Plot straddle delta
plot(straddle_delta, "Straddle Delta", color=color.blue, linewidth=2)
hline(0, "Zero Delta", color=color.gray, linestyle=hline.style_dashed)
hline(delta_band, "Upper Band", color=color.red, linestyle=hline.style_dotted)
hline(-delta_band, "Lower Band", color=color.green, linestyle=hline.style_dotted)

// Background color for volatility regime
bgcolor(long_vol_regime ? color.new(color.green, 90) : short_vol_regime ? color.new(color.red, 90) : na)

// Entry signals
plotshape(long_entry and can_trade, "Long Signal", shape.triangleup, location.belowbar, color.green, size=size.small)
plotshape(short_entry and can_trade, "Short Signal", shape.triangledown, location.abovebar, color.red, size=size.small)

// ============================================================================
// ALERTS
// ============================================================================

alertcondition(long_entry and can_trade, "Gamma Long Entry", "BS Strategy: Long entry signal - Delta={{straddle_delta}}")
alertcondition(short_entry and can_trade, "Gamma Short Entry", "BS Strategy: Short entry signal - Delta={{straddle_delta}}")
alertcondition(drawdown_exceeded, "Drawdown Alert", "BS Strategy: Max drawdown exceeded - trading paused")