做市

赚取价差，承担风险

做市商挂出买价和卖价然后等待。当卖家打到买价、买家吃到卖价时，做市商低买高卖，赚取价差。这就是收入。成本是库存风险：在两笔交易之间，做市商持有一个价值随价格波动的头寸，而且更常被知情者（逆向选择）而不是不知情者成交。好的做市是在两者间取得平衡：报价要足够窄，才能经常被成交；也要足够宽，才能为风险获得补偿；同时让库存保持在零附近。

# The income: buy at the bid, sell at the ask, keep the spread
bid, ask, size = 99.95, 100.05, 100
cash = -bid * size + ask * size
print(round(cash, 2))                                # 10.0    a round trip of 100 shares earned the 0.10 spread

# The risk: between the two trades the price moves against the inventory
inventory, mid_after = 100, 99.80
print(round(inventory * (mid_after - bid), 2))       # -15.0   the position lost more than the spread would have earned

Avellaneda-Stoikov：在哪里报价

Avellaneda-Stoikov 模型给出了闭式答案。第一，保留价格 r = mid - q * gamma * sigma^2 * tau 是在库存为 q 时做市商乐于成交的位置：多头做市商（q > 0）应把资产估值压到中间价之下，并报更低的价格来吸引买家，空头做市商则报更高。风险厌恶 gamma、波动率 sigma 和剩余时间 tau 决定倾斜的力度。第二，最优价差 gamma * sigma^2 * tau + (2 / gamma) * ln(1 + gamma / k) 有两部分：持仓到期限的风险费用，以及订单流项，其中 k 表示报价离中间价越远成交枯竭得有多快。报价为 r - spread/2 和 r + spread/2。

# The reservation price leans against inventory: long -> quote lower, short -> quote higher
def reservation_price(mid, inventory, gamma, sigma, tau):
    return mid - inventory * gamma * sigma ** 2 * tau

for inventory in (-50, 0, 50):
    print(inventory, round(reservation_price(100.0, inventory, 0.1, 0.2, 1.0), 4))
# -50 100.2   short: quote higher to buy back
#   0 100.0   flat: centred on the mid
#  50  99.8   long: quote lower to sell

# The optimal spread and the two quotes around the reservation price
import math

def optimal_spread(gamma, sigma, tau, k):
    return gamma * sigma ** 2 * tau + (2 / gamma) * math.log(1 + gamma / k)

def make_quotes(mid, inventory, gamma, sigma, tau, k):
    centre = mid - inventory * gamma * sigma ** 2 * tau
    half = optimal_spread(gamma, sigma, tau, k) / 2
    return centre - half, centre + half

print(round(optimal_spread(0.1, 0.2, 1.0, 1.5), 4))                       # 1.2948
bid, ask = make_quotes(100.0, 0, 0.1, 0.2, 1.0, 1.5)
print(round(bid, 4), round(ask, 4))                  # 99.3526 100.6474   symmetric around the mid when flat
bid, ask = make_quotes(100.0, 50, 0.1, 0.2, 1.0, 1.5)
print(round(bid, 4), round(ask, 4))                  # 99.1526 100.4474   long 50: both quotes move down by 0.2

库存、成交与盈亏

用两个数字跟踪状态：库存和现金。买价被打到时你买入了：库存上升，现金减少 price * quantity；卖价被吃到时你卖出了：反之。任一时刻的财富是现金加上按中间价估值的库存，cash + inventory * mid，它的变化就是你的盈亏。盈亏来自赚到的价差，以及持有库存期间的价格波动；围绕保留价格报价会把库存拉回零，因为能减少库存的那一边变得更有吸引力。实践中限制很重要：库存的硬上限、快市时暂停，以及每次报价时检查买价低于卖价。

# Booking fills and marking the book to market
def apply_fill(state, side, price, quantity):
    inventory, cash = state
    if side == "bid":
        return inventory + quantity, cash - price * quantity
    return inventory - quantity, cash + price * quantity

state = (0, 0.0)
state = apply_fill(state, "bid", 99.95, 100)         # our bid is hit: we buy 100
print(state)                                         # (100, -9995.0)
state = apply_fill(state, "ask", 100.05, 100)        # our ask is hit: we sell 100
print(state)                                         # (0, 10.0)   flat again, and the spread is the profit
inventory, cash = apply_fill(state, "bid", 99.9, 50)
print(round(cash + inventory * 99.9, 2))             # 10.0   marked at the price we just paid, the equity is unchanged

• 做市商赚取价差，但要承担库存风险和逆向选择；目标是足够窄以成交、足够宽以获得补偿，并让库存接近零。
• 保留价格 mid - q * gamma * sigma^2 * tau 逆着库存倾斜；价差 gamma * sigma^2 * tau + (2/gamma) ln(1 + gamma/k) 随风险厌恶加权的波动率和时间而变宽。
• 跟踪库存和现金，按中间价估值，强制执行硬性库存限制，并在实盘使用前用记录数据在模拟器里测试策略。

下一节：Heston 模型，本身也是随机的波动率。

实验：评分内容

• 用 reservation_price 逆库存调整25 分
• 用 optimal_spread 设定价差宽度25 分
• 用 make_quotes 双边报价25 分
• 用 apply_fill 记录一笔成交25 分