Inventory Models
Garman (1976) Model
- order arrive at ask for price $p_a$ with intensity $\lambda_a$ and size 1
- order arrive at bif for price $p_b$ with intensity $\lambda_b$ and size -1
The problem looks like a gambler ruin problem with $Pr(fail) = ((1-q)Loss/(q Gain))^N$, Garman shows that the ruin is avoided for
- $\lambda_a p_a > \lambda_b p_b$
- $\lambda_b \geq \lambda_a$
For the special case of intensity equality, profit is maximised by maximising the area $(p_a - p_b) \lambda_0$.
Amihud Mendelson (1980) Model
A&M show that bid/ask should be skewed with inventory to obtain the preferred size. The model is assuming linear demand and supply functions.
Stoll (1978) Model, and Ho and Stoll (1982)
Stoll proposes a utility of wealth function $U(W)$ for the market maker. The bid/ask is shown to depend on asset volaility and duration of exposure.
Insider / Liquidity Trader Model
Kyle (1985) Model
Total demand $z=x+y$ for a dealer comes from two sources:
- Informed trader knows that price at end of period has distribution $\nu \sim N(p_0,\sigma_0^2)$, their demand $x(\nu)$ depends on expected profit.
- Liquidity traders submit a random demand $y \sim N(0, \sigma_y^2)$
Kyle obtains an equilibrium if price is a linear function of demand: $p(z) = \mu + \lambda z$. In a multi-period setup, the insider splits his order with different sizes. The merit of this model is to look at order size.
Glosten-Milgrom (1985) Model
We assume that value $$V$ can become V_h$ with probability $\theta$ or $V_l$ with $1-\theta$. Liquidity traders have a 0.5 chance of guessing correctly, informed traders have 100% accuracy. The probability of informed trade is $\mu$.
- $a = E(V|B) = \frac{\theta (1-\mu) V_l + (1-\theta)(1+\mu) V_h}{1-\mu(1-2\theta)}$
- $b = E(V|S) = \frac{\theta (1+\mu) V_l + (1-\theta)(1-\mu) V_h}{1-\mu(1-2\theta)}$
- bid/offer spread $s = a-b = \frac{4 \theta (1-\theta) (V_h-V_l)}{1-(1-2\theta)^2\mu^2}$
For $\theta=1/2$, the spread is $s = \mu (V_h-V_l)$ depends linearly on the number of informed traders.
Easley and O Hara (1987) expanded this framework to large and small order size, in general large orders betray that trader is informed. Back and Baruch (2004) model propose two models where liquidity orders arrive as a brownian motion or poisson process. The insider can optimize their time of trading. The optimal strategy for informed trader is to use a mixed strategy of trading and waiting.
Back Testing
- Markov Switching Model, Tsay, 2005
- Bootstrap methods and their applications, Davidson and Hinkley 1997
Performance
- Thorp (2006) kelly ratio $f= \frac{pr+1-p}{r}$ with $p$ probability of gain and $r=G/L$ ratio.
- return t-stat test against 0: $t = \frac{\mu}{(\sigma^2/N)^{1/2}}$
- return t-stat test of A against B: $t = \frac{\mu_A - \mu_B}{(\sigma_A^2/N_A+\sigma_B^2/N_B)^{1/2}}$ with $N_A+N_B-2$ degree of freedom
- $IR = \frac{E(r_i)-r_0}{\sigma_{i0}}$
- a Gaussian process with no drift has MDD $E(MD) = 1.2533 \sigma \sqrt{T}$