Link: http://www.sciencedirect.com/science/article/pii/0304405X76900064
In Garman's inventory model, buying order and selling order are poisson process with order size = 1. Buying price and selling price are denoted by pb and ps, that is, the market maker gets pb when she sells a stock to the others, and spends ps to buy a stock from the others.
Garman than calculates the probability of the inventory of the market maker, says Q(k, t+dt) = probability to get 1 dollar x Q(k-1, t) + probability to lose 1 dollar x Q(k+1, t) + no buying or selling order x Q(k, t), where Q(k, t+dt) = probability to have k money at time t+dt.
In the above equation, I think Garman had split the money received and loss by buying or selling a shock in many sub-poisson process, otherwise, getting 1 dollar or losing 1 dollar are impossible, as market maker receive pb dollar and loses ps dollar in each order, but not 1 dollar. Do my statement correct? Thank you very much.
Inventory models address the problem that flows on either side of the market are not synchronized which lead to the possibility of running out of cash (bankruptcy) or out of inventory (failure). The dealer aims to fill orders while avoiding running out of cash and inventory.
Garman inventory model is based on the principle that lower/higher price increases/decreases demand. The model shows that a risk-neutral profit maximizing dealer must establish a certain bid/ask price spread and manipulate this to maintain the inventory.
In the Garman model, orders follow a Poisson process with arrival rates for buy orders λa(pa) and and sell orders λb(pb); . λa(pa) monotonically decreases and λb(pb) monotonically increases.
Let Qk(t) denote the probability that dealer’s cash supply at time t equals k. Three events at time t-dt determine the cash position of k units :
- dealer had k -1 units of cash and sold one unit of stock
- dealer had k +1 units of cash and bought one unit of stock
- dealer did not trade
This then gives:
Qk(t) = λapadt [1 - λbpbdt] Qk-1(t-dt) + λbpbdt [1 - λapadt] Qk+1(t-dt) +
[1 - λapadt] [1 - λbpbdt ] Qk(t-dt)
Dear Chi,
Your statement may not be correct. Pa is greater than pb, it can be provided liquidity crash is there and bankruptcy reached the market size. In that case λa < λb, hence split position may occur and possibilities are there.
Earlier models of dealership markets are inventory-based models. Demsetz (1968) seems to be the first work emphasizing order imbalance. Trading has a time dimension and at each point in time, there may be an imbalance in buy and sell orders. The presence of dealers allows submitted orders to be executed immediately. The bid-ask spread is therefore a price that public investors must pay in order to obtain immediacy in order execution. Unlike Demsetz (1968) where the focus is on the trading desires of individual traders, Garman (1976) switches the spot light to market clearing mechanisms. Hence Garman’s research is considered by many to be the first formal analysis of market microstructure. Two mechanisms were considered in Garman’s model research, a double auction and a monopolistic dealer exchange.
Model describes a monopolistic dealer must first set an ask price and a bid price, receive and execute orders from public traders, and the bid and ask prices are chosen to maximize the dealer’s profit at each point in time subject to the constraint that bankruptcy or failure must not take place (which would occur if the dealer ran out of the traded security or cash). In Garman’s model, the bid-ask spread exists in order that specialist will not be ruined with probability one. That is, market viability dictates the existence of a bid-ask spread
Most importantly, The Garman's Inventory Model deals the problem of dealer's inventory imbalance was first addressed by Garman (1976). In his model, a monopolistic dealer assigns ask (pa) and bid (pb) prices, and fills all orders. Each order size is assumed to be one unit. The dealer's goal is, as a minimum, to avoid bankruptcy.
By equation risk probability could be understood,
Let Qk(t) be the probability that the dealer has k units of cash at time t, and Rk(t) be the probability that the dealer has k units of stock at time t. For sufficiently small time interval ∆t, we have from the assumed Poisson processes
∂Qk(t) ∂t = Qk−1(t)[λa(pa)pa]+Qk+1(t)[λb(pb)pb]−Qk(t)[λa(pa)pa+λb(pb)pb],
Similarly, one can show that
lim t→+∞ R0(t) { ∼ [ λa(pa) λb(pb) ] Is(0) , if λa(pa) < λb(pb); 1, otherwise.
By solving, Garman concludes that to avoid a sure failure, it must be that
paλa > pbλb
And
λa < λb
This implies a spread.
To guarantee that the dealer can make the market indefinitely, the dealer that starts with a fixed amount of cash and a fixed number of shares of the stock at time 0 must set the ask price higher than the bid price.
Garman’s model exhibits the couple of important features like monopolistic risk neutral specialist; liquidity traders only; market orders following Poisson processes; specialist can only set prices once and for all; and specialist is facing the ‘ruin’ problem.
Amihud and Mendelson (1980) extend Garman’s study to allow price adjustments. Prices change according to inventory positions. Unlike in Garman (1976), spread in Amihud and Mendelson simply reflects monopoly power of the specialist. One can show that the spread tends to zero, as competition gets in and increases. Inventory plays the role of a ‘buffer.’ Stoll (1978) takes a different view about the bid-ask spread: market makers are those selling insurance to liquidity traders, and the spread is risk premium.
Ashish
Dear Chi,
Your statement may not be correct. Pa is greater than pb, it can be provided liquidity crash is there and bankruptcy reached the market size. In that case λa < λb, hence split position may occur and possibilities are there.
Earlier models of dealership markets are inventory-based models. Demsetz (1968) seems to be the first work emphasizing order imbalance. Trading has a time dimension and at each point in time, there may be an imbalance in buy and sell orders. The presence of dealers allows submitted orders to be executed immediately. The bid-ask spread is therefore a price that public investors must pay in order to obtain immediacy in order execution. Unlike Demsetz (1968) where the focus is on the trading desires of individual traders, Garman (1976) switches the spot light to market clearing mechanisms. Hence Garman’s research is considered by many to be the first formal analysis of market microstructure. Two mechanisms were considered in Garman’s model research, a double auction and a monopolistic dealer exchange.
Model describes a monopolistic dealer must first set an ask price and a bid price, receive and execute orders from public traders, and the bid and ask prices are chosen to maximize the dealer’s profit at each point in time subject to the constraint that bankruptcy or failure must not take place (which would occur if the dealer ran out of the traded security or cash). In Garman’s model, the bid-ask spread exists in order that specialist will not be ruined with probability one. That is, market viability dictates the existence of a bid-ask spread.
Most importantly, The Garman's Inventory Model deals the problem of dealer's inventory imbalance was first addressed by Garman (1976). In his model, a monopolistic dealer assigns ask (pa) and bid (pb) prices, and fills all orders. Each order size is assumed to be one unit. The dealer's goal is, as a minimum, to avoid bankruptcy.
By equation risk probability could be understood,
Let Qk(t) be the probability that the dealer has k units of cash at time t, and Rk(t) be the probability that the dealer has k units of stock at time t. For sufficiently small time interval ∆t, we have from the assumed Poisson processes
∂Qk(t) ∂t = Qk−1(t)[λa(pa)pa]+Qk+1(t)[λb(pb)pb]−Qk(t)[λa(pa)pa+λb(pb)pb],
Similarly, one can show that
lim t→+∞ R0(t) { ∼ [ λa(pa) λb(pb) ] Is(0) , if λa(pa) < λb(pb); 1, otherwise.
By solving, Garman concludes that to avoid a sure failure, it must be that
paλa > pbλb
And
λa < λb
This implies a spread.
To guarantee that the dealer can make the market indefinitely, the dealer that starts with a fixed amount of cash and a fixed number of shares of the stock at time 0 must set the ask price higher than the bid price.
Garman’s model exhibits the couple of important features like monopolistic risk neutral specialist; liquidity traders only; market orders following Poisson processes; specialist can only set prices once and for all; and specialist is facing the ‘ruin’ problem.
Amihud and Mendelson (1980) extend Garman’s study to allow price adjustments. Prices change according to inventory positions. Unlike in Garman (1976), spread in Amihud and Mendelson simply reflects monopoly power of the specialist. One can show that the spread tends to zero, as competition gets in and increases. Inventory plays the role of a ‘buffer.’ Stoll (1978) takes a different view about the bid-ask spread: market makers are those selling insurance to liquidity traders, and the spread is risk premium.
Ashish
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability