Option Market Making

7/31/2019 Option Market Making

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find that the net demand for options exerts pressure on option prices, whenperfect replication is impossible. They derive the sensitivity of option priceswith respect to net demand and they find that it depends on the standardGreeks, Gamma and Vega. The price effect of risk due to discrete time

hedging is dependent on the options Gamma, while the price effect of riskdue to stochastic volatility depends on Vega. A related insight may be foundin Jameson and Wilhelm (1992), who show that Gamma and Vega contributesignificantly to the bid-ask spread.

From a theoretical perspective, our stochastic control problem can be viewedas an inventory management problem, where the order flow depends on the bidand ask prices. This type of problem was first studied by Ho and Stoll (1981)and an empirical analysis was applied to the options market by Ho and Macris(1984). These authors show that if a dealers goal is to maximize expectedutility, he will adjust his quotes in response to inventory positions. As onewould expect, when the dealers inventory is positive, the quotes should be

lowered and when the inventory is negative, the quotes should be raised. Thissimple risk-management mechanism helps the dealer keep inventory positionsunder control.

There are some important differences between our model and the one de-scribed above. First, we consider a dealer in both the stock and the optionmarket. This approach allows us to model the relative liquidity of the optionand stock markets and to study how a dealer may transfer liquidity across them.

Second, rather than using a utility function, we choose a mean-variance ob-jective. This type of objective has enjoyed a revival in the optimal order execu-tion literature (see Almgren and Chriss (2001), Engle and Ferstenberg (2006))and is a natural choice to address optimal market making strategies. Indeed,we find that a fast numerical procedure can be implemented to determine theoptimal bid and ask prices, without resorting to asymptotic expansions, as isoften the case in stochastic control problems of this nature.

Third, we choose an investment horizon of one day, and explicitly separatethe problem into a day of trading, followed by an overnight move in the stockand option prices. The advantage of our approach is that we can capture theoften observed fact that dealers tend to go home flat (see Hasbrouck (2007)).Indeed, by framing our problem in a twenty four hour horizon, we find that theoptimal dealer will move his quotes so as to avoid keeping excessive inventoryovernight.

In Section 2, we present the model: the dynamics of the mid prices, theevolution of the dealers wealth, the order flow and the dealers objective. In

Section 3, Delta Risk, we focus on the role of the net Delta on the dealers bidand ask quotes. We solve a version of the model where the option and the stockare related through the Black-Scholes formula, first in a one-period setting, thenin a multi-period setting where all the price moves occur overnight, and finallyin a multi-period setting where prices evolve continuously. We illustrate thefact that our mean-variance problem allows us to trace out the efficient frontierfor a dealer, in analogy with the classic Markowitz framework. In Section

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4, Gamma and Vega Risk, we introduce a version of the model where theoption inventory is delta-hedged over the course of the trading day, but wherethe dealer is subject to residual risks due to stochastic volatility and overnightmoves in the stock price. Once again, we present the one-period and the multi-

period versions of the model. We find that the bid and ask quoting policydepends on the net Gamma and Vega of the dealer. This affects long and shortmaturity options differently and we illustrate this by numerically computingthe dealers optimal policies. We present our conclusions in Section 5, and theproofs are all included in the appendix.

2 The model

The time horizon. We divide the trading day into n sessions 0 = t0 < t1 >T and strike K, whose mid price follows

dCt = tdt + tdSt +1

2t(dSt)

2

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where the function C(S, t) is given by the Black Scholes formula and t, tand t are the standard greeks, Theta, Delta and Gamma, respectively. Usingthe Black Scholes PDE, we obtain the following approximation

Ci = iSiutwhere u is a standard normal random variable and the time step t may rep-resent one of the trading sessions or an overnight move.

The dealers state variables and controls. The wealth Yi of the marketmaker is

Yi = Xi + qsi Si + q

oi Ci

where Xi is the dollar amount of cash and qsi and q

oi are the number of stocks

and options in inventory. We assume without loss of generality that X0 = 0,qs0 = 0, q

o0 = 0 and therefore the initial wealth Y0 is also zero. The change in

Yi over period (ti, ti+1) will depend on (i) the arrival of a new stock or optiontransaction and (ii) returns on the stocks and options held in inventory. Wetherefore decompose the change in wealth into two components

Yi = Yi+1 Yi = Zi + Iiwhere

Zi Xi + qsi Si + q

oi Ci

is the change in wealth due to transactions and

Ii qsi+1Si + q

oi+1Ci (2.1)

is the change in the market value of the inventory. Note that since no tradingoccurs in the period (tn, T),

Zn ZT Zn = 0

andIn IT In = qsn(ST Sn) + qon(CT Cn). (2.2)

Rather than crossing the spread, or even worse, impacting the price adverselyby trading with market orders, the dealer will deal exclusively with limit orders.In other words, the dealer quotes bid and ask prices around the stocks midprice,

pb,s

i = Si bi

pa,si = Si + ai

around the options mid price,

pb,oi = Ci bipa,oi = Ci +

ai

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and stands ready to trade one unit of stock or option at the prices above.If we only admit controls ai ,

bi ,

ai and

bi that are greater than zero, these

prices represent an improvement over the mid price. As long as the mid pricespecifications are arbitrage free, the dealers quotes cannot be arbitraged.

The liquidity. The bid and ask quotes, or equivalently the premiums ai , bi ,

ai and bi , indirectly influence the inventory held by the market maker, since

they affect the arrival rates of orders. In the sequel, we let

s() =

A B if 0 < A/B0 if A/B (2.3)

and

o() =

C D if 0 < C/D0 if C/D (2.4)

be the Poisson arrival rates of stock and option orders, respectively. This choice

of piecewise linear functions simply reflects the fact that the closer the quotesare to the mid price, the higher the intensity of arrival of orders. The constantsA, B, C and D will allow us to capture the relative liquidity of the stock andoption markets.

Assuming that the trading sessions t are small enough for only one event(purchase or sale of a stock or option) to occur, the probabilities associatedwith transactions returns and changes in inventory are given by

P(Zi = 0) = P(qsi = 0, q

oi = 0) = 1

s(bi ) +

s(ai ) + o(bi ) +

o(ai )

tP(Zi =

bi ) = P(q

si = 1, q

oi = 0) =

s(bi )tP(Zi =

ai ) = P(q

si = 1, qoi = 0) = s(ai )t

P(Zi = b

i

) = P(qsi

= 0, qoi

= 1) = o(bi

)tP(Zi = ai ) = P(q

si = 0, q

oi = 1) = o(ai )t

(2.5)

The objective. We consider the stochastic control problem of a dealer whosets bid and ask prices throughout the trading day. The value function is

v(X0, S0, qs0, q

o0, t0) = max

ai,bi,ai,bi,0in1

E[ZT]

ni=0

V ar[Ii]

(2.6)

The dealer wishes to maximize profit from transactions, with a penalty pro-portional to the variance of inventory value. Notice that the variance penalty

affects changes in the inventory value over each trading session, as well as theovernight variance. This forces the dealer to keep the inventory under control.

3 Delta risk

In this section, we solve problem (2.6), first in a one-period model (Section3.1), then in a multi-period model where all the price moves occur overnight

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(Section 3.2), and finally in a multi-period model where prices evolve continu-ously (Section 3.3). Since the stock and option prices are related through theBlack-Scholes formula, there is essentially only one source of risk, captured bythe net Delta of the dealers position (qs + qo). This net Delta affects the

optimal premiums ai , bi , ai and bi that the dealer charges around the midprices.

3.1 One-period model

Let us first solve the one-period version of our model. To keep notation coherentwith the rest of the paper, we consider the dealers problem at the beginningof the last trading session, assuming that he may only trade in the interval(tn1, tn). The dealer chooses bid and ask quotes at time tn1, defined throughthe controls an1,

bn1,

an1 and

bn1. These quotes influence the probabilities

of the four possible transactions (see (2.5)) over the time interval (tn1, tn).

The stock and option then move, without the possibility of any further tradinguntil time T.The dealers objective is to maximize expected marked to market wealth at

time T, with a penalty term that is proportional to the variance of the change invalue of the inventory. This mean-variance objective therefore strikes a balancebetween the desire to maximize the marked to market profits made from thebid ask spread and the risk associated with changes in market prices.

The value function can be written as

v(Xn1, Sn1, qsn1, q

on1, tn1) = max

an1

,bn1

,an1

,bn1

(E[ZT] V ar[IT In1]) .(3.1)

The dealers objective is to determine the optimal bid and ask quotes on thestock and options market.

Theorem 3.1. The optimal policy for the dealer is given by

an1 = max

0, minAB

, A2B 2(T tn1)S2n1

qsn1 + qon1n1 12

bn1 = max

0, min

AB ,

A2B +

2(T tn1)S2n1

qsn1 + qon1n1 +

12

an1 = max

0, min

CD

, C2D 2(T tn1)S2n1n1

qsn1 + qon1n1 12n1

an1 = max

0, min

CD

, C2D + 2(T tn1)S2n1n1

qsn1 + q

on1n1 +

12n1

(3.2)

Remark 3.1. In the risk neutral case where = 0, the objective is to maximizeterminal wealth and the optimal premiums are

=A

2B

and

=C

2D

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regardless of inventory.

Remark 3.2. Notice that if the dealer is risk-averse, i.e. > 0, he adjusts ortilts all his quotes away from the profit maximizing solution by an amountproportional to

qsn1 + qon1n1

which is the net Delta. Also note that the solution (3.2) is less sensitive toinventory as one approaches the horizon T.

3.2 Multi-period model, no intraday movement

We now consider a multi-period model where the stock price does not moveduring the day and the only inventory risk comes from the possibility of anovernight move. This simplifies the problem, since most of the variance termsin (2.6) drop out and the only source of uncertainty during the trading daycomes from the transactions. In this setting the stock and option premiums a,b, a and b can be computed recursively as functions of time and inventory(see Theorem 3.2).

The stock price is fixed at Si = S for i n and

ST = S+ S(WT Wtn).

Likewise, the option price remains constant at Ci = C throughout the day andthen becomes

CT = C+ Sn(WT Wtn).The objective (2.6) simplifies to:

v0(X0, S0, qs0, qo0, t0) = maxai,bi,ai,bi,0in1

(E[ZT] Var[In])

If we define

vj(Zj, qsj , q

oj ) = max

ai,bi,ai,bi,jin1

E[ZT|Fj] Var[In|Fj]

we may use the dynamic programming principle

vj(Zi, qsi , q

oi ) = max

ai,bi,ai,bi

E[vi+1(Zi+1, qsi+1, q

oi+1)|Fi]

to find the optimal bid and ask quotes by working backward in a tree.The computational task of building a non-recombining 5-nomial tree to

solve such a problem seems daunting. Fortunately, the value function vi canbe expressed as the linear combination of (i) the marked to market wealth Ziaccumulated until time i and (ii) a piecewise quadratic function of inventory. Itfollows that the optimal bid and ask premiums are piecewise linear in inventory.This is summarized in the following theorem.

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Theorem 3.2. The value function at time ti is given by

vi(Zi, qsi , q

oi ) = Zi + wi(q

si , q

oi )

where wi is a quadratic function of inventory

wi(qsi , q

oi ) = mi

qsi + nq

oi

2+ fi

where mi and fi are given recursively by

mi = mi+1 + t

Bx + D2ny

m2i+1

fi = fi+1 + tA2

4B+

1

4Bm2i+1 +

1

2Ami+1

x +

C2

4D+

1

4Dm2i+1

4n +

1

2Cmi+1

2n

y

Ami+1

11{ai=0} + 11{b

i=0}

Cmi+12n

11{a

i=0} + 11{b

i=0}

where x =

11{0


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Figure 1: No intraday movement, low risk aversion ( = 0.006, qs = 0, A = 40,B = 2000, C = 40, D = 200, S= 100, K = 100, = 0.01, n = 100, t0 = 0, tn = 1,T = 1.5, Tmat = 400.)

Figure 2: No intraday movement, high risk aversion ( = 0.1, qs = 0, A = 40,B = 2000, C = 40, D = 200, S= 100, K = 100, = 0.01, n = 100, t0 = 0, tn = 1,T = 1.5, Tmat = 400.)

Also worthy of note is the time-of-the-day effect: near the end of the daytn = 1, the ask quotes are more sensitive to small changes in inventory, thanat the beginning of the day. This reflects the fact that at the beginning of theday, the dealer has more chances to trade in and out of a position, while at theend of the day, he is most likely to get stuck with his marginal inventory.

For each level of risk aversion, there is an optimal quoting strategy, givenby the premiums ai ,

bi ,

ai and

bi . By simulating 1000 days with initial inven-

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Standard deviation of overnight change in wealth

Expected

Profit

= 0.1

= 0.006 =0

Figure 3: Efficient frontier (A = 40, B = 2000, C= 40, D = 200, S= 100, K= 100, = 0.01, n = 100, t0 = 0, tn = 1, T = 1.5, Tmat = 400.)

tory equal to zero, we obtain an efficient frontier (see Figure 3). This frontierdescribes the tradeoff between the expected daily profit and the standard de-viation of the inventory value, for various levels of risk aversion. The discon-tinuities in the frontier are due to our particular modeling of the functions s

and o in (2.3) and (2.4). The profit maximizing point ( = 0) is obtained bya risk neutral dealer whose bid-ask quotes are symmetric around the mid price,regardless of inventory.

We identify two qualitatively different regions of the frontier, of which our

choices of = 0.006 and = 0.1 are representative. The low risk aversiondealer, = 0.006 essentially treats the stock and the option as two similarinstruments by slightly moving quotes on the stock and the option, for moderatelevels of inventory (as illustrated in Figure 1). The highly risk-averse dealer, = 0.1, uses the stock mainly as a hedging instrument by aggressively movingthe stock quotes when his Delta is non-zero (as illustrated in Figure 2). Thistype of dealer would be likely to hedge options very frequently, even at the costof crossing the spread, essentially abandoning his role as a market maker in thestock. We will in fact revisit dealers of this type in Section 4, by modeling adealer who only makes markets in the options and delta-hedges his position atevery time-step. This will require a more sophisticated modeling of the option

dynamics and will highlight the role of higher order risks, Gamma and Vega.

3.3 Multi-period model, constant volatility

We now turn to the general multi-period problem (2.6) with constant volatility.The stock and option prices fluctuate during the course of the n trading sessions.In this case, the optimal bid and ask premiums depend on the price path and

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we describe a Montecarlo procedure to compute the optimal stock and optionpremiums. We first condition on the entire path of the stock price:

u(X0,{

Sj}(0jn), q

s

0

, q0

0

, t0) = maxai ,bi ,ai ,bi ,0in1

E[ZT] n

i=0

V ar[Ii|Sj] .

Once the function u has been computed for a stock path, the value function issimply an average of the form

v(X0, S0, qs0, q

00, t0) = E

u(X0, {Sj}(0jn), qs0, q00, t0)

For each stock path, the dynamic programming principle is now

ui(Zi, qsi , q

oi ) = max

ai,bi,ai,bi

E[ui+1(Zi+1, qsi+1, q

oi+1)|Fi] V ar[Ii|qsi , qoi , Si]

where V ar[Ii|qsi , qoi , Si] = 2S2i t (qsi + qoi i)2

which yields a result similar to Theorem 3.2.


ui(Zi, qsi , q

oi ) = Zi + wi(q

si , q

oi )

where the function is a quadratic function of inventory

wi(qsi , q

oi ) = ai(q

si )2 + bi(q

oi )

2 + ciqsi q

oi + fi

with coefficients given recursively by

ai = ai+1 + t

Ba2i+1x +

1

4Dc2i+1y 2S2i

bi = bi+1 + t

Db2i+1y +

1

4Bc2i+1x 2S2i 2i

ci = ci+1 + t

Bai+1ci+1x + Dbi+1ci+1y 22S2i i

fi = fi+1 + t

A24B

+1

4Ba2i+1 +

1

2Aai+1

x +

C2

4D+

1

4Db2i+1 +

1

2Cbi+1

y

Aai+1 11{a

i=0} + 11{b

i=0}

Cbi+1 11{a

i=0} + 11{b

i=0}

and with terminal conditions

an = 2S2n(T tn)bn = 2S2n (T tn) 2ncn = 22S2n (T tn) nfn = 0.

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Figure 4: Constant volatility ( = 0.006, A = 40, B = 2000, C = 40, D = 200,S= 100, K= 100, = 0.01, n = 100, t0 = 0, tn = 1, T = 1.5, Tmat = 400.)

Moreover, for i < n, the optimal bid and ask premiums at time ti are given by

ai = max

0, min

A

B,

A

2B+ ai+1q

si +

1

2ci+1q

oi

1

2ai+1

bi = max

0, min

A

B,

A

2B ai+1qsi

1

2ci+1q

oi

1

2ai+1

ai = max

0, min

CD ,

C

2D + bi+1qoi +

1

2 ci+1qsi

1

2 bi+1

bi = max

0, min

C

D,

C

2D bi+1qoi

1

2ci+1q

si

1

2bi+1

In Figure 4, we illustrate the effect of the option inventory on the ask pre-miums on the stock and the option for = 0.006. Compared to Figure 1, thesensitivity of quotes with respect to inventory is larger, especially towards thebeginning of the day. Inventory held at the beginning of the day is now viewedas more risky from the point of view of the dealer, since he is penalized for the

variance of inventory throughout the day.

4 Gamma and Vega risk

In this section, we introduce Gamma risk, by considering risk due to discretehedging and Vega risk, by directly modeling the stochastic implied volatility ofthe option. However, we also simplify the problem, by abandoning the dealers

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control of the bid and ask quotes (or equivalently the controls a and b) andassuming that the delta of his option inventory is set to zero at the beginningof every trading period ti.

The market dynamics. The continuous time dynamics of the stock mid priceis given by

dSt = tStdWt

where the volatility t is stochastic. Here we follow the approach of Schonbucher(1999) by directly modeling the implied volatility of the option as

d = dW1t (4.1)

where the Brownian motions Wt and W1t are independent.

The dealer makes markets in a European call option with maturity Tmat >>T and strike K, whose mid price follows

dCt = tdt + tdSt +12

t(dSt)2 + Cdt +

12

C(dt)2

where the function C(S, t) is given by the Black Scholes formula and t, t andt are the standard greeks. Schonbucher (1999) shows that for the choice (4.1),the stochastic volatility of the stock must be related to the implied volatilitythrough the relation

2t = 2t

2

2t

ln

S

K

2 1

4(Tmat t)24t

to avoid arbitrages. Our expression for the change in option value thus becomes

Ci = iiSiu

t +1

2i

2i S

2i (u

2 1)t + C

t (4.2)

where u and are standard normal random variables. The second term is astandard second order approximation (see Boyle and Emanuel (1980)), whichcaptures the risk of discrete hedging. The third term captures the risk of astochastic implied volatility. Using (4.2), the relationship between Gamma andVega

C = S2(Tmat t) (4.3)

and assuming that the options are delta-hedged at every time step

qsi = qoi iwe may use (2.1) to obtain

Ii = qoi+1

1

2i

2i S

2i (u

2 1)t + iiS2i (Tmat ti)

t

. (4.4)

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and

In = qon

1

2n

2nS

2n(u

2 1)t + nnS2n(Tmat tn)

t

. (4.5)

Expression (4.4) highlights the fact that for long maturity options the stochasticvolatility contribution is likely to dominate, while for short maturity options,the risk contribution of discrete hedging will be more important.

The advantage of directly modeling the implied volatility is that it is observ-able and that statistics on its volatility can readily be obtained. Moreover,this approach can be generalized to a model of the entire volatility surface (seeCont and Da Fonseca (2002)).

The objective. The new value function

v(X0, S0, qs0, q

o0, t0) = max

a

i ,b

i ,0in1E[ZT]

n

i=0

V ar[Ii]now only depends on two controls ai and

bi , but the inventory risk now depends

on higher order terms given in (4.4) and (4.5).

4.1 One-period model

The value function can be written as

v(Xn1, Sn1, qsn1, q

on1, tn1) = max

an1

,bn1

(E[ZT] V ar[IT In1])

The dealers objective is to determine the optimal bid and ask quotes on thestock and options market.

Theorem 4.1. The optimal policy for the dealer is given by

an1 = max

0, minCD ,

C2D k

qon1 12

an1 = max

0, min

CD

, C2D + k

qon1 +12

where k =

12

2n1(T tn1) + 2(Tmat tn1)2

2n1S

4n1

2n1(T tn1)

Remark 4.1. In the risk neutral case where = 0, the objective is to maximizeterminal wealth and the optimal premium is

=C

2D

regardless of inventory.

Remark 4.2. Notice that if the dealer is risk-averse, i.e. > 0, he adjusts ortilts all his quotes away from the profit maximizing solution by an amountproportional to the net option position, qon1.

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4.2 Multi-period model

Much like in Section 3.2, we may use the dynamic programming principle tosolve the multi-period model in the setting with no intraday movement. Thevalue function simplifies to

v(X0, S0, qs0, q

o0, t0) = max

ai,bi,0in1

(E[ZT] V ar[In])

and can be computed by applying the following result, analogous to Theorem3.2.


vi(Zi, qsi , q

oi ) = Zi + wi(q

oi )

where wi is a quadratic function of inventory

wi(qoi ) = mi (q

oi )

2 + fi

where ai,bi, ci and fi are given recursively by

mi = mi+1 + t

Dm2i+1y

fi = fi+1 + tC2

4D+

1

4Dm2i+1 +

1

2Cmi+1

y Cmi+1

11{a

i=0} + 11{b

i=0}

where y =

11{0


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Figure 5: Long maturity option for = 0 and = 0.00035 ( = 0.1, A = 40,B = 2000, C = 40, D = 200, S = 100, 0 = 0.01, n = 100, t0 = 0, tn = 1, T = 1.5,Tmat = 400.)

Figure 6: Short maturity option for = 0 and = 0.00035 ( = 0.1, A = 40,B = 2000, C = 40, D = 200, S = 100, 0 = 0.01, n = 100, t0 = 0, tn = 1, T = 1.5,Tmat = 2.)

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In Figure 5 and Figure 6, we illustrate the effect of the maturity, Tmat, andthe volatility of implied volatility, , on the option ask premiums. In Figure 5,we notice that for long maturity options, the magnitude of has an importantimpact on the sensitivity of the quotes with respect to inventory. This reflects

the fact Vega risk is more important than Gamma risk for long maturity options.However, when time-to-maturity is short, Gamma risk dominates and the parameter has virtually no effect on the optimal quoting policy (see Figure 6).

5 Conclusion

We solve two related mean-variance optimization problems for a market makerin European options. The dealer has control over bid and ask quotes andmaximizes expected returns over a one day period, with a penalty that is pro-portional to the variance of the inventory value.

In the first problem, we model the dealer as an agent who simultaneously

updates bid and ask quotes on a European call option and its underlying stock.The mid prices of the stock and the option are related by the Black-Scholesformula. This modeling choice highlights the importance of keeping the netDelta of the inventory under control. Since we model the liquidity of the stockto be greater than that of the option, we find that the optimal strategy is toset a large bid-ask spread around the option mid price and move the stockquotes aggressively as the dealers net Delta departs from zero. By varyingthe coefficient of risk aversion, we are able to numerically compute the efficientfrontier of the dealer. Furthermore, we generalize our problem to a settingwhere the penalty term also depends on the variance of intraday returns.

In the second problem, we model a dealer who Delta hedges the option

inventory with the stock at every time step, but is subject to residual risksdue to stochastic volatility and overnight moves in the stock price. These riskshighlight the need to keep the Vega and Gamma of the dealers inventory undercontrol, and this is reflected in the dealers quoting strategy. We illustrate thisoptimal quoting policy for options of varying maturity. We find that the dealersnet Gamma dominates the quoting policy for short maturity options, while thenet Vega dominates the quoting policy for the long maturity options.

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References

[1] Almgren, R. and Chriss, N. (2001), Optimal Execution of Portfolio Trans-actions, Journal of Risk, 3: 5-39.

[2] Boyle, P. and Emanuel, D. (1980), Discretely Adjusted Option Hedges,Journal of Financial Economics, 8: 259-282.

[3] Cont, R. and da Fonseca, J. (2002), Dynamics of Implied Volatility Sur-faces, Quantitative Finance, 2: 45-60.

[4] Engle, R. and Ferstenberg, R. (2006), Execution Risk, Working paper.

[5] Garleanu, N., Pedersen, L. H. and Poteshman, A. M. (2006), DemandBased Option Pricing, Working Paper, Wharton School of Business.

[6] Hasbrouck, J. (2007), Empirical Market Microstructure, Oxford Univer-sity Press.

[7] Ho, T. S. Y. and Macris, R. G. (1984), Dealer Bid-Ask Quotes and Trans-action Prices: An Empirical Study of Some AMEX Options, Journal ofFinance, 39(1), 23-45.

[8] Ho, T. S. Y. and Stoll H. R. (1981), Optimal Dealer Pricing under Trans-actions and Return Uncertainty, Journal of Financial Economics, 9, 47-73.

[9] Jameson, M. and Wilhelm, W. (1992), Market Making in the OptionsMarkets and the Costs of Discrete Hedge Rebalancing, Journal of Finance,47, 765-779.

[10] Schonbucher, P. J. (1999) A Market Model for Stochastic Implied Volatil-ity, Philosophical Transactions: Mathematical, Physical and EngineeringSciences, 357(1758), 2071-2092.

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Appendix

Proof of Theorem 3.1. Lets first write down the expected wealth

E[ZT|F

i] = Zn1 + E[Zn1|F

n1] + E[In1 + In|F

n1]

= Zn1 + t

as(a) + bs(

b) + ao(a) + bo(

b)

We then compute the variance terms

V ar

In1 + In|Fn1

= E

V ar

In1 + In|qsn, qon, Sn1|qsn1, qon1, Sn1

= E

V ar

qsn(ST Sn1) + qon(CT Cn1)|qsn, qon, Sn1

|qsn1, qon1, Sn1

= E

qsnSn1 + q

onSn1n1

2(T tn1)|qsn1, qon1, Sn1

= s(

a)t(qsn1 1)Sn1 + q

on1Sn1n1

2(T

tn1)

+s(b)t

(qsn1 + 1)Sn1 + q

on1Sn1n1

2(T tn1)

+o(a)t

qsn1Sn1 + (q

on1 1)Sn1n1

2(T tn1)

+s(b)t

qsn1Sn1 + (q

on1 + 1)Sn1n1

2(T tn1)

+(1 s(a)t s(b)t o(a)t o(b)t)

(qsn1Sn1 + qon1Sn1n1)

2(T tn1)

which equals after cancelations:

= (T tn1)

s(a)t

(2qsn1 + 1)2S2n1 2qon12S2n1n1

+s(

b)t(2qsn1 + 1)2S2n1 + 2qon12S2n1n1+o(

a)t

(2qon1 + 1)2S2n12n1 2qsn12S2n1n1

+o(b)t

(2qon1 + 1)

2S2n12n1 + 2q

sn1

2S2n1n1

+

(qsn1)22S2n1 + (q

on1)

22S2n12n1 + 2q

sn1q

on1

2S2n1n1

Using the above formulas, then v(Xn1, Sn1, q

sn1, q

on1, tn1) equals:

maxan1

,bn1

,an1

,bn1

Yn1 + t

as(a) + bs(

b) + ao(a) + bo(

b)

(T

tn1)s(a)t(

2qsn1 + 1)2S2n1

2qon1

2S2n1n1+s(

b)t

(2qsn1 + 1)2S2n1 + 2q

on1

2S2n1n1

+o(a)t

(2qon1 + 1)2S2n12n1 2qsn12S2n1n1

+o(

b)t

(2qon1 + 1)2S2n1

2n1 + 2q

sn1

2S2n1n1

+

(qsn1)22S2n1 + (q

on1)

22S2n12n1 + 2q

sn1q

on1

2S2n1n1

19


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By taking the first order conditions, we can solve for each bid and ask quotefor both the option and the stock:

an1 = max0, minAB , A2B 2(T tn1)S2n1qsn1 + qon1n1 12bn1 = max

0, min

AB

, A2B + 2(T tn1)S2n1

qsn1 + q

on1n1 +

12

an1 = max

0, min

CD ,

C2D 2(T tn1)S2n1n1

qsn1 + q

on1n1 12n1

an1 = max

0, min

CD

, C2D + 2(T tn1)S2n1n1

qsn1 + q

on1n1 +

12n1

Proof of Theorem 3.2. We need the following lemmas to prove Theorem 3.2.

Lemma 5.1. Letx =

11{0


21/27

and

ak = max

0, min

A

B,

A

2B+ ak+1q

sk +

1

2ck+1q

ok

1

2ak+1 +

1

2dk+1

bk = max

0, min

A

B,

A

2B ak+1qsk

1

2ck+1q

ok

1

2ak+1 1

2dk+1

ak = max

0, min

C

D,

C

2D+ bk+1q

ok +

1

2ck+1q

sk

1

2bk+1 +

1

2ek+1

bk = max

0, min

C

D,

C

2D bk+1qok

1

2ck+1q

sk

1

2bk+1 1

2ek+1

Proof. This lemma can be proven by induction.Base case: If i = n, then

vn(Zn, qsn, q

on) = Zn 2S2n(T tn)

qsi + q

oi n

2which implies that

an = 2S2(T tn)bn = 2S22n(T tn)cn = 22S2n(T tn)dn = 0

en = 0.

Inductive step: Suppose Lemma 5.1 holds for i = k + 1 where 0 < k + 1 n.Then for i = k, we use

vk(Zk, qsk, q

ok) = max

ak,bk,ak,bk

E[vk+1(Zk+1, qsk+1, q

ok+1)|Fk]

Using induction hypothesis, vk(Zk, qsk, q

ok) equals

= maxak,bk,ak,bk

E

Zk+1 + ak+1(q

sk)

2 + bk+1(qok)

2 + ck+1qskq

ok + dk+1q

sk + ek+1q

ok + fk+1|Zk, qsk, qok

= Zk +

aks(

ak) +

bks(

bk) +

ako(

ak) +

bko(

bk)

t

+s(ak

)tak+1

(qsk

1)2 + bk+1

(qok

)2 + ck+1

(qsk

1)qok

+ dk+1

(qsk

1) + ek+1

qok

+ fk+1

+s(

bk)t

ak+1(q

sk + 1)

2 + bk+1(qok)

2 + ck+1(qsk + 1)q

ok + dk+1(q

sk + 1) + ek+1q

ok + fk+1

+o(

ak)t

ak+1(q

sk)

2 + bk+1(qok 1)2 + ck+1qsk(qok 1) + dk+1qsk + ek+1(qok 1) + fk+1

+s(

bk)t

ak+1(q

sk)

2 + bk+1(qok + 1)

2 + ck+1qsk(q

ok + 1) + dk+1q

sk + ek+1(q

ok + 1) + fk+1

+(1 s(ak)t s(bk)t o(ak)t o(bk)t)

ak+1(qsk)

2 + bk+1(qok)

2 + ck+1qskq

ok + dk+1q

sk + ek+1q

ok + fk+1

21


22/27

= Zk +

aks(ak) +

bks(

bk) +

ako(

ak) +

bko(

bk)

t

+s(ak)t

ak+1(2qsk + 1) ck+1qok dk+1

+s(

bk)tak+1(2qsk + 1) + ck+1qok + dk+1

+o(ak)t

bk+1(2qok + 1) ck+1qsk ek+1

+s(bk)t

bk+1(2q

ok + 1) + ck+1q

sk + ek+1

+

ak+1(qsk)

2 + bk+1(qok)

2 + ck+1qskq

ok + dk+1q

sk + ek+1q

ok + fk+1

taking first order conditions in

ak(A Bak) + (A Bak) 2ak+1qsk ck+1qok + ak+1 dk+1

ak(A Bbk) + (A Bbk)

2ak+1qsk + ck+1q

ok + ak+1 + dk+1

ak(C D

ak) + (C D

ak) 2bk+1qok ck+1qsk + bk+1 ek+1

bk(C Dbk) + (C Dbk)

2bk+1qok + ck+1q

sk + bk+1 + ek+1

obtains

ak = max

0, min

A

B,

A

2B+ ak+1q

sk +

1

2ck+1q

ok

1

2ak+1 +

1

2dk+1

bk = max

0, min

A

B,

A

2B ak+1qsk

1

2ck+1q

ok

1

2ak+1 1

2dk+1

ak = max

0, min

CD , C2D + bk+1qok + 12 c

k+1qsk 12 bk+1 + 12 ek+1

bk = max

0, min

C

D,

C

2D bk+1qok

1

2ck+1q

sk

1

2bk+1 1

2ek+1

22


23/27

substituting back in the value function yields

vk(Zk, qsk, q

ok) = Zk + 11{0


24/27

Proof. This lemma can also be proven by induction.Base case: If i = n, then

an = 2S2(T tn)bn = an2n

cn = 2ann

dn = 0

en = 0.

Inductive step: Suppose Lemma 5.2 holds for i = k + 1 where 0 < k + 1 n.Then for i = k, we use the following identities from Lemma 5.1.

bk = bk+1 + t

Db2k+1y +

1

4Bc2k+1x

ck = ck+1 + t (Bak+1ck+1x + Dbk+1ck+1y)

dk = dk+1 + t

Bak+1dk+1x +

1

4Dck+1ek+1y

ek = ek+1 + t

Dbk+1ek+1x +

1

4Bck+1dk+1y

Using induction hypothesis,

bk = ak+12n + t

yD

ak+1

2n

2+

1

4xB (2ak+1n)

2

= 2nak+1 + txBa2k+1 +

1

4yDc2k+1

= 2nak

ck = ak+1n + t

xBak+1ak+1n + yDak+12nak+1n

= 2n

ak+1 + t

xBa2k+1 +

1

4yc2k+1

= 2nak

dk = dk+1 + t

xBak+1dk+1 +

1

4yDck+1ek+1

= 0 + t

xBak+10 +

1

4yDck+10

= 0

ek = ek+1 + t

yDbk+1ek+1 + 14

xBck+1dk+1

= 0 + t

yDbk+10 +

1

4xck+10

= 0

24


25/27

By using Lemma 5.1 and Lemma 5.2, and letting mi = ai, we get

mi = mi+1 + t

Bm2i+1

11{0


26/27

plying (4.4) and (4.5) and simplifying

V ar

In + In1|Fn1

= EV arIn + In1|qon, Sn1|

n1, qon1, Sn1

= E

12q

onn1S

2n1

2n1(u

2 1)(T tn1)2|n1qon1, Sn1

+E

qonn1n1S

2n1(Tmat tn1)

2(T tn1)|n1, qon1, Sn1

= o(

a)t

(qon1 1)2n1S2n1n12

12(T tn1)2

+

(qon1 1)2n1S2n1n1(Tmat tn1)2

(T tn1)

+o(b)t

(qon1 + 1)

2n1S

2n1n1

212(T tn1)2

+

(qon1 + 1)2n1S

2n1n1(Tmat tn1)

2(T tn1)

+1

o(a)t

o(

b)tqon12n1S2n1n1212(T

tn1)

2

+

qon12n1S

2n1n1(Tmat tn1)

2

(T tn1)

which equals after cancelations:

= 2n1S4n1

2n1(T tn1)

o(

a)t

(2qon1 + 1)12


+o(

b)t

(2qon1 + 1)12


+ (qon)

2 12


Using the above formulas, v(Xn1, Sn1, q

sn1, q

on1, tn1) equals:

maxan1

,bn1 Yn1 + t

ao(a) + bo(

b)2n1S4n12n1(T tn1)o(a)t (2qon1 + 1) 122n1(T tn1) + 2(Tmat tn1)2+o(

b)t

(2qon1 + 1)12


+ (qon)

2 12


By taking the first order conditions, we can solve for each bid and ask quotefor the option:

an1 = max

0, minCD

, C2D k

qon1 12

an1 = max0, minCD

, C2D + kqon1 +12

where k =12


2n1S

4n1

2n1(T tn1)

Proof of Theorem 4.2. This proof is very similar to that of Theorem 3.2 and itcan also be proven by induction.Base case: If i = n, then

vn(Zn, qon) = Zn(qon)2 2nS22n (T tn)

1

22nS

2(T tn) + 2S2(Tmat tn)2

26


27/27

which implies that

mn = 2nS22n (T tn)

1

22nS

2(T tn) + 2S2(Tmat tn)2

Inductive step: Suppose Theorem 4.2 holds for i = k + 1 where 0 < k + 1 n.Then for i = k, we usevk(Zk, q

ok) = max

ak,bk

E[vk+1(Zk+1, qok+1)|Fk]

Using induction hypothesis, vk(Zk, qok) equals

= maxak,bk

E

Zk+1 + mk+1(q

ok)

2 + fk+1|Zk, qok

= Zk +

ako(ak) +

bko(

bk)

t + o(ak)t

mk+1(q

ok 1)2 + fk+1

+s(

bk)t

mk+1(q

ok + 1)

2 + fk+1

+ (1 o(ak)t o(bk)t)

mk+1(q

ok)

2 + fk+1

= maxak,bk

EZk+1 + mk+1(q

o

k)2

+ fk+1|Zk, qo

k

= Zk +

ako(ak) +

bko(

bk)

t + o(ak)t

mk+1(2qok + 1)

+s(

bk)t

mk+1(2q

ok + 1)

+

mk+1(qok)

2 + fk+1

taking first order conditions in

ak(C Dak) + (C Dak) 2mk+1qok

bk(C Dbk) + (C Dbk)

2mk+1q

ok

obtains

a

k

= max0, minCD

,C

2D+ mk+1q

oi

1

2mk+1

bk = max

0, min

C

D,

C

2D mk+1qoi

1

2mk+1

substituting back in the value function yields

vk(Zk, qok) = Zk + 11{0

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