SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing

TerminologyOne-Period Binomial Model

Multiple Periods: Binomial ModelMore General Situations

Black-Scholes TheoryAlternate Models for Asset Prices

SYSM 6304: Risk and Decision AnalysisLecture 6: Pricing and Hedging Financial

Derivatives

M. Vidyasagar

Cecil & Ida Green ChairThe University of Texas at DallasEmail: [email protected]

October 31, 2015

M. Vidyasagar Pricing and Hedging Financial Derivatives




Outline

1 Terminology

2 One-Period Binomial Model

Problem Formulation

The Replicating Portfolio

3 Multiple Periods: Binomial Model

4 More General Situations

5 Black-Scholes Theory

6 Alternate Models for Asset Prices





Outline

1 Terminology


Problem Formulation










Portfolio

A market consists of a “safe” asset usually referred to as a “bond,”together with one or more “uncertain” assets usually referred to as“stocks.”

A “portfolio” is a set of holdings, consisting of the bond and thestocks. If there are n stocks, we think of the portfolio as an(n+ 1)-dimensional vector (a0, a1, . . . , an), where each ai is a realnumber. a0 is the quantity of the bond we hold and ai is thequantity of stock i that we hold.





Portfolio (Cont’d)

We can think of a portfolio as a vector in Rn+1. Each ai can bepositive or negative. Negative “holdings” correspond to borrowingmoney or “shorting” stocks.

Note: In a multi-period investment problem, ai is actually ai(t),for i = 0, . . . , T − 1, where T is the duration of the planningperiod. So we keep adjusting our portfolio from one time instantto the next.

Note: We take our last investment decision at time T − 1, whoseoutcome will become known at time T .





Self-Financing

Our investment strategy has to be “self-financing.” In other words,we cannot introduce fresh money into the system. The logic isthat any “extra” money we had at the start would be reflected inthe holding a0(0) of the “safe” asset.

This introduces a constraint on the portfolios at successive times,as shown next.





Self-Financing (Cont’d)

The sequence of events is as follows: At time 0, we have a certainamount of money available to us. The prices of the safe assetS0(0) and of the “risky” assets S1(0), . . . , Sn(0) are known to uswhen we make the initial investment decision.

Suppose the amount of money we have available to us is V (0). Sothe initial portfolio vector a(0) ∈ Rn+1 must satisfy

n∑i=0

ai(0)Si(0) = V (0).





Self-Financing (Cont’d)

At time t = 0, the risky assets S1, . . . , Sn are to be viewed asrandom variables. But at time t = 1, they are no longer random,but have known values S1(1), . . . , Sn(1). So at time t = 1, thevalue of our portfolio is

V (1) =

n∑i=1

ai(0)Si(1).

At time t = 1 we are free to reallocate the money as we wish. Butwe must stay within the available funds. This can be expressed as

n∑i=1

ai(1)Si(1) = V (1) =

n∑i=1

ai(0)Si(1).

This is called the self-financing constraint.M. Vidyasagar Pricing and Hedging Financial Derivatives




Options

A “call option” is an instrument that gives the buyer the right, butnot the obligation, to buy a stock a prespecified price called the“strike price” K. (A “put” option gives the right to sell at a strikeprice.)

A “European” option can be exercised only at a specified time T .An “American” option can be exercised at any time prior to aspecified time T .

If St is the price of the stock as a function of time, T is thematurity date, and K is the strike price, then the value of theEuropean option is {ST −K}+, the nonnegative part of ST −K.

The value of the American option at time t ≤ T is {St −K}+.





European vs. American Options

The European option is worthless even though St > K for someintermediate times. The American option has positive value atintermediate times but is worthless at time t = T .





Derivatives or Contingent Claims

An option (call or put) is a special case of a “contingent claim”whereby the value of the instrument is “contingent” upon thevalue of an underlying asset at (or before) a specified time.

The terminology ”derivative” is also used in the place of“contingent claim” because value of the instrument is “derived”from that of an underlying asset.





Typical Questions

Suppose there is only one risky asset (the stock) and one safe asset(the bond). Suppose the period T and the strike price K for anoption have been mutually agreed upon by the buyer and seller ofthe option.

What is the minimum price that the seller of an option shouldbe willing to accept?

What is the maximum price that the buyer of an optionshould be willing to pay?

How can the seller (or buyer) of an option “hedge”(minimimze or even eliminate) his risk after having sold (orbought) the option?





Problem FormulationThe Replicating Portfolio

Outline

1 Terminology


Problem Formulation











Outline

1 Terminology


Problem Formulation











One-Period Binomial Model

Many key ideas can be illustrated via “one-period binomial” model.

We have a choice of investing in a “safe” bond or an “uncertain”stock.

B(0) = Price of the bond at time T = 0.

It increases to B(1) = (1 + r)B(0) at time T = 1, where thenumber r (the guaranteed return) is known at time T = 0.

S(0) = Price of the stock at time T = 0.

S(1) =

{S(0)u with probability p,S(0)d with probability 1− p.






Simplification

Assumption: d < 1 + r < u; otherwise problem is meaningless!

Rewrite as d′ < 1 < u′, where d′ = d/(1 + r), u′ = u/(1 + r).Convert everything (bond and stock price) to “constant” currency,and drop the superscript prime.

So: Value of bond at time T = 1 is the same as the value of thebond at time T = 0, or B(1) = B(0).

Stock price at time T = 1 is

S(1) =

{S(0)u with probability p,S(0)d with probability 1− p,

whered < 1 < u.






Explanation of Nomenclature

Why is this called a “one-period binomial” model?

Because

We are following the stock for just one time period, and

The stock price has only two possible values at time T = 1.






Options

An “option” gives the buyer the right, but not the obligation, tobuy the stock at time T = 1 at a predetermined strike price K.

Again, assume S(0)d < K < S(0)u. Otherwise the problem makesno sense.

If stock goes up, the value of the option at time T = 1 isS(1)−K = S(0)u−K. If the stock goes down, the value of theoption at time T = 1 is zero, because the option is worthless.

So if X is the value of the option at time T = 1, then

X =

{{S(1)−K}+ if S(1) = S(0)u,

0 if S(1) = S(0)d.






Contingent Claims

More generally, a “contingent claim” is a random variable X suchthat

X =

{Xu if S(1) = S(0)u,Xd if S(1) = S(0)d.






Option Example

Suppose B(0) = S(0) = 1, u = 1.3, d = 0.9, p = 0.8, 1− p = 0.2.So the stock price S(1) has values

S(1) =

{1.3 with probability 0.8,0.9 with probability 0.2.

So the expected value of the stock price at time T = 1 is

E[S(1)] = 1.3× 0.8 + 0.9× 0.2 = 1.22.

So “on average” the stock offers a positive return.






Option Example (Cont’d)

Suppose the strike price is K = 1.1. Then the value of the optionX satisfies

{S(1)−K}+ =

{0.2 with probability 0.8,0 with probability 0.2.

The expected value of the option is given by

E[{S(1)−K}+] = 0.16.






Option Example (Cont’d)

A more general “contingent claim” is specified by{Xu = 4 if S(1) = S(0)u,Xd = −1 if S(1) = S(0)d.

The expected value of this contingent claim is

E(X) = 4× 0.8− 1× 0.2 = 3.






An Incorrect Intuition

Question: How much should the seller of such a claim charge forthe claim at time T = 0?

Is it the expected value of the claim, namely

E[X,p] = pXu + (1− p)Xd?

NO! The seller of the claim can “hedge” against futurefluctuations of stock price by using a part of the proceeds to buythe stock himself.






An Incorrect Intuition (Cont’d)

Consider the option with strike price K = 1.1, whose expectedvalue is 0.16.

Suppose the buyer of the option pays 0.16 to the option seller. Ifthe seller of the option just “sits” on the stock, hoping that itwon’t go up, then on average he will break even.

If the stock goes up, he has to pay 0.2. But he already has0.16; so he has to pay only 0.04, with probability 0.8.

If the stock goes down, he owes nothing, and gets to keep the0.16 with probability 0.2.

Net profit is zero (break-even).







But the seller can invest the 0.16 in the same stock against whichhe has sold the option.

If the stock goes up, he loses 0.2 on the option, but his stockis now worth 0.16× 1.3 = 0.208, so he makes a net profit of0.08, with probability of 0.8.

If the stock goes down, then he loses nothing on the option,but his stock is now worth 0.16× 0.9 = 0.144, withprobability 0.2.







So the expected return by selling the option and “hedging” byinvesting the proceeds in the stock is

0.08× 0.8 + 0.144× 0.2 = 0.0928 > 0.

More to the point, this profit is risk-free, because the seller ofthe option makes money for every outcome!






Arbitrage-Free Price

The previous calculation shows that the price of 0.16 for the optionis too high, because the seller can make a risk-free profit – hemakes a profit whether the stock goes up or goes down! This isknown as “arbitrage.”

So what is the “fair” or arbitrage-free price for this option?

The answer is: 0.05. The correct strategy for the seller of theoption is to take the 0.05 from selling the option, borrow another0.45, and buy 0.5 of the stock.






Outline

1 Terminology


Problem Formulation











The Replicating Portfolio: General Idea

Build a portfolio at time T = 0 such that its value exactly matchesthat of the claim at time T = 1 irrespective of stock pricemovement.

Choose real numbers ξ and b (quantity of stocks and bondsrespectively) such that (remember that B(1) = B(0))

ξS(0)u+ bB(0) = Xu,

ξS(0)d+ bB(0) = Xd,






The Replicating Portfolio: Algebraic Details

In vector-matrix notation

[ξ b]

[S(0)u S(0)dB(0) B(0)

]= [Xu Xd].

This equation has a unique solution for a, b if u 6= d. It is called a“replicating portfolio”.






Numerical Example: Option

Suppose B(0) = S(0) = 1, and u = 1.3, d = 0.9. The value of theoption with a strike price K = 1.1 is Xu = 0.2 if the stock goesup, and Xd = 0 is the stock goes down. Then the uniquereplicating portfolio is given by

[ξ b] = [Xu Xd]

[S(0)u S(0)dB(0) B(0)

]−1= [0.2 0]

[1.3 0.91 1

]−1= [0.5 − 0.45].

So the option seller borrows 0.45 and buys 0.5 units of the stock.






Numerical Example (Cont’d)

Check:

If stock goes up, then value of portfolio at time T = 1 is0.5× 1.3− 0.45 = 0.2.

If the stock goes down, then at time T = 1, 0.5× 0.9− 0.45 = 0.

The cost of implementing this strategy is 0.5− 0.45 = 0.05, whichis the cost of the option.






Numerical Example: Contingent Option

Suppose B(0) = S(0) = 1, and u = 1.3, d = 0.9. The contingentclaim is Xu = 4, Xd = −1. Then the unique replicating portfolio isgiven by

[ξ b] = [Xu Xd]

[S(0)u S(0)dB(0) B(0)

]−1= [4 − 1]

[1.3 0.91 1

]−1= [12.5 − 12.25].

So option seller “shorts” 12.25 units of bonds (i.e. borrow $ 12.25)and buys 12.5 units of stock.






Numerical Example (Cont’d)

Check:

If stock goes up, then value of portfolio at time T = 1 is12.5× 1.3− 12.25 = 4.

If the stock goes down, then at time T = 1,12.5× 0.9− 12.25 = −1.

The cost of implementing this strategy is 12.5− 12.25 = 0.25,which is the cost of the contingent claim.






Interpretation: Risk-Neutral Measure

The unique solution for a, b is

[ξ b] = [Xu Xd]

[u d1 1

]−1 [1/S(0) 0

0 1/B(0)

].

Amount of money needed at time T = 0 to implement thereplicating strategy is

c = [ξ b]

[S(0)B(0)

]= [Xu Xd]

[quqd

],

where

q :=

[quqd

]=

[u d1 1

]−1 [11

]=

[ 1−du−du−1u−d

]M. Vidyasagar Pricing and Hedging Financial Derivatives





Interpretation: Risk-Neutral Measure (Cont’d)

Note that qu, qd > 0 and qu + qd = 1. So q := (qu, qd) is aprobability distribution on S(1). Moreover it is the uniquedistribution such that

E[S(1),q] = S(0)u1− du− d

+ S(0)du− 1

u− d= S(0),

i.e. such that the stock also becomes “risk-neutral” like the bond.

Important point: q depends only on the returns u, d, and not onthe associated “real world probabilities p, 1− p.






Interpretation of Cost of Replicating Portfolio

Thus the initial cost of the replicating portfolio

c = [Xu Xd]

[quqd

]= E[X,q]

is the discounted expected value of the contingent claim X underthe unique risk-neutral distribution q.






Numerical Example Revisited

We had u = 1.3, d = 0.9. So the unique risk-neutral distribution qsuch that E[S(1),q] = S(0) is given by q = [0.25 0.75].

For the option, the cost of the replicating portfolio is0.2× 0.25 + 0× 0.75 = 0.05.

If [Xu Xd] = [4 − 1], then the cost of the replicating portfolio is

E[X,q] = 4× 0.25 + (−1)× 0.75 = 0.25.






Arbitrage-Free Price for a Contingent Claim

Theorem: The quantity

c = [Xu Xd]

[quqd

]= E[X,q]

is the unique arbitrage-free price for the contingent claim.

Suppose someone is ready to pay c′ > c for the claim. Then theseller collects c′, invests c′ − c in a risk-free bond, uses c toimplement replicating strategy and settle claim at time T = 1, andpockets a risk-free profit of c′ − c. This is called an “arbitrageopportunity”.






Example of Arbitrage

Recall the option with strike price K = 1.1. Suppose someone iswilling to pay, say, 0.07 for the option. The seller can us 0.05 toimplement the replicating strategy, which guarantees that he comesout even at period 1, no matter what happens. The excess of 0.02is his “risk-free profit.” Hence this is an arbitrage opportunity.

On the other side, suppose you are the buyer of the option, andsomeone is foolish enough to sell you the option for 0.04. You canthen sell your own option for 0.05, use the replicating strategy tobreak even at period 1, and the balance of 0.05− 0.04 = 0.01 isyour risk-free profit.

Conclusion: The quantity E[X,q] is the unique price at whichneither the buyer nor seller has an arbitrage opportunity.






Optimal Hedging Strategy

It is not enough to set the right price for the option or contingentclaim (price discovery); it is also necessary to use an optimalhedging strategy.

Recall that the “optimal” hedging strategy is to charge 0.05 forthe option, borrow 0.45, and buy 0.50 of the stock.

If the stock goes up, the seller loses 0.2 on the option, but thevalue of his holding is 0.50× 1.3− 0.45 = 0.20.

If the stock goes down, the seller loses nothing on the option, butthe value of his holding is 0.50× 0.9− 0.45 = 0. This is why it is areplicating portfolio.






Suboptimal Hedging Strategy

What happens if the seller doesn’t follow this strategy?

Suppose the seller borrows only 0.25 and buy 0.30 of the stock.

If the stock goes up, the seller loses 0.2 on the option, and hisportfolio is worth 0.30× 1.3− 0.25 = 0.14 < 0.20. So he loses0.06.

If the stock goes down, the seller loses nothing on the option, andhis portfolio is worth 0.30× 0.9− 0.25 = 0.02. So he makes aprofit of 0.02.

So the expected return is −0.06× 0.8 + 0.02× 0.2 = −0.044.

Similar argument if he borrows too much instead of too little.






Alternate Formulation of Arbitrage-Free Price

V (0) = xdqd+xuqu =xd(u− 1) + xu(1− d)

u− d= xd+

xu − xdu− d

(1−d).

This can be given a pictorial interpretation, as in the next slide.






Pictorial Interpretation of Arbitrage-Free Price

α

x

d u1

xd

xuV (0)

α

x

d u1

xd

xuV (0)

(a) (b)






Summary

Fact 1. There is a unique “synthetic” distribution q on S(1) suchthat E[S(1),q] = S(0), so that the stock also becomesrisk-neutral. This distribution q depends only on the two possibleoutcomes, but not on the associated “real world” probabilities.

Fact 2. The unique arbitrage-free price of a contingent claim(Xu, Xd) is the discounted expected value of the claim under therisk-neutral distribution q.

Fact 3. There is a unique “replicating” strategy that allows theseller of the derivative to hedge his risk completely irrespectiveofoutcome.





Outline

1 Terminology


Problem Formulation










Multiple Periods: Binomial Model

Bond price is deterministic:

Bt+1 = (1 + rt)Bt, t = 0, . . . , T − 1,

where the interest rates r0, . . . , rT−1 are known beforehand. Stockprice can go up or down: St+1 = Stut or Stdt.

Note: No reason to assume that either the rt or the ut, dt areconstant – they can vary with time.





Normalized Multiple Period Binomial Model

Since return on bond at each time is known beforehand, expresseverything in constant currency: Bt = B0 for all t, andSt+1 = Stut or Stdt where the returns ut, dt have been normalizedwith respect to the bond returns. So dt < 1 < ut for all t.

Since stock can go up or down at each time instant, there are 2T

possible sample paths for the stock. Define {u, d}T to be the set ofstrings of length T where each symbol is either u or d. Then eachstring h ∈ {u, d}T defines one possible evolution of the stock price.





Illustration: Three Periods

S0

S0u0

S0d0

S1u1

S1d1

S1u1

S1d1

S2u2

S2d2S2u2

S2d2

S2d2

S2u2

S2d2

S2u2





Problem Formulation

For each sample path h ∈ {u, d}T , at time T there is an associatedpayout Xh. Note: In other words, final payout may depend notonly on the final stock price ST , but also the path to the final price.

Claim becomes due only at the end of the time period (Europeancontingent claim). In an “American” option, the buyer chooses thetime of exercising the option.

Questions

What is the arbitrage-free price for this claim?

How does the seller of the claim “hedge” against variations instock price?





Replicating Strategy for T Periods

We already know to replicate over one period. We can extend theargument to T periods.

Suppose j ∈ {u, d}T−1 is the set of stock price transitions up totime T − 1. Let Sj denote the stock price at time T − 1corresponding to this set of stock price movements. So now thereare only two possibilities for the final sample path: ju and jd. Thestock price at time T can therefore go up by uT or down by dT .As before, compute the risk-neutral probability distribution

qT−1 =

[1− dT−1

uT−1 − dT−1uT−1 − 1

uT−1 − dT−1

]=: [qu,T qd,T ].





Replicating Strategy for T Periods (Cont’d)

Only two possible payouts: Xju and Xjd. Denote these by cju andcjd respectively. At time T − 1, compute a cost cj and a replicatingportfolio [aj bj] to replicate this claim, namely:

cj = (cjuqu,n + cjdqd,n) .

[aj bj] = [cju cjd]

[Sj SjB0 B0

]−1.





Replicating Strategy for T Periods (Cont’d)

Do this for each j ∈ {u, d}T−1. So if we are able to replicate eachof the 2T−1 payouts cj at time T − 1, then we know how toreplicate each of the 2T payouts at time T .

Repeat backwards until we reach time T = 0. Number of payoutsdecreases by a factor of two at each time step.





Two-Period Example

Bond price Bt = 1 for all t, and S0 = 1. S1 can go up to 1.3 ordown to 0.9 times S0. S2 can go up to 1.2 or down to 0.9 times S1.

Option at final time with a strike price of K = 1.05.

Next slide shows possible paths and corresponding payouts.





Two-Period Example (Cont’d)

1

1.3

0.9

1.56

0.51

1.17

0.12

1.08

0.03

0.81

0.00






Risk-neutral distribution at period t = 1 is given, as before, by

q1 :=

[qu,1qd,1

]=

[u1 d11 1

]−1 [11

]=

[1−d1u1−d1u1−1u1−d1

]=

[1/32/3

].

Similarly risk-neutral distribution at period t = 0 is given, asbefore, by

q0 :=

[qu,0qd,0

]=

[u0 d01 1

]−1 [11

]=

[1−d0u0−d0u0−1u0−d0

]=

[1/43/4

].






1

0.07

1.3

0.25

0.9

0.01

1.56

0.51

1.17

0.12

1.08

0.03

0.81

0.00






At t = 1, top node price equals

(1/3)× 0.51 + (2/3)× 0.12 = 0.25,

while at the bottom top node price equals

(1/3)× 0.03 + (2/3)× 0.00 = 0.01.

At t = 0, node price equals

(1/4)× 0.25 + (3/4)× 0.01 = 0.07.

This is the arbitrage-free price for the option.

If we had more complex payouts, the arbitrage-free price can becomputed entirely similarly.





Replicating Strategy in Multiple Periods

The unique arbitrage-free price is

c0 :=∑

h∈{u,d}TXhqh.

c0 is the expected value of the claim Xh under the syntheticdistribution {qh} that makes the discounted stock pricerisk-neutral. Moreover, c0 is the unique arbitrage-free price for theclaim.





Implementation of Replicating Strategy

Seller of claim receives an amount c0 at time t = 0 and invests a0in stocks and b0 in bonds, where

[ξ0 b0] = [cu cd]

[S0u0 S0d0B0 B0

]−1.

Due to replication, at time t = 1, the portfolio is worth cu if thestock goes up, and is worth cd if the stock goes down. At timet = 1, adjust the portfolio according to

[ξ1 b1] = [ci0u ci0d]

[S1u1 S1d1B1 B1

]−1,

where i0 = u or d as the case may be. Then repeat.






Arbitrage-free price at time t = 0 is 0.07. The optimal hedgingstrategy at time t = 0 is given by

[ξ0 b0] = [0.25 0.01]

[1.3 0.91 1

]−1= [0.60 − 0.53].

So at time t = 0, seller of the option takes 0.07 from the buyer,borrows 0.53, and buys 0.60 units of stock.






Suppose that stock goes up at time t = 1. Then value of portfoliois 0.25 (that is how it was chosen). Optimal hedging strategy attime t = 1 is given by

[ξ1 b1] = [0.51 0.12]

[1.2 0.91 1

]−1= [1.30 − 1.05].

So the option seller borrows 1.05, adds to the 0.25 on hand, andbuys 1.3 units of stock.






Suppose that stock goes down at time t = 1. Then value ofportfolio is 0.01 (that is how it was chosen). Optimal hedgingstrategy at time t = 1 is given by

[ξ1 b1] = [0.03 0.00]

[1.2 0.91 1

]−1= [0.10 − 0.09].

So the option seller borrows 0.09, adds to the 0.01 on hand, andbuys 0.10 units of stock.

This is known as portfolio rebalancing at each time step.





Self-Financing

It is true thatξ0S1 + b0B1 = ξ1S1 + b1B1

whether S1 = S0u0 or S1 = S0d0 (i.e. whether the stock goes up ordown at time t = 1). So no fresh infusion of funds is required aftertime n = 0. This property has no analog in the one-period case.

It is also replicating from that time onwards.

Observe: Implementation of replicating strategy requiresreallocation of resources T times, once at each time instant.

Therefore this theory assumes no transaction costs.





Complete Markets

The multi-period binomial model is an example of a completemarket, because for every sample path, there is a hedging strategythat can ensure that the final payout is zero.

If this is not the case, then we get an “incomplete” market.”





Outline

1 Terminology


Problem Formulation










More General Situations (Not Discussed)

Multinomial model: At each time instant, the stock can takemore than two values.

Stochastic interest rates: The “guaranteed” returns on bondsrn are themselves random variables.

Transactions: Moving from one type of asset to anotherentails costs.

Multiple stocks and/or multiple bonds.

Each of these situations can be handled, but not with elementaryalgebra.

Finally, if make time continuous instead of discrete, we get thefamous Black-Scholes theory of option pricing.





Outline

1 Terminology


Problem Formulation










Continuous-Time Processes

Take “limit” at time interval goes to zero and T →∞; binomialasset price movement becomes geometric Brownian motion:

St = S0 exp

[(µ− 1

2σ2)t+ σWt

], t ∈ [0, T ],

where Wt is a standard Brownian motion process. µ is the “drift”of the Brownian motion and σ is the “volatility.”

Bond price is deterministic: Bt = B0ert. Claim is European and a

simple option: XT = {ST −K}+.

What we can do: Make µ, σ, r functions of t and not constants.

What we cannot do: Make σ, r stochastic! (Stochastic µ is OK.)





Black-Scholes Formula

Theorem (Black-Scholes 1973): The unique arbitrage-free optionprice is

C0 = S0Φ

(log(S0/K

∗)

σ√T

+1

2σ√T

)−K∗Φ

(log(S0/K

∗)

σ√T

− 1

2σ√T

),

where

Φ(c) =1√2π

∫ c

−∞e−u

2/2du

is the Gaussian distribution function, and K∗ = e−rTK is thediscounted strike price.





Black-Scholes PDE

Consider a general payout function erTψ(e−rTx) to the buyer ifST = x (various exponentials discount future payouts to T = 0).Then the unique arbitrage-free price is given by

C0 = f(0, S0),

where f is the solution of the PDE

∂f

∂t+

1

2σ2x2

∂2f

∂x2= 0, ∀(t, x) ∈ (0, T )× (0,∞),

with the boundary condition

f(T, x) = ψ(x).

No closed-form solution in general (but available ifψ(x) = (x−K∗)+).





Black-Scholes PDE

Consider a general payout function erTψ(e−rTx) to the buyer ifST = x (various exponentials discount future payouts to T = 0).Then the unique arbitrage-free price is given by

C0 = f(0, S0),

where f is the solution of the PDE

∂f

∂t+

1

2σ2x2

∂2f

∂x2= 0, ∀(t, x) ∈ (0, T )× (0,∞),

with the boundary condition

f(T, x) = ψ(x).

No closed-form solution in general (but available ifψ(x) = (x−K∗)+).





Replicating Strategy in Continuous-Time

Define

C∗t = C0 +

∫ t

0fx(s, S∗s )dS∗s , t ∈ (0, T ),

where the integral is a stochastic integral, and define

αt = fx(t, S∗t ), βt = C∗t − αtS∗t .

Then hold αt of the stock and βt of the bond at time t.

Observe: Implementation of self-financing fully replicating strategyrequires continuous trading and no transaction costs.





Practical Considerations

Black-Scholes theory gives closed-form formulas when thederivative is a simple option. For more complicated derivatives, agood approach is to divide the total time [0, T ] into N equalintervals, create a binomial model, and then let N →∞.

The Financial Instruments Toolbox of Matlab containsseveral tools for approximating continuous-time processes bybinomial trees, and for price determination.

The Cox-Ross-Rubinstein (CRR) tree converges very slowly, andthe Leisen-Reimer tree is to be preferred.

Other approaches are to solve the Black-Scholes PDE usingnumerical techniques.





Extensions to Multiple Assets

Binomial model extends readily to multiple assets.

Black-Scholes theory extends to the case of multiple assets of theform

S(i)t = S

(i)0 exp

[(µ(i) − 1

2[σ(i)]2

)t+ σW

(i)t

], t ∈ [0, T ],

where W(i)t , i = 1, . . . , d are (possibly correlated) Brownian

motions.

Analog of Black-Scholes PDE: C0 = f(0, S(1)0 , . . . , S

(d)0 ) where f

satisfies a PDE. But no closed-form solution for f in general.





American Options

An “American” option can be exercised at any time up to andincluding time T .

So we need a “super-replicating” strategy: The value of ourportfolio must equal or exceed the value of the claim at all times.

If Xt = {St −K}+, then both price and hedging strategy are sameas for European claims.

Very little known about pricing American options in general.Theory of “optimal time to exercise option” is very deep anddifficult.





Sensitivities and the “Greeks”

Recall C0 = f(0, S0) is the correct price for the option underBlack-Scholes theory. (We need not assume the claim to be asimple option!)

∆ =∂C0

∂S0,Γ =

∂∆

∂S0=∂2C0

∂S20

,

Vega = ν =∂C0

∂σ, θ = −∂f(t,Xt)

∂t, ρ =

∂f(t,Xt)

∂r.

“Delta-hedging”: A strategy such that ∆ = 0 – return isinsensitive to initial stock price (to first-order approximation).

“Delta-gamma-hedging”: A strategy such that ∆ = 0,Γ = 0 –return is insensitive to initial stock price (to second-orderapproximation).





Outline

1 Terminology


Problem Formulation










Can Geometric Brownian Motion Model be Changed?

Black-Scholes theory is predicated on the assumption that asssetsfollow a “geometric Brownian motion” model, or asset returnshave a Gaussian distribution.

We have seen that, even with “respectable” stocks such as theDow Jones 30 stocks, this is simply not true. A stable distributionwith α around 1.8 or 1.7 provides better approximation.

So what happens to option pricing theory in this situation?

Short Answer: It pretty much falls apart.





Incomplete Markets

The binomial model had the very useful feature that there exists areplicating portfolio.

In other words, ignoring transaction costs and permittingcontinuous trading, it is possible to rebalance one’s portfolioconstantly so as to ensure that the current value of the portfolioexactly equals the value of the derivative.

Such a situation is called a complete market.

Unfortunately, geometric Brownian assets are the only model thatleads to complete markets. All others lead to “incomplete”markets.





Incomplete Markets (Cont’d)

This means that there is no unique arbitrage-free price. Instead,there is a range of prices at which neither the buyer nor the sellercan gain arbitrage (i.e., risk-free profit.

However, there are various possible hedging strategies, and it is notclear which one(s) is (are) best.

This a subject of current research.


Documents

SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing