The document provides an introduction to quantitative finance concepts including option pricing models. It begins with an outline and terminology. It then covers the Black-Scholes option pricing model, which uses stochastic calculus to derive a partial differential equation for pricing European options. The document also discusses replicating strategies in discrete and continuous time models, as well as extensions like American options and the Greeks.
Risk-Aversion, Risk-Premium and Utility TheoryAshwin Rao
This lecture helps understand the concepts of Risk-Aversion and Risk-Premium viewed from the lens of Utility Theory. These are foundational economic concepts used widely in Financial applications - Portfolio problems and Pricing problems, to name a couple.
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
Risk-Aversion, Risk-Premium and Utility TheoryAshwin Rao
This lecture helps understand the concepts of Risk-Aversion and Risk-Premium viewed from the lens of Utility Theory. These are foundational economic concepts used widely in Financial applications - Portfolio problems and Pricing problems, to name a couple.
In this paper we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach we define random market price for each market scenario. The spot price then is interpreted as a one that reflect balance between profit-loss expectations of the market participants
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
HJB Equation and Merton's Portfolio ProblemAshwin Rao
Deriving the solution to Merton's Portfolio Problem (Optimal Asset Allocation and Consumption) using the elegant formulation of Hamilton-Jacobi-Bellman equation.
Stochastic Control/Reinforcement Learning for Optimal Market MakingAshwin Rao
Optimal Market Making is the problem of dynamically adjusting bid and ask prices/sizes on the Limit Order Book so as to maximize Expected Utility of Gains. This is a stochastic control problem that can be tackled with classical Dynamic Programming techniques or with Reinforcement Learning (using a market-learnt simulator)
. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE) contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
In this paper, we present somewhat alternative point of view on early exercised American options. The standard valuation of the American options the exercise moment is defined as one, which guarantees the maximum value of the option. We discuss the standard approach in the first two sections of the paper. The standard approach was initially presented in the papers [3] - [7]. Our idea is that the exercise moment of the American call / put options is defined by maximum / minimum value of underlying. It was shown that at this moment exercise and sell prices are equal.
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 1.
More info at http://summerschool.ssa.org.ua
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
HJB Equation and Merton's Portfolio ProblemAshwin Rao
Deriving the solution to Merton's Portfolio Problem (Optimal Asset Allocation and Consumption) using the elegant formulation of Hamilton-Jacobi-Bellman equation.
Stochastic Control/Reinforcement Learning for Optimal Market MakingAshwin Rao
Optimal Market Making is the problem of dynamically adjusting bid and ask prices/sizes on the Limit Order Book so as to maximize Expected Utility of Gains. This is a stochastic control problem that can be tackled with classical Dynamic Programming techniques or with Reinforcement Learning (using a market-learnt simulator)
. In some papers it have been remarked that derivation of the Black Scholes Equation (BSE) contains mathematical ambiguities. In particular in [2,3 ] there are two problems which can be raise by accepting Black Scholes (BS) pricing concept. One is technical derivation of the BSE and other the pricing definition of the option.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
In this paper, we show how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
Solution to Black-Scholes P.D.E. via Finite Difference Methods (MatLab)Fynn McKay
Simple implementable of Numerical Analysis to solve the famous Black-Scholes P.D.E. via Finite Difference Methods for the fair price of a European option.
In this paper, we present somewhat alternative point of view on early exercised American options. The standard valuation of the American options the exercise moment is defined as one, which guarantees the maximum value of the option. We discuss the standard approach in the first two sections of the paper. The standard approach was initially presented in the papers [3] - [7]. Our idea is that the exercise moment of the American call / put options is defined by maximum / minimum value of underlying. It was shown that at this moment exercise and sell prices are equal.
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 1.
More info at http://summerschool.ssa.org.ua
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The presentation is part of a conference conducted by QuantInsti Quantitative Learning Pvt. Ltd. along with a multinational investment banking firm that engages in global investment banking, securities, investment management, and other financial services.
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Connect with us:
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Twitter - http://twitter.com/quantinsti
Youtube - http://youtube.com/quantinsti
In this paper, it is shown how the ambiguities in derivation of the BSE can be eliminated.
We pay attention to option as a hedging instrument and present definition of the option price based on market risk weighting. In such approach, we define random market price for each market scenario. The spot price then is interpreted as a one that reflects balance between profit-loss expectations of the market participants.
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Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling!
Any and all corrections are welcome!
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In this paper we present a critical point on connections between stock volatility, implied volatility, and local volatility. The essence of the Black Sholes pricing model is based on assumption that option piece is formed by no arbitrage portfolio. Such assumption effects the change of the real underlying stock by its risk neutral counterpart. Market practice shows even more. The volatility of the underlying should be also changed. Such practice calls for implied volatility. Underlying with implied volatility is specific for each option. The local volatility development presents the value of implied volatility.
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Vidyasagar rocond09
1. A Tutorial Introduction to
Quantitative Finance
M. Vidyasagar
Executive Vice President
Tata Consultancy Services
Hyderabad, India
sagar@atc.tcs.com
ROCOND 2009, Haifa, Israel
Full tutorial introduction to appear in Indian Academy of Sciences
publication, available upon request.
3. Outline
•
•
•
•
•
•
Preliminaries: Introduction of terminology, problems studied, etc.
Finite market models: Basic ideas in an elementary setting.
Black-Scholes theory: What is it and why is it so (in-)famous?
Some simple modifications of Black-Scholes theory
What went wrong?
Some open problems that would interest control theorists
5. Some Terminology
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 the stocks.
We can think of it as a vector in Rn+1 where n is the number of
stocks. Negative ‘holdings’ correspond to borrowing money or ‘shorting’ stocks.
An ‘option’ is an instrument that gives the buyer the right, but not the
obligation, to buy a stock a prespecified price called the ‘strike price’
K.
A ‘European’ option can be exercised only at a specified time T . An
‘American’ option can be exercised at any time prior to a specified
time T .
6. European vs. American Options
The value of the European option is {ST −K}+. In this case it is worthless even though St > K for some intermediate times. The American
option has positive value at intermediate times but is worthless at time
t = T.
7. The Questions Studied Here
• What is the minimum price that the seller of an option should be
willing to accept?
• What is the maximum price that the buyer of an option should be
willing to pay?
• How can the seller (or buyer) of an option ‘hedge’ (minimimze or
even eliminate) his risk after having sold (or bought) the option?
9. One-Period Model
Many key ideas can be illustrated via ‘one-period’ 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.
(1 + r)B(0) at time T = 1.
It increases to B(1) =
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.
Assumption: d < 1 + r < u; otherwise problem is meaningless!
Rewrite as d < 1 < u , where d = d/(1 + r), u = u/(1 + r).
10. Options and Contingent Claims
An ‘option’ gives the buyer the right, but not the obligation, to buy
the stock at time T = 1 at a predetermined strike price K. Again,
assume S(0)d < K < S(0)u.
More generally, a ‘contingent claim’ is a random variable X such that
X=
Xu if S(1) = S(0)u,
Xd if S(1) = S(0)d.
To get an option, set X = {S(1) − K}+. Such instruments are called
‘derivatives’ because their value is ‘derived’ from that of an ‘underlying’
asset (in this case a stock).
Question: How much should the seller of such a claim charge for the
claim at time T = 0?
11. An Incorrect Intuition
View the value of the claim as a random variable.
X=
Xu with probability pu = p,
Xd with probability pd = 1 − p.
So
E[X, p] = (1 + r)−1[pXu + (1 − p)Xd].
Is this the ‘right’ price for the contingent claim?
NO! The seller of the claim can ‘hedge’ against future fluctuations of
stock price by using a part of the proceeds to buy the stock himself.
12. The Replicating Portfolio
Build a portfolio at time T = 0 such that its value exactly matches
that of the claim at time T = 1 irrespective of stock price movement.
Choose real numbers a and b (investment in stocks and bonds respectively) such that
aS(0)u + bB(0)(1 + r) = Xu, aS(0)d + bB(0)(1 + r) = Xd,
or in vector-matrix notation
[a b]
S(0)u
S(0)d
B(0)(1 + r) B(0)(1 + r)
= [Xu Xd].
This is called a ‘replicating portfolio’. The unique solution is
[a b] = [Xu Xd]
S(0)u
S(0)d
B(0)(1 + r) B(0)(1 + r)
−1
.
13. Cost of the Replicating Portfolio
So how much money is needed at time T = 0 to implement this replicating strategy? The answer is
c = [a b]
S(0)
B(0)
= [Xu Xd]
S(0)u
S(0)d
B(0)(1 + r) B(0)(1 + r)
= (1 + r)−1[Xu Xd]
qu
qd
−1
S(0)
B(0)
,
where with u = u/(1 + r), d = d/(1 + r), we have
q :=
qu
qd
=
u d
1 1
−1
1
1
1−d
= u −d
u −1
u −d
Note that qu, qd > 0 and qu + qd = 1. So q := (qu, qd) is a probability
distribution on S(1).
14. Martingale Measure: First Glimpse
Important point: q depends only on the returns u, d, and not on the
associated probabilities p, 1 − p.
Moreover,
−1 S(1), q] = S(0)u 1 − d + S(0)d u − 1 = S(0).
E[(1 + r)
u −d
u −d
Under this synthetic distribution, the discounted expected value of the
stock price at time T = 1 equals S(0). Hence {S(0), (1 + r)−1S(1)} is
a ‘martingale’ under the synthetic probability distribution q.
Thus
c = (1 + r)−1[Xu Xd]
qu
qd
is the discounted expected value of the contingent claim X under the
martingale measure q.
15. Arbitrage-Free Price of a Claim
Theorem: The quantity
c = (1 + r)−1[Xu Xd]
qu
qd
is the unique arbitrage-free price for the contingent claim.
Suppose someone is ready to pay c > c for the claim. Then the
seller collects c , invests c − c in a risk-free bond, uses c to implement
replicating strategy and settle claim at time T = 1, and pockets a riskfree profit of (1 + r)(c − c). This is called an ‘arbitrage opportunity’.
Suppose someone is ready to sell the claim for c < c. Then the buyer
can make a risk-free profit.
16. Multiple Periods: Binomial Model
The same strategy works for multiple periods; this is called the binomial model.
Bond price is deterministic:
Bn+1 = (1 + rn)Bn, n = 0, . . . , N − 1.
Stock price can go up or down: Sn+1 = Snun or Sndn.
There are 2N possible sample paths for the stock, corresponding to
each h ∈ {u, d}N . For each sample path h, at time N there is a payout
Xh.
Claim becomes due only at the end of the time period (European
contingent claim). In an ‘American’ option, the buyer chooses the
time of exercising the option.
17. Define dn = dn/(1 + rn), un = un/(1 + rn),
1 − dn
un − 1
.
qu,n =
, qd,n =
un − dn
un − dn;
Introduce a modified stochastic process
Sn+1 =
Snun with probability qu,n,
Sndn with probability qd,n.
Then
E{(1 + rn)−1Sn+1|Sn, Sn−1, . . . , S0} = Sn, for n = 0, . . . , N − 1.
Thus the discounted process
n−1
(1 + ri)−1Sn
i=0
where the empty product is taken as one, is a martingale.
18. Replicating Strategy for N Periods
We already know to replicate over one period. Extend argument to N
periods.
Suppose j ∈ {u, d}N −1 is the set of stock price transitions up to time
N − 1. So now there are only two possibilities for the final sample path:
ju and jd, and only two possible payouts: Xju and Xjd. Denote these
by cju and cjd respectively. We already know how to compute a cost cj
and a replicating portfolio [aj bj] to replicate this claim, namely:
cj = (1 + rn−1)−1 cjuqu,n + cjdqd,n .
[aj bj] = [cju cjd]
Sjuj
Sjdj
Bj(1 + rN −1) Bj(1 + rN −1)
−1
.
19. Do this for each j ∈ {u, d}N −1. So if we are able to replicate each of
the 2N −1 payouts cj at time T = N − 1, then we know how to replicate
each of the 2N payouts at time T = N .
Repeat backwards until we reach time T = 0.
decreases by a factor of two at each time step.
Number of payouts
20.
21. Arbitrage-Free Price for Multiple Periods
Recall earlier definitions:
dn
un
1 − dn
un − 1
dn =
,u =
, qu,n =
.
, qd,n =
1 + rn n
1 + rn
un − dn
un − dn;
Now define
N −1
qh =
qh,n, ∀h ∈ {u, d}N ,
n=0
−1
N −1
c0 :=
(1 + rn)
Xhqh.
n=0
h∈{u,d}N
c0 is the expected value of the claim Xh under the synthetic distribution
{qh} that makes the discounted stock price a martingale. Moreover,
c0 is the unique arbitrage-free price for the claim.
22. Replicating Strategy in Multiple Periods
Seller of claim receives an amount c0 at time T = 0 and invests a0 in
stocks and b0 in bonds, where
[a0 b0] = [cu cd]
S0u0
S0 d 0
B0(1 + r0) B0(1 + r0)
−1
.
Due to replication, at time T = 1, the portfolio is worth cu if the stock
goes up, and is worth cd if the stock goes down. At time T = 1, adjust
the portfolio according to
[a1 b1] = [ci0u ci0d]
S1u1
S1 d 1
B1(1 + r1) B1(1 + r1)
where i0 = u or d as the case may be. Then repeat.
−1
,
23. Important note: This strategy is self-financing:
a0S1 + b0B1 = a1S1 + b1B1
whether S1 = S0u0 or S1 = S0d0 (i.e. whether the stock goes up or
down at time T = 1). This property has no analog in the one-period
case.
It is also replicating from that time onwards.
Observe: Implementation of replicating strategy requires reallocation
of resources N times, once at each time instant.
25. Continuous-Time Processes: Black-Scholes Formula
Take ‘limit’ at time interval goes to zero and N → ∞; binomial asset
price movement becomes geometric Brownian motion:
1
µ − σ 2 t + σWt , t ∈ [0, T ],
2
where Wt is a standard Brownian motion process. µ is the ‘drift’ of
the Brownian motion and σ is the volatility.
St = S0 exp
Bond price is deterministic: Bt = B0ert.
simple option: XT = {ST − K}+.
Claim is European and a
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.)
26. Theorem (Black-Scholes 1973):
price is
log(S0/K ∗)
1 √
√
C0 = S0Φ
+ σ T
2
σ T
The unique arbitrage-free option
− K ∗Φ
log(S0/K ∗) 1 √
√
− σ T
2
σ T
,
where
c
1
−u2 /2 du
√
e
Φ(c) =
2π −∞
is the Gaussian distribution function, and K ∗ = e−rT K is the discounted
strike price.
27. Black-Scholes PDE
Consider a general payout function erT ψ(e−rT x) to the buyer if ST = 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
1 2 2 ∂ 2f
+ σ x
= 0, ∀(t, x) ∈ (0, T ) × (0, ∞),
2
∂t
2
∂x
with the boundary condition
f (T, x) = ψ(x).
No closed-form solution in general (but available if ψ(x) = (x − K ∗)+).
28. Replicating Strategy in Continuous-Time
Define
∗
Ct = C0 +
t
0
∗
∗
fx(s, Ss )dSs , t ∈ (0, T ),
where the integral is a stochastic integral, and define
∗
∗
∗
αt = fx(t, St ), βt = Ct − αtSt .
Then hold αt of the stock and βt of the bond at time t.
Observe: Implementation of self-financing fully replicating strategy
requires continuous trading.
30. Extensions to Multiple Assets
Binomial model extends readily to multiple assets.
Black-Scholes theory extends to the case of multiple assets of the form
(i)
St
(i)
= S0 exp
1
(i)
µ(i) − [σ (i)]2 t + σWt
, t ∈ [0, T ],
2
(i)
where Wt , i = 1, . . . , d are (possibly correlated) Brownian motions.
(1)
(d)
Analog of Black-Scholes PDE: C0 = f (0, S0 , . . . , S0 ) where f satisfies a PDE. But no closed-form solution for f in general.
31. American Options
An ‘American’ option can be exercised at any time up to and including
time T .
So we need a ‘super-replicating’ strategy: The value of our portfolio
must equal or exceed the value of the claim at all times.
If Xt = {St − K}+, then both price and hedging strategy are same as
for European claims.
Very little known about pricing American options in general. Theory
of ‘optimal time to exercise option’ is very deep and difficult.
32. Sensitivities and the ‘Greeks’
Recall C0 = f (0, S0) is the correct price for the option under BlackScholes theory. (We need not assume the claim to be a simple option!)
∂C0
∂∆
∂ 2C0
∆=
,
,Γ =
=
2
∂S0
∂S0
∂S0
∂C0
∂f (t, Xt)
∂f (t, Xt)
,θ = −
,ρ =
.
∂σ
∂t
∂r
‘Delta-hedging’: A strategy such that ∆ = 0 – return is insensitive to
initial stock price (to first-order approximation).
Vega = ν =
‘Delta-gamma-hedging’: A strategy such that ∆ = 0, Γ = 0 – return
is insensitive to initial stock price (to second-order approximation).
34. What Went Wrong?
My view: The current financial debacle owes very little to poor modeling of risks. Most significant factors were:
• Complete abdication of oversight responsibility by US government –
led to over-leveraging and massive conflicts of interests
• OTC trading of complex instruments – made ‘price discovery’ impossible
⇒ Net outstanding derivative positions: $ 760 trillion!
⇒ U.S. Annual GDP: $ 13 trillion, World’s: $ 60 trillion
⇒ 90% of derivatives traded OTC – No regulation whatsoever!
• ‘One-way’ rewards for traders: Heads the traders win – tails the depositors and shareholders lose
• Real bonuses paid on virtual profits, and so on
Please read full paper for elaboration.
36. Some Open Problems
• Incomplete markets: Replication is impossible
• Multiple martingale measures: Which one to choose?
• Computing the ‘greeks’ when there are no closed-form formulas
• Alternate models for asset price movement: Why geometric Brownian
motion?
37. Incomplete Markets
Consider the one-period model where the asset takes three, not two,
possible values.
S(0)u
with probability pu,
S(1) = S(0)m with probability pm,
S(0)d with probability p ,
d
To construct a replicating strategy, we need to choose a, b such that
[a b]
S(0)u
S(0)m
S(0)d
(1 + r)B(0) (1 + r)B(0) (1 + r)B(0)
No solution in general – replication is impossible!
= [Xu Xm Xd].
38. Martingale measures
Let us try to find a synthetic measure q = (qu, qm, qd) so that
E[(1 + r)−1S(1), q] = S(0).
Need to choose qu, qm, qd such that
qu + qm + qd = 1, quu + qmm + qdd = 1 + r.
Infinitely many martingale measures!
Is there a relationship?
39. Finite Market Case: DMW Theorem
Consider d assets over N time instants, each assuming values in R.
We allow infinitely many values for each asset, correlations, etc. Let
˜
P denote the law of these assets (law over RdN ).
Theorem (Dalang-Morton-Willinger):
are equivalent:
The following statements
˜
˜
• There exists a probability measure Q on RdN that is equivalent to P
˜
such that, under Q, the process {Sn}N
n=0 is a martingale.
• There does not exist an arbitrage opportunity.
Moreover, if either of these equivalent conditions holds, then it is pos˜
sible to choose the measure Q in such a way that the Radon-Nikodym
˜ ˜
derivative (dQ/dP ) is bounded.
40. Existence of a Replicating Strategy
Define
˜
˜
V−(X) := min E[X, Q], V+(X) := max E[X, Q],
˜
Q∈M
˜
Q∈M
where M is the set of martingale measures.
[V−(X), V+(X)] is arbitrage-free.
Then every price in
Note: V−(X) is the maximum that the buyer of the claim would (normally) be willing to pay. Similarly, V+(X) is the minimum that the
seller of the claim would (normally) be willing to accept.
Theorem: A European contingent claim with the payout function X
is replicable if and only if V−(X) = V+(X).
Modulo technical conditions, replicability ≈ unique martingale measure.
41. Minimum Relative Entropy Martingage Measures
˜
In incomplete markets (M is infinite), choose Q to minimize the relative
entropy:
˜
˜
Q∗ = arg min H(Q
˜
Q∈M
˜
P ),
where
n
˜
H(Q
˜
P) =
qi
qi log .
pi
i=1
˜
Choose martingale measure that is ‘closest’ to the real-world law P .
˜
Q∗ has a nice interpretation in terms of maximizing utility. Since M is
˜ ˜
˜
a convex set and H(Q P ) is convex in Q, this is a convex optimization
problem.
42. Computing the Greeks
Even for simple geometric Brownian motion models, no closed form
for fair price C0 = f (0, S0) unless claim X is a simple option. So how
to compute the ‘greeks’ ?
One possibility: Evaluate fair price(s) using stochastic (Monte Carlo)
simulation; then differentiate numerically. Absolutely fraught with numerical instabilities and sensitivities.
Answer: Malliavin calculus.
Originally developed for Brownian motion, now extented to arbitrary
L´vy processes.
e
43. Alternate Models for Asset Prices
Geometric L´vy processes: Assume
e
St = S0 exp(Lt),
where {Lt} is a L´vy process.
e
A L´vy process is the most general process with independent incree
ments. For all practical purposes, a L´vy process consists of a Wiener
e
process (Brownian motion) plus a countable number of ‘jumps’.
Sad fact: Replication is possible only for geometric Brownian motion
asset prices!
44. Ergo: GBM (Geometric Brownian Motion) is the only asset price model
for which there exists a unique martingale measure and the theory is
very ‘pretty’.
Unfortunately, real world distributions are quite far from Gaussian, in
the range [µ − 10σ, µ + 10σ]. Moreover, they are also ‘skewed’ in the
positive direction.
A tractable theory of pricing claims with real world probabilities is still
waiting to be discovered.
45. Conclusions
Quantitative finance has definitely made pricing and trading more ‘scientific’, but several poorly understood issues still remain:
• Irrational behavior
• Non-equilibrium economics (e.g. incomplete information)
• Correlated increments in asset price returns
• Effects of alternate asset price models on option pricing and/or trading strategies.
In short, what is unknown dwarfs what is known.
Thank You!