OverviewPermalink

For asset pricing, we have the following approaches:

  • Replication
  • Adjusting for risk premium (Market Price for Risk)
  • Risk-neutral Pricing
    • Discrete: Tree model
    • Continuous: Martingale approach
  • PDE approach

In fact, these are equivalent ways of pricing. There is no “best” method, and we should choose a method depending on the context.

FrameworkPermalink

Binomial Model FrameworkPermalink

For simplicity, let’s assume a binomial model framework, which can be easily extended to multi-steps or continuous models (e.g. Black-Scholes models). It is also extensively used by practitioners to price real securities.

The binomial model: We assume that we have a risk-free asset Dt, and a risky asset St. For simplicity, we call St as stock price.

A short tick later(Δt),the stock price can jump up to Su with probability p and jump down to Sd with probability (1-p).

The bond price is not random. Let r be the continuously compounded risk-free rate. If the initial price of a cash bond is D0=1, then a short tick later, the bond price will grow to D0erΔt=erΔt

A derivative paysoff fu if the stock price goes up and fd if the stock price goes down.

-w708

Question: What is the initial price of the derivative?

The unconscious statistician might pay f=erΔt(pfu+(1p)fd) ,i.e.the expectation of future payoffs discounted by the risk-free rate.

But, this answer is wrong. Since there is uncertainty in the future price of f, investors require risk-premium to take the risk. Therefore, we need to add risk-premium to the discounted rate.

Market price of risk. The market price of risk is compensation per unit of risk. In the binomial model: λ0= risk premium  amount of risk =er0ΔE0[P1]P0P1,uP1,d
where E0[P1]=puP1,u+pdP1,d

Equivalent definition in terms of returns:

P0=er0ΔE0[P1]λ0(P1,uP1,d)

So far, we have no idea how to “price” this MPR term.

Black-Scholes FrameworkPermalink

In the original Black-Scholes economy:
1. There are no market imperfections. That is, there are no taxes, no transactions costs, and no short-sale constraints. Security trading is frictionless;
2. security trading is continuous;
3. there is unlimited risk-free borrowing and lending at the constant continuously compounded risk-free rate r .
4. the stock price follows a geometric Brownian motion with constant growth rate μ and constant volatility parameter σ dSt=μStdt+σStdBt,
where Bt is a standard Brownian motion under P-measure;
5. there are no dividends during the life of the derivative to be evaluated;
6. there are no risk-less arbitrage opportunities.

Ito’s lemma: a formula used to calculate explicitly the stochastic differential equation (SDE) that governs the dynamics of an arbitrary function given the stochastic process of the function’s arguments.

Let x follows the SDE dx=μ1dt+σ1dBxt
, then the SDE of f(x,t) given the stochastic process of x is: df(x,t)=ftdt+fxdx+122fx2(dx)2

Let x and y follows the SDEs dx=μ1dt+σ1dBxt
and dy=μ2dt+σ2dByt
, with dBxtdByt=ρdt
, then the SDE of f(x,y,t) given the stochastic process of x and y is: df(x,y,t)=ftdt+fxdx+fydy+12[2fx2(dx)2+2fy2(dy)2+22fxy(dxdy)]

Ito’s lemma basically says that we need to do Taylor’s expansion to the second order. The multiplication rules for stochastic differentials:

xdBxdBydtdBxdtρdt0dBydt0dt0

Modeling of the Stock Price

We typically model stock prices as a geometric Brownian motion or lognormal diffusion under measure P:

dSt=μdSt+σStdBt

with a constant growth rate μ and a constant diffusion parameter σ.

If the underlying stock does not distribute dividends, then dStSt=The return of investment over the period of [t,t+dt] is given by:

expected return μdt+unexpected return σdBt.

In a risk-neutral world: dSt=rStdt+σStdBQt

The following two proposition is equivalent:
1. The log-price pt=lnSt follows a Brownian motion dpt=(μ12σ2)dt+σdBt
2. St follows a geometric Brownian motion dSt=μStdt+σStdBt

It is easy to prove with Ito’s lemma. In fact:

dpt=pttdt+ptStdSt+122ptS2t(dSt)2=1St(μStdt+σStdBt)+12(1S2t)(μStdt+σStdBt)2=(μ12σ2)dt+σdBt

Similarly,we can show that p follows dpt=(r12σ2)dt+σdBQt under the risk-neutral measure Q

The log-price pt=lnSt follows a geometric Brownian

A random variable Y is log-normally distributed if the natural logarithm of it, X, is a normal random variable. XN(μ,σ)
Y=eXLN(μ,σ)
  • The probability density function of a lognormal random variable is:
f(y)=12πσye(lnyμ)22σ2,0y<+
  • The mean and the variance of a lognormal random variance are
E(Y)=eμ+12σ2
V(Y)=e2μ+σ2(eσ21)

Application

Since dlnSt=dpt=(r12σ2)dt+σdBQt, we have

lnST=lnS0+(r12σ2)T+σ(BTB0)

Therefore, log price at maturity lnSTN(μ,σ2)

where μ=lnS0+(r12σ2)T and σ2=σ2T.

Therefore, the density function of ST is

f(S(T))=12πσTS(T)e(lnS(T)S(0)(r12σ2)T)22σ2T
E(ST)=eμ+12σ2=S(0)erT

ReplicationPermalink

First let’s introduce some important concepts:

The simultaneous buying and selling of a security at two different prices in two different markets, resulting in profits without risk. Perfectly efficient markets present no arbitrage opportunities. Perfectly efficient markets seldom exist, but, arbitrage opportunities are often precluded because of transactions costs.
There is no arbitrage opportunity in financial markets.

Based on this principle, we have the following law:

If two securities have the same payoffs in the future, then they must have the same price today.

A replicating portfolio of a security with payoffs Pu and Pd in the two node u and d at time 1 is a portfolio of D and S that exactly replicates the values of the security at time 1.

We construct a portfolio out of the stock and the cash bond to mimic the payoffs of the derivative. This portfolio is called “replicating portfolio.” Then the price of the derivative must be the same as the price of the replicating portfolio by the law of one price.

Let x be the stock holding strategy and let y be the bond holding strategy. The unit of x and y is number of shares. Then the price and the payoffs of the portfolio are as shown in the following diagram:

{xsu+yerΔt=fuxsd+yerΔt=fd

By the law of one price, this must also be the price of the derivative otherwise there exists an arbitrage opportunity.

f0=xs0+y=erΔt(erΔts0sdsusdfu+(1erΔts0sdsusd)fd)

The above procedure implies several points:

  • To construct a replicating portfolio, we should buy x=fufdsusd shares of the stock
  • For any risky-asset, we can always form x,y, s.t. xS0+yD0 will replicate the payoffs of the risky-asset

Risk-neutral pricingPermalink

In last part, we show:

f0=xs0+y=erΔt(erΔts0sdsusdfu+(1erΔts0sdsusd)fd)

Let q=erΔts0sdsusd, then

f0=xs0+y=erΔt(qfu+(1q)fd)

This Pricing formula looks like what we want — the price of the derivative is the expectation of the future payoffs discounted at the risk-free rate!

The following are equivalent:
1. There is No Arbitrage
2. There exist valid risk-neutral probabilities
(That is, q[0,1] in the case of the one-step binomial model.)

For q=erΔts0sdsusd, we can prove that q[0,1] by non-arbitrage assumption.

For example, erΔts0sd, otherwise there is an arbitrage opportunity by short D and long S.

So, q is the “risk-neutral probability.” A collection of risk-neutral probabilities q,1q on the set of all possible outcomes {up, down} is called risk-neutral probability measure or simply risk-neutral measure, denoted by Q.

The price of the derivative is the expected future payoffs under the risk-neutral measure Q discounted by the risk-free rate.

f0=xs0+y=erΔt(qfu+(1q)fd)

Note, q depends only on the risk-free rate r, current stock price s0 and the future possible stock prices su and sd. It does not depend on the true probability p at all.

-w534

Summary: 5-step risk-neutral derivative pricing rule in the binomial branch framework

  • Step 1: Discount the stock price, SΔt , by using risk-free rate to have discounted stock price, erΔtSΔt ;
  • Step 2: Find a measure Q that makes the discounted stock price a martingale;
  • Step 3: Discount the derivative, fΔt , by using risk-free rate to have discounted stock price, erΔtfΔt ;
  • Step 4: Calculate the expectation of the discounted derivative price under measure Q.
  • Step 5: Calculate hedge ratio x=fufdsusd, s.t. f=xS+yD

Martingale ApproachPermalink

A process Xt is a martingale with respect to a measure Ρ if it satisfies the following two conditions:
(i) EP(XTXt,,X0)=Xt, for all T>t, and
(ii) EP(|XT|)<, for all T
A martingale measure of a process is the measure that makes the process a martingale.

Therefore, the risk-free discounted stock price is a martingale under the risk-neutral measure Q.

Since any risky asset can be expressed as f=xtSt+ytDt, he risk-free discounted derivative is also a martingale under the risk-neutral measure Q. So the risk-neutral measure Q is also called “equivalent martingale measure.”

The 5-step risk-neutral pricing rule works in the continuous-time. The difference from the discrete-time is that we now have to identify the density function of the underlying under the risk-neutral probability measure and then do an integration to evaluate the risk-neutral expectation of the discounted derivative payoff.

Specifically, we have the Harrison-Pliska (1981) risk-neutral valuation formula.

c(S0,0)=erTEQ(c(ST,T))c(St,t)=er(Tt)EQt(c(ST,T))

Market Price of RiskPermalink

The market price of risk is compensation per unit of risk. In the binomial model: λ0= risk premium  amount of risk =er0ΔE0[P1]P0P1,uP1,d
where E0[P1]=puP1,u+pdP1,d

Equivalent definition in terms of returns:

P0=er0ΔE0[P1]λ0(P1,uP1,d)

So far, we have no idea how to “price” this MPR term.

We can easily prove that all risky asset has a same MPR λ, by the replication approach f=xS+yD.

Risk-neutral vs. physical probabilitiesPermalink

Physical probabilities (pu and pd in the context of the binomial model) describe the actual likelihood of events occurring.

Risk-neutral probabilities (qu and qd in the context of the binomial model) follow from No Arbitrage and is a construct used for pricing.

What is the relationship between the two? How do we interpret this relationship?

Consider an asset which only pays 1 in state u and zero otherwise. -w576

We can price asset using the MPR approach:

P0=puer0Δλ0(10)

We can also use risk-neutral pricing:

P0=quer0Δ

Since the two are equal:

pu:=qu=puλ0er0Δ

Similarly,

pd:=qd=pd+λ0er0Δ

Summary:

pu=puλ0er0Δpd=pd+λ0er0Δ

For λ0=0, they are the same.

PDE approachPermalink

The Black-scholes formula for the price of vanilla call/put options is a function of five arguments.

Let ct be the price of a option at time t; Let St be the underlying stock price at time t; Let τ=Tt be the time-to-maturity, where T is the maturity time; Let K be the strike price; Let r be the constant risk-free rate; Let σ be the return volatility of the underlying stock; Then ct=C(St,τ=Tt;K,r,σ). The option price ct is the dependent variable. The current stock price St and the current time t are the independent variables. The rest are parameters.

Assume the stock price St follows a geometric Brownian motion:

dSt=μStdt+σStdBt

By Ito’s lemma, we can derive the differential expression of the option price:

dct=cttdtt+ctStdSt+122ctS2t(dSt)2=cttdtt+ctSt(μStdt+σStdBt)+122ctS2tσ2S2tdt=(ctt+122ctS2tσ2S2t+μStctSt)dt+σStctStdBt

Let us long one unit of the option and short Δ units of the stock. The price of the portfolio at time t is:

Pt=ctΔSt

The instantaneous dollar return on the portfolio is:

dPt=dctΔdSt=(ctt+122ctS2tσ2S2t+μSt(ctStΔ))dt+σSt(ctStΔ)dBt

By choosing Δ=CtSt, we ge want a risk-neutral portfolio. Such a portfolio is said to be perfectly hedged or Delta-neutral portfolio

Since P becomes risk-free asset, now we have: dPt=rPtdt

After regrouping, we have the well-known Black and Scholes partial differential equation (PDE):

ctt+12σ2S2t2ctS2t+rStctStrct=0

This PDE is a second-order linear parabolic PDE for the option price ct subject to the following boundary condition:

CT(ST,T)=Max(STK,0)

The unique solution to this PDE is the famous Black-Scholes formula:

c(St,t)=StΦ(d1)Ker(Tt)Φ(d2)d1=ln(StK)+(r+12σ2)(Tt)σTtd2=ln(StK)+(r12σ2)(Tt)σTt

where Φ() is the standard normal cumulative distribution function.

How to solve the PDE for different Derivatives?

Heat equation: The temperature in a metal rod satisfies the standard heat equation with an initial condition f(x0,0)

{ft12σ22fx2=0f(x0,0)

Delta function δ(xx0) is defined as:

δ(xx0)={0xx0+x=x0 and δ(xx0)dx=1

Green’s function G(x,t;x0) : Green’s function is a solution of the standard heat equation with a particular initial condition, which is the delta function δ(xx0) :

{Gt12σ22Gx2=0G(x,0;x0)=δ(xx0)

The formula for Green’s function, which is the solution of the above PDE system, is

G(x,t;x0)=12πσte(xx0)22σ2t

As shown, Green’s function is like a density function of a normal random variable x with mean x0 and variance σ2t conditional on the initial state x .

The solution of the standard heat equation with an arbitrary initial condition f(x0,0) can be written in terms of Green’s function as

f(x,t)=+f(x0,0)G(x,t;x0)dx0

Summary of PDE approach:

Step 1: Transformation.

Let

τ=Ttc(S,t)=erτKf(x,τ)S=Ke(r12σ2)τ+xx=lnSK+(r12σ2)τ

TheBlack-Scholes PDE:

ctt+12σ2S2t2ctS2t+rStctStrct=0

becomes:

{fτ12σ22fx2=0f(x0,0)

The given boundary condition c(ST,T) becomes:

f(x0,0)=er0Kc(ST,T)

Example for vanilla European call, the new PDE system after transformation is

{fτ12σ22fx2=0f(x0,0)=Max(ex01,0)

Step 2: Solve the new PDE

The transformed PDE system is a standard heat equation with an arbitrary initial condition. We know the solution to the transformed PDE system is

f(x,τ)=+f(x0,0)G(x,τ;x0)dx0
G(x,t;x0)=12πσte(xx0)22σ2t

Comments