Option Pricing

Lecture 3: Option Pricing

August 2014

Cvitanic Lecture 3: Option Pricing 1

OPTION PRICING


No Arbitrage Price in Complete Markets

I In complete markets the option payoff can be replicated by trading inthe underlying risky assets and in the bank account

I Price of the option = Cost of the replicating portfolio

I Otherwise, there is arbitrage

I Finding the replicating portfolio is the basis for hedging and pricing


Pricing and Hedging in the Binomial Tree

I Consider a single-period setting in which the stock S(0) = s only cantake two values in the future, su and sd.

I There is also a bond/bank account that pays interest r


I There is no arbitrage if and only if u > (1 + r) > d.

I The market here is complete

⇒ We can price securities by no-arbitrage by finding the cost of thereplicating portfolio that invests in the stock and the bank account


I We buy ∆ shares and invest B in the bank

I Denote by Cu and Cd the price of the derivative when the price of thestock is us and ds, respectively

I For replication, we need to have

∆us+ (1 + r)B = Cu

∆ds+ (1 + r)B = Cd

I The price C(0) of the derivative has to be the same as the cost of thereplicating portfolio,

C(0) = ∆s+B


Risk-Neutral/Martingale Pricing

I After simple algebra, we can see that

C(0) =1

1 + r

(1 + r − d

u− dCu +

u− (1 + r)

u− dCd

)I Denote

p∗ =1 + r − d

u− d; (1− p∗) =

u− (1 + r)

u− d

I Then

C(0) =1

1 + r[p∗Cu + (1− p∗)Cd] = E∗

[1

1 + rC(1)

]⇒ Risk-neutral pricing


Risk-Neutral/Martingale Pricing

I In classical insurance the price of liability payoff C(T ) is

C(0) = E[e−rTC(T )]

However, when there is possibility of hedging, so that X(T ) = C(T ),then, under probability P ∗ under which e−rtX(t) is a martingale,

C(0) = X(0) = E∗[e−rTX(T )] = E∗[e−rTC(T )]


Call Option in the Binomial Tree

I In a multi-period setting with n periods, denoting

p =pu

1 + r

I one can show that

C(0) = sϕ(a, n, p)− K

(1 + r)nϕ(a, n, p)

I where ϕ(·, ·, ·) is a binomial distribution function

I a is the minimum number of “up” jumps so that the option ends upin-the-money

I When n → ∞, this formula converges to the Black and Scholesformula


Merton-Black-Scholes Pricing

I Consider the following stock dynamics

dS(t) = S(t)(µdt+ σdW (t))

I with solution

S(t) = S(0) exp

{(µ− 1

2σ2)t+ σW (t)

}I There is also a bank account with price

dB(t) = rB(t)dt

I orB(t) = B(0)ert


Merton-Black-Scholes Pricing (cont)

I Since E[ekW (t)] = e12k2t, we have

E[S(t)] = S(0)eµt

and thus

µ =1

tlogE

[S(t)

S(0)

]Similarly, since

logS(t)− logS(0) = (µ− 1

2σ2)t+ σW (t)

we get

σ2 =Var[log(S(t))− logS(0)]

t


Merton-Black-Scholes Pricing: PDE approach

I We want to find the price of the European call option on this stock,with strike price K and maturity at T

I For a claim with payoff C(T ) = g(S(T )) it is reasonable to guess thatthe price will be a function C(t, S(t)) of the current time and price ofunderlying.

I If so, from Ito’s lemma, the price at time t C(t, S(t)) satisfies

dC =

[Ct +

1

2σ2S2Css + µSCs

]dt+ σSCsdW

I On the other hand, with π(t) = amount invested in stock at time t, aself-financing wealth process satisfies:

dX(t) =π(t)

S(t)dS(t) +

X(t)− π(t)

B(t)dB(t)

dX(t) = [rX(t) + (µ− r)π(t)]dt+ σ(t)π(t)dW (t)


I If we want replication, C(t) = X(t), we need the dt terms to beequal, and the dW terms to be equal

I Comparing dW terms we get that the number of shares needs tosatisfy

π(t)

S(t)= Cs(t, S(t))

I Using this and comparing the dt terms we get the Black-Scholes PDE:

Ct +1

2σ2s2Css + r(sCs − C) = 0

I subject to the boundary condition,

C(T, s) = (s−K)+


Black and Scholes Formula

I The solution is the Black and Scholes formula for European calloptions:

C(t, S(t)) = S(t)N(d1)−Ke−r(T−t)N(d2)

I where

N(x) := P [Z ≤ x] =1√2π

∫ x

−∞e−

y2

2 dy

d1 =1

σ√T − t

[log(S(t)/K) + (r + σ2/2)(T − t)]

d2 =1

σ√T − t

[log(S(t)/K) + (r − σ2/2)(T − t)]

= d1 − σ√T − t


Black and Scholes Formula Using MartingaleApproach

I Let us find the dynamics of S under martingale probability P ∗

I Denote by W ∗ the Brownian motion under P ∗

I We claim that if we replace µ by r, that is, if the stock satisfies thedynamics

dS(t)

S(t)= rdt+ σdW ∗(t)

I then the discounted stock price is a P ∗-martingale.

I Indeed, this is because Ito’s rule then gives

d(e−rtS(t)) = e−rtdS(t) + S(t)d(e−rt)

= e−rt[rS(t)dt+σS(t)dW ∗(t)]−S(t)re−rtdt = 0×dt+σS(t)dW ∗(t)


Girsanov theorem

I In order to have the above dynamics and also

dS(t)

S(t)= µdt+ σdW (t),

I we need to have

W ∗(t) = W (t) +µ− r

σt

I The famous Girsanov theorem tells us that this is possible: thereexists a probability P ∗ under which so-defined W ∗ is a Brownianmotion.


Computing the price as expected value

I To find expectation under P ∗, we note that we can write

S(T ) = S(0)eσW∗(T )+(r− 1

2σ2)T

I We have to compute

E∗[e−rT (S(T )−K)+]

= E∗[e−rTS(T )1{S(T )>K}]−Ke−rTE∗[1{S(T )>K}]


I For the second term we need to compute the price of a Digital(Binary) option:

E∗e−rT1{S(T )>K} = e−rTP ∗(S(T ) > K)

P ∗(S(T ) > K) = P ∗(S(0)e(r−σ2/2)T+σW ∗(T ) > K)

= P ∗(W ∗(T )√

T> −d2)

= N(d2)

I where the middle equality follows by taking logs and re-arranging.

I The first term is computed in the book using the formula

E

[g

(W ∗(T )√

T

)]=

1√2π

∫ ∞

−∞g(x)e−x2/2dx


An alternative way to find the PDE

I Under risk-neutral probability P ∗, by Ito’s rule

dC(t, S(t)) = [Ct + rS(t)Cs +1

2σ2S2(t)Css]dt

+σCsS(t)dW∗(t)

Then, discounting,d(e−rtC(t, S(t)))

= e−rt[(Ct + rS(t)Cs +1

2σ2S2(t)Css − rC)]dt

+e−rtσCsS(t)dW∗

I This has to be a P ∗ martingale, which means that the dt term has tobe zero.

I This gives the Black-Scholes PDE.


Black and Scholes Formula for Puts

I It can be obtained from put-call parity

I or solving the same PDE with different boundary condition:

P (T, s) = (K − s)+

I or by computing the expected value under P ∗.


Plot of a Call Price

Call Option

0

25

50

0 25 50 75 100 125

Figure 7.3: Black-Scholes values for a call option.


Plot of a Put Price

Put Option

0

25

50

0 25 50 75 100

Figure 7.4: Black-Scholes values for a put option.


Implied Volatility

I It is the value of σ that matches the theoretical Black-Scholes price ofthe option with the observed market price of the option

I In the Black - Scholes model, volatility is the same for all options onthe same underlying

I However, this is not the case for implied volatilities: volatility smile


Plot of Volatility Smile

Volatility Smile

Implied volatility

Strike price

Figure 7.5: The level of implied volatility differs for different strike levels.


Dividend-Paying Underlying

I Assume the stock pays a dividend at a continuous rate q

I This is appropriate when the underlying is an index, for example

I Total value of holding one share of stock is

G(t) := S(t) +

∫ t

0qS(u)du

I Therefore, the wealth process of investing in this stock and the bankaccount is

dX = (X − π)dB/B + πdG/S

I ordX(t) = [rX(t) + π(t)(µ+ q − r)]dt+ π(t)σdW (t)


Dividend-Paying Underlying (cont)

I To get the discounted wealth process to be a martingale,

dX(t) = rX(t)dt+ π(t)σdW ∗(t)

I we need to define a Brownian motion under pricing probability P ∗ as

W ∗(t) = W (t) + t(µ+ q − r)/σ

I which makes the stock dynamics

dS(t) = S(t)[(r − q)dt+ σdW ∗(t)]

I and the pricing PDE is

Ct +1

2σ2s2Css + (r − q)sCs − rC = 0


Dividend-Paying Underlying (cont)

I The solution, for the European call option is obtained by replacing thecurrent underlying price s with se−q(T−t):

C(t, s) = se−q(T−t)N(d1)−Ke−r(T−t)N(d2)

I where

d1 =1

σ√T − t

[log(s/K) + (r − q + σ2/2)(T − t)]

d2 =1

σ√T − t

[log(s/K) + (r − q − σ2/2)(T − t)]


Options on Futures

I Assume that futures satisfy

dF (t) = F (t)[µFdt+ σFdW (t)]

I Since holding futures does not cost any money, the wealth is allinvested in the bank account and it satisfies

dX =π

FdF +

X

BdB =

π

FdF + rXdt

I Then, the discounted wealth process is

dX =π

FdF = πµFdt+ πσFdW


Options on futures (cont)

I The risk-neutral Brownian Motion is

W ∗(t) = W (t) +µF

σFt

I and the PDE for path independent options is

Ct +1

2σ2f2Cff − rC = 0

I The solution for the call option is

C(t, f) = e−r(T−t)[fN(d1)−KN(d2)]

d1 =1

σF√T − t

[log(f/K) + (σ2F /2)(T − t)]

d2 =1

σF√T − t

[log(f/K)− (σ2F /2)(T − t)]


Currency Options

I Consider the payoff, evaluated in the domestic currency, equal to

(Q(T )−K)+

I where Q(T ) is time T domestic value of one unit of foreign currency

I Assume that the exchange rate process is given by

dQ(t) = Q(t)[µQdt+ σQdW (t)]

I If we hold one unit of the foreign currency in foreign bank, we get theforeign interest rate rf

I Pricing formula is the same as in the case of dividend-payingunderlying, but with q replaced by rf


Currency Options (cont.)I This is because if replicating the payoff by trading in the domestic

and foreign bank accounts, denoting B(t) := ert the time t value ofone unit of domestic currency, the dollar value of one unit of theforeign account is

Q∗(t) := Q(t)erf ·t

I Ito’s rule for products gives

dQ∗ = Q∗ [(µQ + rf )dt+ σQ] dW

I The wealth dynamics (in domestic currency) of a portfolio of π dollarsin the foreign account and the rest in the domestic account are

dX =X − π

BdB +

π

Q∗dQ∗ = [rX + π(µQ + rf − r)]dt+ πσQdW

I This is exactly the same as for dividends with q replaced by rfI We have

W ∗(t) = W (t) + t(µQ − (r − rf ))/σQ


Example: Quanto OptionsI - S(t): a domestic equity index

- Payoff: S(T )− F units of foreign currency; quanto forwardI We need to have

dX(t) = rX(t)dt+ π(t)σQdW∗(t) + d(investment in S)

I As in the previous slide, we have

W ∗(t) = W (t) + t(µQ − (r − rf ))/σQ

and thusdQ(t) = Q(t)[(r − rf )dt+ σQdW

∗(t)]

I AssumedS(t) = S(t)[rdt+ σSdZ

∗(t)]

where BMP Z∗ has instantaneous correlation ρ with W ∗. We have

d(S(t)Q(t)) = S(t)Q(t)[(2r−rf+ρσQσS)dt+σQdW∗(t)+σSdZ

∗(t)]


Example: Quantos cont.

I S(T )− F units of foreign currency is the same as (S(T )− F )Q(T )units of domestic currency. The domestic value is

e−rT (E∗[S(T )Q(T )]− FE∗[Q(T )])

I To make it equal to zero

F =E∗[S(T )Q(T )]

E∗[Q(T )](1)

I We have

E∗[S(T )Q(T )] = S(0)Q(0)e(2r−rf+ρσSσQ)T

E∗[Q(T )] = Q(0)e(r−rf )T

I We getF = S(0)e(r+ρσSσQ)T


Merton’s jump-diffusion model

I Suppose the jumps arrive at a speed governed by a Poisson process

I That is, the number N(t) of jumps between moment 0 and t is givenby Poisson distribution:

P [N(t) = k] = e−λt (λt)k

k!

I The stock price satisfies the following dynamics:

dS(t) = S(t)[r − λm]dt+ S(t)σdW ∗(t) + dJ(t) ,

I where m is related to the mean jump size and dJ is the actual jumpsize.


I That is, dJ(t) = 0 if there is no jump at time t, anddJ(t) = S(t)Xi − S(t) if the i-th jump of size Xi occurs at time t

I Therefore,

S(t) = S(0) ·X1 ·X2 · . . . ·XN(t) · e(r−σ2/2−λm)t+σW ∗(t)

I Here,m := E∗[Xi]− 1

I in order to make the discounted stock price a martingale under P ∗.


I We want to price an European option with payoff g(S(T ))

I The price of the option is,

C(0) =∞∑k=0

E∗[e−rT g(S(T ))

∣∣∣∣ N(T ) = k

]P ∗[N(T ) = k]

I which is equal to∑∞k=0 E∗

[e−rT g

(S(0)X1 · . . . Xk · e(r−σ2/2−λm)T+σW ∗(T )

)]× e−λT (λT )

k

k!


I If Xi’s are lognormally distributed, the price of the option can berepresented as

C(0) =

∞∑k=0

e−λT (λT )k

k!BSk

I where λ = λ(1 +m) and BSk is the Black and Scholes formula wherer = rk and σ = σk depend on k (see the book)


Stochastic Volatility; Incomplete Markets

I Consider two independent BMP W1 and W2

dS(t) = S(t)[µ(t, S(t), V (t))dt+ σ1(t, S(t), V (t))dW1(t)

+σ2(t, S(t), V (t))dW2(t)]

dV (t) = α(t, S(t), V (t))dt+ γ(t, S(t), V (t))dW2(t)

I Under a risk-neutral probability measure

dS(t) = S(t)[r(t)dt+ σ1(t)dW∗1 (t) + σ2(t)dW

∗2 (t)]

I Denote by κ(t) any (adapted) stochastic process


Stochastic Vol with Inc Markets (cont)

I There are many measures P ∗. In particular, for any such process κ wecan set

dW ∗1 (t) = dW1(t) +

1

σ1(t)[µ(t)− r(t)− σ2(t)κ(t)]dt

dW ∗2 (t) = dW2(t) + κ(t)dt

I It can be checked that discounted S is then a P ∗ martingale and

dV (t) = [α(t)− κ(t)γ(t)]dt+ γ(t)dW ∗2 (t)

I Thus, for constant κ, the PDE becomes

Ct +1

2Csss

2(σ21 + σ2

2) +1

2Cvvγ

2

+ Csvγσ2 + r(sCS − C) + Cv(α− κγ) = 0


Examples

I Complete market – CEV model

dS/S = rdt+ σSpdW ∗

I Incomplete market – Heston’s model:

dS(t) = S(t)[rdt+√

V (t)dW ∗(t)]

dV (t) = A(B − V (t))dt+ γ√V (t)dZ∗(t)

for some other risk-neutral Brownian motion Z∗ having correlation ρwith W ∗. Price is a function C(t, s, v) satisfying

0 = Ct +1

2v[s2Css + γ2Cvv] + r(sCs −C)+A(B− σ2)Cv + ργvsCsv


Pricing theory for American options

I Denote by g(τ) the discounted payoff of the option when exercised attime τ .

I The price A(t) of the American option at time t is given by

A(t) = maxt≤τ≤T

E∗t [e

−r(τ−t)g(τ)]

I where τ is a stopping time

I When early exercise is not optimal, we have

A(t) > g(t)

I At the moment it is optimal to exercise the option, we have that

A(τ) = g(τ)


I The Black-Scholes equation holds in the continuation region whereA(t, s) > g(t, s):

At +1

2σ2s2Ass + r(sAs −A) = 0

I In the exercise region, where if A(t, s) = g(t, s), we have:

At +1

2σ2s2Ass + r(sAs −A) < 0

I This is because if the holder does not optimally exercise it, the valueof the option falls down.


Pricing in Binomial Tree

I Find p∗ = er∆t−du−d

I Given values Cu(T ), Cd(T ), compute

C(T −∆t) = e−r∆t[p∗Cu(T ) + (1− p∗)Cd(T )]

and so on, to C(0).

I For an American option with payoff g(S(t)),

A(T −∆t) = max{g(S(T −∆T )), e−r∆t[p∗Au(T ) + (1− p∗)Ad(T )]}


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