Option Pricing Formulas Derivations

Ramesh Kadambi

March 20, 2013

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Contents

0.1 The Black Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.1.1 CAPM Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.2 Modeling Futures Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.2.1 Futures Price Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.3 Options On Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.3.1 Blacks Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

0.3.2 Margined options on Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

0.3.3 Formula Derivation MOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

0.3.4 Cash Settled Option Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

0.4 Currency Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

0.4.1 Modeling Currency Price Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

0.4.2 Call Option Price and PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

0.5 Parities and Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

0.5.1 Put Call Parity Stock Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

0.5.2 Put Call Parity Option Paying Dividend Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

0.5.3 PCP Option on Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

0.5.4 PCP Margined Options On Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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4 CONTENTS

0.1 The Black Scholes Formula

In this section we derive the Black Scholes formula for pricingvanilla european options. The following is similar to the deriva-tion of BS from the original paper1. The formula assumes thefollowing as stated in the original paper:

1. The short-term interest rate is known and is constantthrough time.

2. The stock price follows a random walk in continuous timewith a variance rate proportional to the square of thestock price. Thus the distribution of possible stock pricesat the end of any finite interval is log- normal. The vari-ance rate of the return on the stock is constant.

3. The stock pays no dividends or other distributions.

4. The option is ”European,” that is, it can only be exercisedat maturity.

5. There are no transaction costs in buying or selling thestock or the option.

6. It is possible to borrow any fraction of the price of a se-curity to buy it or to hold it, at the short-term interestrate.

7. There are no penalties to short selling. A seller who doesnot own a security will simply accept the price of the secu-rity from a buyer, and will agree to settle with the buyeron some future date by paying him an amount equal tothe price of the security on that date.

The idea is to form a risk-less position using the option and thestock. We create the following portfolio.

1. Long one stock.

2. Short 1Cs(S,t)

option on the stock at strike K and expira-

tion T.

Where S(t) is the price of the stock, C(S, t) is the value of theoption.

We note that the change in value of the portfolio is 0 forsmall changes in the value of the stock price. If the stock pricechanges by a value dS the value of the portfolio Π(C, S, t) isgiven by.

dΠ(C, S, t) = dS(t)− 1

Cs(S, t)dC(S, t)

Using first order approximation we have

dC(S, t) = Cs(S, t)dS(t)2

dΠ(C, S, t) = dS(t)− 1

Cs(S, t)Cs(S, t)dS(t) = 0

Quoting from the original paper again:

”As the variables S(t) and t change, the number of optionsto be sold short to create a hedged position with one share ofstock changes. If the hedge is maintained continuously, thenthe approximations mentioned above become exact, and thereturn on the hedged position is completely independent of thechange in the value of the stock. In fact, the return on thehedged position becomes certain.”

The hedged portfolio is risk-less and grows at a risk-free rate.The original paper contends that in the absence of continuoushedging the risk can be diversified away and the risk is mainlydue to the fact that the hedge is not continuously adjusted.

Substituting for dC(S, t) using Ito’s lemma we get the fol-lowing.

dC(S, t) = Ctdt+ Cs(S, t)dS +1

2CssdS

2

substituting into equation (1) we have

dΠ(S, t) = dS(t)− 1

Cs(S, t)[Ctdt+ Cs(S, t)dS +

1

2CssdS

2]

dΠ(S, t) = − 1

Cs(S, t)[Ctdt+

1

2CssdS

2]

r(S(t)− 1

Cs(S, t)C(S, t))dt = − 1

Cs(S, t)[Ctdt+

1

2CssdS

2]

r(CsS(t)− C(S, t))dt = −dt[Ct +1

2σ2S2Css]

canceling dt and rearranging terms

Ct +1

2σ2S2Css + rSCs − rC = 0 (0.1.1)

The boundary conditions to the BS equation:

C(S, T ) = (S(T )−K)+

C(Smax, t) = S(t)−Ke−r(T−t)

C(Smin, t) = 0

The analytical solution the PDE given BC’s

C(S, t) = S(t)N(d1)−Ke−r(T−t)N(d2)

where

d1 =ln(F/K) + 1

2σ2(T − t)

σ√T − t

d2 = d1 − σ√T − t

0.1.1 CAPM Derivation

CAPM relates the excess returns of a security to excess returnon the market portfolio. The CAPM equation is given by

E[ri]− rfβi

= E[rm]− rf1Pricing of Options and Corporate Liabilities, Fischer Black, Myron Scholes2Note this is not applying Ito’s lemma on C(S, t) which it really should. It so happens it works out fine if you do so.

0.2. MODELING FUTURES DYNAMICS 5

. Considering the option as a investment security we have thefollowing for the option excess returns.

E[rc] = rf + βf (E[rm]− rf )

Similarly for the stock we have.

E[rs] = rf + βs(E[rm]− rf )

The question now is how are the beta of the option andthe beta of the stock related. From the relation dC(S, t) =Ctdt+ CsdS + Css

2 dS2 we have.

dC

C=

1

C[Ct +

1

2σ2S2]dt|+ Cs

CdS

denoting the return on the stock as rs we have

dC

C=

1

C[Ct +

1

2σ2S2]dt|+ SCs

Crs

rC =1

C[Ct +

1

2σ2S2]dt|+ SCs

Crs (0.1.2)

The beta of the option is given by Cov(rc,rm)var(rm) . Using (0.1.2) we

can show that βC = βSCsSC . Now we have the following two

equations from CAPM model.

E[dC

C] = (rf + (E[rm]− rf )βC)dt (0.1.3)

E[dS

S] = (rf + (E[rm]− rf )βS)dt

multiplying (0.1.3) by C and substituting for βC we have

E[dC] = (rfC + (E[rm]− rf )βSSCs)dt

substituting for dC using ito’s lemma and

simplifying yields (0.1.1)

0.2 Modeling Futures Dynamics

A futures contract is an exchange listed product that settles allcash-flows at the end of the day. The value of the contract atexpiration is the value of the underlying at expiration. Con-sider a time [0,T] which are divided into distinct time points0 = t0 < t1 < t2 < t3 < t4... < tn = T . The futures settles attimes tk ∈ {t1, t2, · · · , tn} by paying Futs(tk+1)− Futs(tk, T ).We show here that futures price is a martingale and propose aprice for an option on Future. Consider the sum of paymentsfrom holding a futures contract from [t0, T ].

Futs(t1, T )− Futs(t, T ) + · · ·+ Futs(T, T )− Futs(T − 1, T )

= S(T )− Futs(t, T )

The futures price process by construction is a F(t) measur-able martingale. Let Futs(t, T ) be the price of a futures con-tract at time t ∈ {t0, t1}. Since the payout as we saw earlier

Futs(T, T )− Futs(t0, T ) we have for the price of the future:

Futs(t, T ) = E[D(t)(Futs(T, T )− Futs(t, T ))|F(t)]

but Futs(t, T ) = 0

E[D(t)(Futs(T, T )− Futs(t, T ))|F(t)] = 0

Futs(t, T ) = E[Futs(T, T )|F(t)]

= E[S(T )|F(t)]

0.2.1 Futures Price Process

Given the underlying price process once can determine the dy-namics of the futures prices process given that it is a martingale.Let dS = rSdt+σSdW and F(t) the filtration for the B.M W .We have:

Futs(t, T ) = E[S(T )|F(t)]

= S(t)er(T−t)

dFuts(t, T ) = −rS(t)er(T−t)dt+ er(T−t)dS(t)

dFuts(t, T ) = σer(T−t)S(t)dW (t)

= σFuts(t, T )dW (t)

0.3 Options On Futures

Consider an option on a future paying C(Futs(t, T ), t) at timeT . We can use the same hedging argument we used to obtainthe price of the option. We will however proceed using he mar-tingale approach. We assume there exists a martingale measureP. Since all processes grow at risk free rate under martingalemeasure. We have using Ito’s lemma:

dC(Futs(t, T ), t) = Ctdt+ CF dFuts(t, T ) +1

2CFF dFuts(t, T )2

= Ctdt+ σCFFuts(t, T )dW (t) +σ2

2CFFFuts(t, T )2dt

= (Ct +σ2

2CFFFuts(t, T )2)dt+ σCFFuts(t, T )dW (t)

since the drift of all price processes under P is rPdt we have

(Ct +σ2

2CFFFuts(t, T )2)dt = rCdt

Ct +σ2

2CFFFuts(t, T )2 − rC = 0

6 CONTENTS

0.3.1 Blacks Formula

Consider a call option on a future paying (Futs(T, T )−K)+ attime T . The price of the option is given by the formula.

C(Futs, t) = e−r(T−t)(Futs(t, T )N(d1)−KN(d2))

where:

d1 =ln(Futs(t, T )/K) + σ2

2 (T − t)σ√T − t

d2 = d1 − σ√T − t

The above formula is the result of the following calculation:

Consider the process ln(Futs(t, T )) we have dln(Futs(t, T ))

dln(Futs(t, T )) = −σ2s

2dt+ σsdW (t)

ln(Futs(T, T )) = ln(Futs(t0, T ))− σ2s

2dt+ σsdW (t)

let Z(T,t) =T∫t

−σ2s

2 dt+T∫t

σsdW (t) then Z is a normal Random

variable N(−σ2s(T−t)

2 , σ2s(T − t))

E[e−rtC(Futs, t)|F(t)] = E[e−rT (Futs(T )−K)+|F(t)]

C(Futs, t) = ertE[e−rT (Futs(T )−K)+|F(t)]

C(Futs, t) = ertE[e−rT (eln(Futs(t)+Z(T,t) −K)+|F(t)]

= ertE[e−rT (Futs(t)eZ(T,t) −K)+|F(t)]

= ert∫

I{Futs(t)eZ(T,t)≥K}

e−rT (Futs(t)eZ(T,t) −K)e

−(Z+

σ2s(T−t)2

)2

2σ2s(T−t) dZ

= e−r(T−t)∫

I{Futs(t)eZ(T,t)≥K}

Futs(t)eZ(T,t)e

−(Z+

σ2s(T−t)2

)2

2σ2s(T−t) dZ

− e−r(T−t)∫

I{Futs(t)eZ(T,t)≥K}

Ke−

(Z+σ2s(T−t)

2)2

2σ2s(T−t) dZ

consider∫

I{Futs(t)eZ(T,t)≥K}

Futs(t)eZ(T,t)e

−(Z+

σ2s(T−t)2

)2

2σ2(T−t) dZ we

will complete the squares to compute integral.

Z(T, t)−(Z +

σ2s(T−t)

2 )2

2σ2s(T − t)

=2σ2

s(T − t)Z − (Z +σ2s(T−t)

2 )2

2σ2s(T − t)

=2σ2

s(T − t)Z − Z2 − σ2s(T − t)Z − (

σ2s(T−t)

2 )2

2σ2s(T − t)

=−(Z − σ2

s(T−t)2 )2

2σ2s(T − t)∫

I{Futs(t)eZ(T,t)≥K}

Futs(t)eZ(T,t)e

−(Z+

σ2s(T−t)2

)2

2σ2s(T−t) dZ

=

∫I{Futs(t)eZ(T,t)≥K}

Futs(t)e−(Z−

σ2s(T−t)2

)2

2σ2s(T−t) dZ (0.3.1)

define a variable Y =Z−σ2s(T−t)

2

σ2s(T−t) , Y is a normally distributed

variable N(0,1).

Futs(t)eZ(T,t) ≥ K

Z(T, t) ≥ ln(K

Futs(t))

Y ≥ln( K

Futs(t))− σ2

s(T−t)2

σ2s(T − t)

substituting the transformations into (0.3.1)∫I{Futs(t)eZ(T,t)≥K}

Futs(t)e−(Z−

σ2s(T−t)2

)2

2σ2s(T−t) dZ

=

∫IY≥

ln( KFuts(t)

)−σ2s(T−t)

2

σ2s(T−t)

Futs(t)e−Y 2/2dY

=

∫IY≤

−ln(Futs(t)

)+σ2s(T−t)

2

σ2s(T−t)

Futs(t)e−Y 2/2dY

= Futs(t)N(d1)

0.4. CURRENCY OPTIONS 7

where d1 = ln(Futs(t)K ) +σ2s(T−t)

2 σ2s(T − t), we also have

∫I{Futs(t)eZ(T,t)≥K}

Ke−

(Z+σ2s(T−t)

2)2

2σ2s(T−t) dZ (0.3.2)

define a new variable Y2 =Z+

σ2s(T−t)2

σ2s(T−t) , we then have.

Futs(t)eZ(T,t) ≥ K

Z(T, t) ≥ ln(K

Futs(t))

Y2 ≥ln( K

Futs(t)) +

σ2s(T−t)

2

σ2s(T − t)

rewriting (0.3.2) in terms of Y2 we have

∫I{Futs(t)eZ(T,t)≥K}

Ke−

(Z+σ2s(T−t)

2)2

2σ2s(T−t) dZ

=

∫I{Y2≥

ln( KFuts(t)

)+σ2s(T−t)

2

σ2s(T−t)}

Ke−Y 2

22 dY2

=

∫I{Y2≤

−ln( KFuts(t)

)−σ2s(T−t)

2

σ2s(T−t)}

Ke−Y 2

22 dY2

=

∫I{Y2≤

ln(Futs(t)K

)−σ2s(T−t)

2σ2s(T−t)

}

Ke−Y 2

22 dY2

= KN(d2)

where d2 =ln(

Futs(t)K )−σ2s(T−t)

2

σ2s(T−t) . Putting it all together we have

the price of the option on a future is:

C(Futs(t),K) = e−r(T−t)[Futs(t)N(d1)−KN(d2)]

where d1 =ln(

Futs(t)K )+

σ2s(T−t)2

σ2s(T−t) and d2 =

ln(Futs(t)K )−σ2s(T−t)

2

σ2s(T−t) .

0.3.2 Margined options on Futures

The margined options on futures are traded on Sydney futuresexchange. The option premium is not exchanged at the timeof purchase i.e. The options contract works very similar to thefutures contract. The option premium is cash settled. The dif-ference in the premium is either credited (debited) based on the

option price moving up (down). Upon expiration no money isexchanged either. Given times t, t1, · · · , tT . The pay out on acash settled Options contract is as follows.

C(Ft1 , t1)− C(Ft, t) + · · ·+ C(FT , T )− C(FT−1, T − 1)

= C(FT , T )− C(Ft, t) (0.3.3)

As seen in (0.3.3) we see that the final payout is the value ofthe option at time T minus the initial premium. As in the caseof the future price we set the expected value of the premiumdifference to be zero. We have:

E[D(t)(C(FT , T )− C(Ft, t))|Ft] = 0

C(Ft, t) = E[C(FT , T )|Ft]

Given that the option price by design is a martingale w/o dis-counting we obtian the PDE of the option as.

Ct +σ2

2CFFFuts(t, T )2 = 0

0.3.3 Formula Derivation MOF

Using the previous definitions for Z,F we have.

C(F, t) = E[C(FT , T )|Ft]

=

∫I{FT≥K}

(FT −K)p(F, T : f, t)dp

=

∫I{eZ≥K}

(Fs(t)eZ −K)e

−(Z+

σ2s(T−t)2

)2

2σ2s(T−t) dZ

This is the same as what we had when we solved for the futuresoptions. There fore we have from (0.3.1).

C(Futs(t),K) = Futs(t)N(d1)−KN(d2)

0.3.4 Cash Settled Option Dynamics

As we saw the Cash Settled Option Process is a martingale. Theoption price process is a function of the Futures price C(FT , T ).Applying ito’s lemma to C(FT , T ) we have:

dC(FT , t) =

[Ct +

σ2

2CFFFuts(t, T )2

]dt+ σfF (T, t)dW (t)

Using the fact that the drift is zero we have:

dC(FT , t) = σfCFF (T, t)dW (t)

0.4 Currency Options

Currency Options are priced using a modified version of BSPM.The model is similar to pricing stock options paying dividendyield. Quoting Bjork3: ”When we buy a foreign currency (sayUS Dollars) we will not just keep the physical bills until wesell them. Instead we typically put the dollars into an accountwhere they will grow at a certain rate of interest”. This implies

3”Arbitrage Theory in Continuous Time”, Thomas Bjork

8 CONTENTS

that the currency options can be modeled like an option onstock that pays a dividend yield. Mathematically, the currencyoption pricing involves two different measures. A foreign RNMPf , and a domestic RNM Pd. When we priced options on stockwe did so in the domestic RNM. In this section we will lookat the different prices processes and the different option pricingformulas.

0.4.1 Modeling Currency Price Dynamics

We take as given the following dynamics for the the spot ex-change rate X(t) quoted as units of domestic currency

units of foreign currency . We consideras given a risk free domestic rate rd and a foreign rate rf . Cor-responding to the risk free rates we have Bd and Bf as riskfree assets. We there fore assume the following dynamics forthe spot rate X(t) and the risk free assets under the physicalmeasure P.

dX = Xαxdt+XσxdW̃

dBd = rdBddt

dBf = rfBfdt

Our objective is to price a pay-off of the type f(X(T )). Wemake the assumption that all foreign assets are invested at therisk-free foreign rate of rf . We have the price of the securitygiven as,

Xt = e−rd(T−t)EQ[f(X(T ))|Ft]. Where Q is the RNM under domestic currency rate rd. Underthis measure all domestic assets have RND rd. Our next goalis to translate the investment Bf into domestic currency. Con-sider the value of the foreign risk free asset in domestic terms,we have.

B̃f (t) = Bf (t)X(t)

dB̃f (t) = dBf (t)X(t) +Bf (t)dX(t) + dBf (t)dX(t)

dB̃f (t) = rfX(t)Bfdt+Bf (t)[αxX(t)dt+ σxX(t)dW̃

dB̃f (t) = (rf + αx)B̃f (t)dt+ σxB̃f (t)dW̃ (0.4.1)

We apply grisanov’s theorem and a change of measure to moveto domestic RNM.

dW̃ = dW − λ(t)dt

substituting into (0.4.1) we have,

dB̃f (t) = rdB̃f (t)dt+ σxB̃fdW

we also have X(t) =B̃fBf

, applying ito

dX(t) = X(rd − rf )dt+XσxdW (0.4.2)

0.4.2 Call Option Price and PDE

Proposition 0.4.1. (Pricing Formula) The arbitrage free pricef(t,X) for the T -claim Z = f(X(T )) is given by f(t,X) =e−rd(T−t)EQ[f(X(T )]. The Q-dynamics of X is given by(0.4.2). The PDE for the pay off is given by:

ft + (rd − rf )Xfx +1

2σ2xX

2fxx − rdf = 0

subject to boundary condition:

f(T, x) = f(x)

The closed form solution to price a call option C(X,T ) =(X(T )−K)+ is given by.

C(X, t) = X(t)e−rf (T−t)N(d1)−Ke−rd(T−t)N(d2)

where

d1 =ln(F )K + σ2

2 (T − t)σ√T − t

F = X(t)e(rd−rf )(T−t)

d2 = d1 − σ√

(T − t)

0.5 Parities and Symmetries

In this section we will derive the different put-call symmetriesand parity relationships for options on different underlying.

0.5.1 Put Call Parity Stock Options

The put call parity comes from looking at the terminal payoff of a forward and noting that at expiration a forward con-tract can be constructed using a long call option and a shortput option. This relation is a pure arbitrage relationship. Wetherefore have:

F (T ) = C(T )− P (T )

Under risk neutral valuation we can compute the forward priceF(t) by taking discounted expectations.

E[e−rTF (T )|F0] = E[e−rT (C(T )− P (T ))|F0]

substituting the value of the forward contract is

F (T ) = S(T )−Ke−rTE[S(T )−K|F0] = E[e−rT (C(T )− P (T ))|F0]

E[e−rTS(T )|F0]−Ke−rT ] = C(0)− P (0)

S(0)−Ke−rT = C(0)− P (0)

for an time t the put call parity for stock options is

S(t)−Ke−r(T−t) = C(t)− P (t)

0.5.2 Put Call Parity Option Paying DividendYield

When an option pays a dividend yield the forward price is ob-tained by the following arbitrage argument. The idea of a for-ward contract is to deliver a unit of stock for a pre-agreed strikeprice K. Given that the stock yields a dividend of q, we havethe following cash flows.

0.5. PARITIES AND SYMMETRIES 9

Table 1: Cash flow tableTime Bank Stock Portfolio Value

0 −S0e−qT S0e

−qT 0T −S0e

−qT erT +K ST K − S0e−qT erT

At time T the value of the portfolio is K −S0e−qT erT . The

strike of the forward contract is the value that makes the for-ward price zero. We therefore have, for the forward price.

K = S0e(r−q)T

Moving on to the put call parity. We have the same situation,as before:

F (T ) = C(T )− P (T )

taking discounted expectations under RNM we have

e−r(T−t)E[S(T )−K|Ft] = e−r(T−t)E[C(T )− P (T )|Ft]E[e−r(T−t)S(T )|Ft]−Ke−r(T−t) = C(t)− P (t)

e−q(T−t)S(t)−Ke−r(T−t) = C(t)− P (t) (0.5.1)

Note that in (0.5.1), e−r(T−t)E[S(T )|Ft] = e−q(T−t)S(t) sincee−(r−q)(T−t)S(t) is a martingale and not just discounted stockprice.

0.5.3 PCP Option on Futures

The put-call parity for options on futures give a strike K andexpiration T is given as follows. At time T we have.

C(T )− P (T ) = F (T )−Ktaking discounted expectation at time t we have.

E[e−rT−t(C(T )− P (T ))|F(t)] = E[e−r(T−t)(F (T )−K)|F(t)]

C(t)− P (t) = e−r(T−t)(E[F (T )|F(t)]−K)

C(t)− P (t) = e−r(T−t)(F (t)−K) (0.5.2)

Note that the fact that the future price is martingale:E[F (T )|F(t)] = F (t), is used in (0.5.2).

0.5.4 PCP Margined Options On Futures

The put-call parity for margined options is arrived at usingthe same principle as before. However, since margined optionprices are martingales we do not take discounted expectations.We therefore have the following:

C(T )− P (T ) = F (T )−Ke−r(T−t)E[C(T )− P (T )|F(t)] = e−r(T−t)E[F (T )−K|F(t)]

C(t)− P (t) = F (t)−K (0.5.3)

Note, since the price processes are martingales the discount fac-tors cancel each other leaving us with (0.5.3).