# Black scholes volatility surface

c 2009 by Martin Haugh

Fall 2009

Black-Scholes and the Volatility Surface

When we studied discrete-time models we used martingale pricing to derive the Black-Scholes formula for European options. It was clear, however, that we could also have used a replicating strategy argument to derive the formula. In this part of the course, we will use the replicating strategy argument in continuous time to derive the Black-Scholes partial diﬀerential equation. We will use this PDE and the Feynman-Kac equation to demonstrate that the price we obtain from the replicating strategy argument is consistent with martingale pricing.

We will also discuss the weaknesses of the Black-Scholes model, i.e. geometric Brownian motion, and this leads us naturally to the concept of the volatility surface which we will describe in some detail. We will also derive and study the Black-Scholes Greeks and discuss how they are used in practice to hedge option portfolios. We will also derive Black’s formula which emphasizes the role of the forward when pricing European options. Finally, we will discuss the pricing of other derivative securities and which securities can be priced uniquely given the volatility surface. Change of numeraire / measure methods will also be demonstrated to price exchange options.

1

The Black-Scholes PDE

We now derive the Black-Scholes PDE for a call-option on a non-dividend paying stock with strike K and maturity T . We assume that the stock price follows a geometric Brownian motion so that dSt = µSt dt + σSt dWt

(1)

where Wt is a standard Brownian motion. We also assume that interest rates are constant so that $1 invested in the cash account at time 0 will be worth Bt := $ exp(rt) at time t. We will denote by C(S, t) the value of the call option at time t. By Itˆ’s lemma we know that

o

dC(S, t) =

µSt

∂C

1

∂C

∂2C

+

+ σ2 S 2 2

∂S

∂t

2

∂S

dt + σSt

∂C

dWt

∂S

(2)

Let us now consider a self-ﬁnancing trading strategy where at each time t we hold xt units of the cash account and yt units of the stock. Then Pt , the time t value of this strategy satisﬁes Pt = xt Bt + yt St .

(3)

We will choose xt and yt in such a way that the strategy replicates the value of the option. The self-ﬁnancing assumption implies that

dPt

=

xt dBt + yt dSt

=

rxt Bt dt + yt (µSt dt + σSt dBt )

=

(rXt Bt + yt µSt ) dt + yt σSt dWt .

(4)

(5)

Note that (4) is consistent with our earlier deﬁnition1 of self-ﬁnancing. In particular, any gains or losses on the portfolio are due entirely to gains or losses in the underlying securities, i.e. the cash-account and stock, and not due to changes in the holdings xt and yt .

1 It is also worth pointing out that the mathematical deﬁnition of self-ﬁnancing is obtained by applying Itˆ’s Lemma to (3) o

and setting the result equal to the right-hand-side of (4).

Black-Scholes and the Volatility Surface

2

Returning to our derivation, we can equate terms in (2) with the corresponding terms in (5) to obtain yt

=

rxt Bt

=

∂C

∂S

∂C

1

∂2C

+ σ2 S 2 2 .

∂t

2

∂S

(6)

(7)

If we set C0 = P0 , the initial value of our self-ﬁnancing strategy, then it must be the case that Ct = Pt for all t since C and P have the same dynamics. This is true by construction after we equated terms in (2) with the corresponding terms in (5). Substituting (6) and (7) into (3) we obtain rSt

∂C

∂C

1

∂2C

+

+ σ 2 S 2 2 − rC = 0,

∂S

∂t

2

∂S

(8)

the Black-Scholes PDE. In order to solve (8) boundary conditions must also be provided. In the case of our call option those conditions are: C(S, T ) = max(S − K, 0), C(0, t) = 0 for all t and C(S, t) → S as S → ∞. The solution to (8) in the case of a call option is

= St Φ(d1 ) − e−r(T −t) KΦ(d2 )

C(S, t)

(9)

St

K

+ (r + σ 2 /2)(T − t)

√

σ T −t

√

= d1 − σ T − t

where d1

=

and d2

log

and Φ(·) is the CDF...

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