Black scholes volatility surface
IEOR E4707: Financial Engineering: Continuous-Time Models 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
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