Risk-neutrality in derivatives pricing University of Oulu - Department of Finance Spring 2018
Portfolio of two assets Value at time t = 0 Expected return Value at time t = 1 Asset A Asset B 10.00 30.00 7% 10% 10.70 33.00 } E [ P 1 = 43.70 Let us consider a portfolio of two assets with their current values of A 0 = e10.00 and B 0 = e30.00. We assume the risk-free rate of 3% and the market risk premium of 5%, both in terms of a simple annual rate. The beta of the asset A is β A = 0.80, and thus the expected return of the asset A is ER A ) = 3% + 5% 0.80 = 7%. Correspondingly, the expected value of the asset A at the end of a one-year period is EA 1 ) = 1 + 7%) 10.00 = 10.70. The beta of the asset B is β B = 1.40, and thus the expected return of the asset B is ER B ) = 3% + 5% 1.40 = 10%. Correspondingly, the expected value of the asset B at the end of a one-year period is EB 1 ) = 1 + 10%) 30.00 = 33.00. The expected value of the portfolio is E [ P 1 = 10.70 + 33.00 = e43.70.
Portfolio of two assets Value at time t = 0 Expected return Value contribution at time t = 1 Asset A Asset B 10.00 30.00 9.25% 9.25% 10.925 32.775 } E [ P 1 = 43.70 Let us apply an alternative approach to evaluate the same portfolio. The beta of the portfolio of the two assets is β P = w A β A + w B β B = 10 10 + 30 0.80 + 30 1.40 = 1.25. 10 + 30 Correspondingly, the expected return of the portfolio of the two assets is ER P ) = 3% + 5% 1.25 = 9.25%. We are able to calculate the portfolio-specific value contributions of the two assets as follows: ÊA 1 ) = 1 + 9.25%) 10.00 = 10.925. ÊA 2 ) = 1 + 9.25%) 30.00 = 32.775. The value contributions sum to the expected value of portfolio: E [ P 1 = ÊA1 ) + ÊB 1 ) = 10.925 + 32.775 = e43.70.
Portfolio of two assets Asset A Asset B Value at time t = 0 Expected return Value contribution at time t = 1 } 10.00 9.25% 10.925 E [ P 1 = 43.70 30.00 32.775 Assume that we know the current price A 0 of the asset A, but do not know the current price B 0 of the asset B. Assume further that we know the expected return of the portfolio. Furthermore, somehow we know the expected end-of-period value E [ P 1 of the portfolio. E [ P 1 = 43.70 The expected return of the portfolio is 9.25% and thus the contribution of the asset A to the value of the portfolio is ÊA 1 ) = 1 + 9.25%) 10.00 = 10.925. The contribution of the asset B to the value of the portfolio depends on Ê[A 1, and is Ê [ P 1 A 1 = P1 Ê[ A 1 = 43.70 10.925 = 32.775. The current value of the asset B is obtained by discounting the contribution with the expected return of the portfolio: B 0 = Ê[B 1 1 + 9.25% = 32.775 1 + 9.25% = e30.00.
Portfolio of a forward contract and the underlying asset Underlying asset Shorted forward contract Value at time t = 0 Expected return Value contribution at time t = T } S 0 r Ê [ S T = S0 e rt P T = K Ê [ e rt Ê [ K S T K S T Let us follow the same approach in a case of a portfolio of an asset and a shorted forward contract. The current price of the underlying asset is S 0. The end-of-period value P T of the portfolio is known to be P T = S T + K S T ) = K. The portfolio is risk-free and thus the contribution of the underlying asset to the value of the portfolio is Ê [ S T = S0 e rt. The contribution of the shorted forward contract to the value of the portfolio depends on Ê[S T, and is Ê[K S T, where Ê[S T = S 0 e rt. The current value of the contract is obtained by discounting the contribution with the expected return of the portfolio: f0 = e rt Ê [ K S T.
Portfolio of options and the underlying asset Value at time t = 0 Expected return Value contribution at time t = T Underlying asset Put option Shorted call option S 0 e rt Ê [ max0, X S T ) e rt Ê [ max0, S T X ) r Ê [ S T = S0 e rt Ê [ max0, X S T ) Ê [ max0, S T X ) } P T = X Let us follow the same approach in a case of a portfolio of an asset, a put option and a shorted call option. The end-of-period value P T of the portfolio is known to be P T = S T + max0, X S T ) max0, S T X ) = X. The portfolio is risk-free and thus the contribution of the underlying asset to the value of the portfolio is Ê [ S T = S0 e rt. The contributions of the options to the value of the portfolio depend on Ê[S T, and are Ê [ max0, X S T ) and Ê [ max0, S T X ), where Ê [ S T = S0 e rt. The current values of the options is obtained by discounting the contributions with the expected return of the portfolio: p 0 = e rt Ê [ max0, X S T ) and c 0 = e rt Ê [ max0, S T X ), where Ê [ S T = S0 e rt.
Risk-neutrality in derivatives pricing When pricing any derivative security we 1. Determine the risk-neutral expected payoff Ê[. of the contract, 2. Calculate the current value of the security by discounting the risk-neutral expected payoff with the risk-free rate. Risk-neutral does not mean risk-free, but it means that we are able to replace the expected return µ of the underlying asset with the risk-free rate r. The risk, or volatility σ, of the underlying asset still remains and has to be considered in the determination of the expected payoff), but no reward is paid on it or any risk), because it can be eliminated by an appropriate diversification. In the case of a forward contract the risk-neutral expected payoff is Ê [ S T K = Ê[ S T K = S0 e rt K. Correspondingly, the current value of the risk-neutral expected payoff is f 0 = e rt Ê [ S T K = e rt [ S 0 e rt K = S 0 Ke rt. In this case, the risk-neutrality appears explicitly in the risk-neutral expected price S 0 e rt of the underlying asset, where the asset-specific expected return µ is replaced with the risk-free rate r. The volatility σ of the asset does not play any role in the pricing of a forward contract, but is not ignored by any means.
Risk-neutrality in derivatives pricing When pricing any derivative security we 1. Determine the risk-neutral expected payoff Ê[. of the contract, 2. Calculate the current value of the security by discounting the risk-neutral expected payoff with the risk-free rate. Risk-neutral does not mean risk-free, but it means that we are able to replace the expected return µ of the underlying asset with the risk-free rate r. The risk, or volatility σ, of the underlying asset still remains and has to be considered in the determination of the expected payoff), but no reward is paid on it or any risk), because it can be eliminated by an appropriate diversification. In the case of an option the risk-neutral expected payoffs are c T = Ê[ max0, S T X ) max 0, Ê[ ) p T = Ê[ max0, X S T ) S T X, max 0, X Ê [ ) S T. In this case the solution is not as straightforward as in the case of a forward contract, and we are to make considerably much more effort to determine an explicit solution for the risk-neutral expected payoff. The payoff will be evaluated under an assumption that the underlying asset earns the risk-free rate r instead of the expected return µ of its own, and in this case the volatility σ of the underlying asset plays an important role. Correspondingly, the current values of the risk-neutral expected payoffs are c 0 = e rt Ê [ max0, S T X ), p 0 = e rt Ê [ max0, X S T ).
Risk-neutrality in derivatives pricing When pricing derivatives portfolio of a derivative and the underlying asset is considered, underlying asset is evaluated and priced as a part of the created portfolio, derivative is evaluated and priced as a part of the created portfolio. The underlying asset price expectation is based on the portfolio s expected return, portfolio is risk-free, and the expected return of it equals to the risk-free rate, analysis results to a risk-neutral price expectation of the asset price. The derivative derivative is priced on the basis of the underlying asset s risk-neutral price expectation, derivative s risk-neutral price expectation is discounted with the risk-free rate. the resulted derivatives price is a fair price in terms of reward-to-risk.
Itô s Lemma Risk-neutrality in derivatives pricing If a random variable x follows an Itô-process dx = ax, t)dt + bx, t)dz, then, according to Itô s lemma, any random variable y, the value of which is determined by the random variable x and time t, follows the Itô-process dy = y x a + y t + 1 2 2 ) y x 2 b2 dt + y x bdz. If asset price S follow the process ds = µsdt + σsdz, then, any derivative price y, the value of which is determined by the underlying asset price S and time t, follows y y dy = µs + S t + 1 2 ) y 2 S 2 σ2 S 2 dt + S σsdz.
Portfolio of an underlying asset and a derivative security Portfolio of an underlying asset and a derivative security n s shares of an underlying asset, n f contracts of a derivative security on the asset. The underlying asset: n s = S ds = µsdt + σsdz The derivative security: n f = 1 df = µs + S t + 1 2 ) f 2 S 2 σ2 S 2 dt + S σsdz. The portfolio: P = n s S + n f f [ dp = n s [µsdt + σsdz + n f µs + S t + 1 2 2 ) f S 2 σ2 S 2 dt + S σsdz
Portfolio of an underlying asset and a derivative security The number n s of shares needed varies over time. The number n f of derivatives remains constant over time. The stochastic terms dz behave identically in the both components. dp = [ [ µsdt + σsdz 1 µs + S S t + 1 2 ) f 2 S 2 σ2 S 2 dt + S σsdz = µsdt + σsdz µsdt S S S t dt 1 2 = t + 1 2 ) f 2 S 2 σ2 S 2 dt 2 f S 2 σ2 S 2 dt S σsdz The resulting price process does not contain dz, and is thus purely deterministic in nature. The result is based on a process of the form ds = µsdt + σsdz, but is independent of the level µ of the expected return of the asset. A practical application of the above is the dynamic delta-hedging of a sold derivative position.
Black-Scholes differential equation The risk-free portfolio of the underlying asset and a derivative is allowed to earn the risk-free rate r only! dp = t + 1 2 ) f 2 S 2 σ2 S 2 dt = rpdt The risk-free rate r is earned on the portfolio value P over a period of dt. t + 1 2 ) f 2 S 2 σ2 S 2 ) dt = r n s S + n f f dt t + 1 2 ) f 2 S 2 σ2 S 2 dt = n s rsdt + n f rfdt }{{} t + 1 2 ) f 2 S 2 σ2 S 2 dt = rsdt rfdt S The original price process of the underlying asset must be replaced with the risk-neutral price process ds = rsdt + σsdz. t + 1 2 f 2 S 2 σ2 S 2 + rs rf = 0 S
Derivatives pricing under risk-neutrality Forward contract: Price process of the underlying asset: Price process of the underlying asset under risk-neutrality: ds = µsdt + σsdz ds = rsdt + σsdz Expected payoff under risk-neutrality: ÊS T K) = ÊS T ) K = S t e rt t) K Current value of the contract: f t = e rt t) rt t) ÊS T K) = S t Ke Call option: Price process of the underlying asset: Price process of the underlying asset under risk-neutrality: ds = µsdt + σsdz ds = rsdt + σsdz Expected payoff under risk-neutrality: Ê[max0, S T X ) =? Current value of the contract: c t = e rt t) Ê[max0, S T X ) =? Put option: Price process of the underlying asset: Price process of the underlying asset under risk-neutrality: ds = µsdt + σsdz ds = rsdt + σsdz Expected payoff under risk-neutrality: Ê[max0, X S T ) =? Current value of the contract: p t = e rt t) Ê[max0, X S T ) =?
Forward contract and Black-Scholes differential equation Forward contract: Pricing model: rt t) f = S Ke Partial derivatives: S = 1 2 f S 2 = 0 t = rke rt t) Black-Scholes differential equation: t + 1 2 f 2 S 2 σ2 S 2 + rs rf S = rke rt t) + 1 2 0 σ2 S 2 rt t)) + rs r S Ke = rke rt t) + rs rs + rke rt t) = 0
An exotic forward contract An exotic forward contract pays off the difference of the squared asset price ST 2 and the contracted delivery price K at the maturity T of the contract. Price processes and expected values: µt t) ds = µsdt + σsdz ES T ) = S t e ds 2 ) = 2µ + σ 2 )S 2 dt + 2σS 2 dz ES 2 T ) = S2 t e2µ+σ2 )T t) Price processes and expected values under risk-neutrality: rt t) ds = rsdt + σsdz ÊS T ) = S t e ds 2 ) = 2r + σ 2 )S 2 dt + 2σS 2 dz ÊS 2 T ) = S2 t e2r+σ2 )T t) Expected payoff under risk-neutrality: ÊS 2 T K) = ÊS2 T ) K = S2 t e2r+σ2 )T t) K Current value of the contract: f t = e rt t) ÊST 2 K) = e rt t)[ St 2 e2r+σ2 )T t) K = St 2 er+σ2 )T t) Ke rt t)
An exotic forward contract and BS differential equation An exotic forward contract: Pricing model: f = S 2 e r+σ2 )T t) Ke rt t) Partial derivatives: S = 2 )T t) 2 f 2Ser+σ S 2 = 2 )T t) 2er+σ t = r + σ2 )S 2 e r+σ 2 )T t) rt t) rke Black-Scholes differential equation: t + 1 2 f 2 S 2 σ2 S 2 + rs rf = 0 S = r + σ 2 )S 2 e r+σ2 )T t) rt t) 1 rke + 2 2 )T t)σ 2er+σ 2 2 r+σ 2 )T t) S + 2Se rs r S 2 e r+σ2 )T t) rt t) ) Ke = 0